Content uploaded by Anjali Goswami
Author content
All content in this area was uploaded by Anjali Goswami on Mar 16, 2022
Content may be subject to copyright.
Springer Proceedings in Mathematics & Statistics
Praveen Agarwal · Dumitru Baleanu ·
YangQuan Chen · Shaher Momani ·
José António Tenreiro Machado Editors
Fractional
Calculus
ICFDA 2018, Amman, Jordan, July 16–18
Springer Proceedings in Mathematics &
Statistics
Volume 303
goyal.praveen2011@gmail.com
Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
workshops and conferences in all areas of current research in mathematics and
statistics, including operation research and optimization. In addition to an overall
evaluation of the interest, scientific quality, and timeliness of each proposal at the
hands of the publisher, individual contributions are all refereed to the high quality
standards of leading journals in the field. Thus, this series provides the research
community with well-edited, authoritative reports on developments in the most
exciting areas of mathematical and statistical research today.
More information about this series at http://www.springer.com/series/10533
goyal.praveen2011@gmail.com
Praveen Agarwal •Dumitru Baleanu •
YangQuan Chen •Shaher Momani •
JoséAntónio Tenreiro Machado
Editors
Fractional Calculus
ICFDA 2018, Amman, Jordan, July 16–18
123
goyal.praveen2011@gmail.com
Editors
Praveen Agarwal
Department of Mathematics
Anand International College of Engineering
Jaipur, Rajasthan, India
Harish-Chandra Research Institute (HRI)
Allahabad, India
International Centre for Basic
and Applied Sciences
Jaipur, India
Dumitru Baleanu
Department of Mathematics
and Computer Science
Çankaya University
Ankara, Turkey
YangQuan Chen
School of Engineering (MESA Lab)
University of California, Merced
Merced, CA, USA
Shaher Momani
Department of Mathematics
The University of Jordan
Amman, Jordan
JoséAntónio Tenreiro Machado
Institute of Engineering
Polytechnic Institute of Porto
Porto, Portugal
ISSN 2194-1009 ISSN 2194-1017 (electronic)
Springer Proceedings in Mathematics & Statistics
ISBN 978-981-15-0429-7 ISBN 978-981-15-0430-3 (eBook)
https://doi.org/10.1007/978-981-15-0430-3
Mathematics Subject Classification (2010): 26-XX, 33-XX, 35-XX, 39-XX, 41-XX, 45-XX, 65-XX,
68-XX
©Springer Nature Singapore Pte Ltd. 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
herein or for any errors or omissions that may have been made. The publisher remains neutral with regard
to jurisdictional claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
goyal.praveen2011@gmail.com
Preface
These are the keynote and invited talks of the International Conference on
Fractional Differentiation and its Applications (ICFDA-2018), which was held at
the Amman Marriot Hotel, Sheissani in Amman, The Hashemite Kingdom of
Jordan from July 16 to July 18, 2018. The ICFDA18 is a specialized conference on
fractional-order calculus and its applications. It is a generalization of the
integer-order ones. The fractional-order differentiation of arbitrary orders takes into
account the memory effect of most systems. The order of the derivatives may also
be variable, distributed, or complex. Recently, fractional-order calculus became a
more accurate tool to describe systems in various fields in mathematics, biology,
chemistry, medicine, mechanics, electricity, control theory, economics, and signal
and image processing.
For this edition, we were happy to have 23 invited speakers who gave talks on a
subject for which they are internationally known experts. Thirteen of these talks are
collected in this volume. Throughout this book, the fractional calculus concepts have
been explained very carefully in the simplest possible terms, and illustrated by a
number of complete solved examples. This book contains some theorems and their
proofs.
The book is organized as follows. In chapter “Closed-Form Discretization of
Fractional-Order Differential and Integral Operators”, a closed-form concretization
of fractional-order differential or integral Laplace operators is introduced. The pro-
posed method depends on extracting the necessary phase requirements from the
phase diagram. The magnitude frequency response follows directly due to the
symmetry of the poles and zeros of the finite z-transfer function. Unlike the con-
tinued fraction expansion technique, or the infinite impulse response of second-order
IIR-type filters, the proposed technique generalizes the Tustin operator to derive a
first-, second-, third-, and fourth-order discrete-time operators (DTO) that were
stable and of minimum phase. The proposed method depends only on the order
of the Laplace operator. The resulted discrete-time operators enjoy flat phase
response over a wide range of discrete-time frequency spectrum. The closed-form
DTO enables one to identify the stability regions of fractional-order discrete-time
systems or even to design discrete-time-fractional-order PI
k
D
l
controllers.
v
goyal.praveen2011@gmail.com
The effectiveness of this work was demonstrated via several numerical simulations.
In chapter “On Fractional-Order Characteristics of Vegetable Tissues and Edible
Drinks”, we are concerned about frequency response techniques to characterize
vegetable tissues and edible drinks. In the first phase, the impedance of the distinct
samples is measured and fractional-order models are applied to the resulting data. In
the second phase, hierarchical clustering and multidimensional scaling tools are
adopted for comparing and visualizing the similarities between the specimen.
In chapter “Some Relations Between Bounded Below Elliptic Operators and
Stochastic Analysis”, we apply Malliavin calculus tools to the case of a bounded
below elliptic right-invariant pseudo-differential operators on a Lie group. We give
examples of bounded below pseudo-differential elliptic operators on Rdby using
the theory of the Poisson process and the Garding inequality. In the two cases, there
are no stochastic processes because the considered semi-groups do not preserve
positivity. In chapter “Discrete Geometrical Invariants: How to Differentiate the
Pattern Sequences from the Tested Ones?”based on the new method (defined
below as the discrete geometrical invariants—DGI(s)), one can show that it enables
to differentiate the statistical differences between random sequences that can be
presented in the form of 2D curves. We generalized and considered the
Weierstrass–Mandelbrot function and found the desired invariant of the fourth order
that connects the WM-functions with different fractal dimensions. Besides, we
consider an example based on real experimental data. A high correlation of the
statistically significant parameters of the DGI obtained from the measured data
(associated with reflection optical spectra of olive oil) with the sample temperature
is shown. This new methodology opens wide practical applications in the differ-
entiation of the hidden interconnections between measured by the environment and
external factors.
In chapter “Nonlocal Conditions for Semi-linear Fractional Differential
Equations with Hilfer Derivative”, we study the existence of solutions and some
topological proprieties of solution sets for nonlocal semi-linear fractional differ-
ential equations of Hilfer type in Banach space by using noncompact measure
method in the weighted space of continuous functions. The main result is illustrated
with the aid of an example. In chapter “Offshore Wind System in the Way of
Energy 4.0: Ride Through Fault Aided by Fractional PI Control and VRFB”,we
present a simulation about a study to improve the ability of an offshore wind system
to recover from a fault due to a rectifier converter malfunction. The system com-
prises: a semi-submersible platform; a variable speed wind turbine; a synchronous
generator with permanent magnets; a five-level multiple point diode clamped
converter; a fractional PI controller using the Carlson approximation. Recovery is
improved by shielding the DC link of the converter during the fault using as further
equipment a redox vanadium flow battery, aiding the system operation as desired in
the scope of Energy 4.0. Contributions are given for: (i) the fault influence on
the behavior of voltages and currents in the capacitor bank of the DC link; (ii) the
drivetrain modeling of the floating platform by a three-mass modeling; (iii) the
vanadium flow battery integration in the system.
vi Preface
goyal.praveen2011@gmail.com
In chapter “Soft Numerical Algorithm with Convergence Analysis for Time-
Fractional Partial IDEs Constrained by Neumann Conditions”, a soft numerical
algorithm is proposed and analyzed to fitted analytical solutions of PIDEs with
appropriate initial and Neumann conditions in Sobolev space. Meanwhile, the
solutions are represented in series form with accurately computable components.
By truncating the n-term approximate solutions of analytical solutions, the solution
methodology is discussed for both linear and nonlinear problems based on the
nonhomogeneous term. Analysis of convergence and smoothness is given under
certain assumptions to show the theoretical structures of the method. Dynamic
features of the approximate solutions are studied through an illustrated example.
The yield of numerical results indicates the accuracy, clarity, and effectiveness
of the proposed algorithm as well as provide a proper methodology in handling
such fractional issues. Chapter “Approximation of Fractional-Order Operators”
deals with the several comparisons in the time response and Bode results between
four well-known methods; Oustaloup’s method, Matsuda’s method, AbdelAty’s
method, and El-Khazali’s method are made for the rational approximation of
fractional-order operator (fractional Laplace operator). The various methods along
with their advantages and limitations are described in this chapter. Simulation
results are shown for different orders of the fractional operator. It has been shown in
several numerical examples that the El-Khazali’s method is very successful in
comparison with Oustaloup’s, Matsuda’s, and AbdelAty’s methods.
In chapter “Multistep Approach for Nonlinear Fractional Bloch System Using
Adomian Decomposition Techniques”, we discuss a superb multistep approach,
based on the Adomian decomposition method (ADM), which is successfully
implemented for solving nonlinear fractional Bolch system over a vast interval,
numerically. This approach is demonstrated by studying the dynamical behavior
of the fractional Bolch equations (FBEs) at different values of fractional order ain
the sense of Caputo concept over a sequence of the considerable domain. Further,
the numerical comparison between the proposed approach and implicit Runge–
Kutta method is discussed by providing an illustrated example. The gained results
reveal that the MADM is a systematic technique in obtaining a feasible solution for
many nonlinear systems of fractional order arising in natural sciences.
The chapter “Simulation of the Space–Time-Fractional Ultrasound Waves with
Attenuation in Fractal Media”deals with the simulation of the space–time-fractional
ultrasound waves with attenuation in fractal media. In chapter “Certain Properties of
Konhauser Polynomial via Generalized Mittag-Leffler Function”, we establish
several new properties of generalized Mittag-Leffler function via Konhauser
polynomials. Properties like mixed recurrence relations, differential equations, pure
recurrence relations, finite summation formulae, and Laplace transform have been
obtained. In chapter “An Effective Numerical Technique Based on the Tau Method
for the Eigenvalue Problems”, we consider the (presumably new) effective
numerical scheme based on the Legendre polynomials for approximate solution of
eigenvalue problems. First, a new operational matrix, which can be represented by
sparse matrix is defined by using the Tau method and orthogonal functions. Sparse
data is by nature more compressed and thus require significantly less storage.
Preface vii
goyal.praveen2011@gmail.com
A comparison of the results for some examples reveals that the presented method is
convenient and effective, also we consider the problem of column buckling to show
the validity of the proposed method. Finally, in chapter “On Hermite–Hadamard-
Type Inequalities for Coordinated Convex Mappings Utilizing Generalized
Fractional Integrals”, we obtain the Hermite–Hadamard-type inequalities for
coordinated convex function via generalized fractional integrals, which generalize
some important fractional integrals such as the Riemann–Liouville fractional
integrals, the Hadamard fractional integrals, and Katugampola fractional integrals.
The results given in this chapter provide a generalization of several inequalities
obtained in earlier studies.
Jaipur, India Praveen Agarwal
Ankara, Turkey Dumitru Baleanu
Merced, USA YangQuan Chen
Amman, Jordan Shaher Momani
Porto, Portugal JoséAntónio Tenreiro Machado
viii Preface
goyal.praveen2011@gmail.com
Contents
Closed-Form Discretization of Fractional-Order Differential
and Integral Operators ..................................... 1
Reyad El-Khazali and J. A. Tenreiro Machado
On Fractional-Order Characteristics of Vegetable Tissues
and Edible Drinks ......................................... 19
J. A. Tenreiro Machado and António M. Lopes
Some Relations Between Bounded Below Elliptic Operators
and Stochastic Analysis ..................................... 37
Rémi Léandre
Discrete Geometrical Invariants: How to Differentiate the Pattern
Sequences from the Tested Ones? ............................. 47
Raoul R. Nigmatullin and Artem S. Vorobev
Nonlocal Conditions for Semi-linear Fractional Differential Equations
with Hilfer Derivative ....................................... 69
Benaouda Hedia
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault
Aided by Fractional PI Control and VRFB ...................... 85
Rui Melicio, Duarte Valério and V. M. F. Mendes
Soft Numerical Algorithm with Convergence Analysis
for Time-Fractional Partial IDEs Constrained
by Neumann Conditions ..................................... 107
Omar Abu Arqub, Mohammed Al-Smadi and Shaher Momani
Approximation of Fractional-Order Operators ................... 121
Reyad El-Khazali, Iqbal M. Batiha and Shaher Momani
ix
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System Using
Adomian Decomposition Techniques ........................... 153
Asad Freihat, Shatha Hasan, Mohammed Al-Smadi, Omar Abu Arqub
and Shaher Momani
Simulation of the Space–Time-Fractional Ultrasound Waves
with Attenuation in Fractal Media ............................. 173
E. A. Abdel-Rehim and A. S. Hashem
Certain Properties of Konhauser Polynomial via Generalized
Mittag-Leffler Function ..................................... 199
J. C. Prajapati, N. K. Ajudia, Shilpi Jain, Anjali Goswami
and Praveen Agarwal
An Effective Numerical Technique Based on the Tau Method
for the Eigenvalue Problems ................................. 215
Maryam Attary and Praveen Agarwal
On Hermite–Hadamard-Type Inequalities for Coordinated Convex
Mappings Utilizing Generalized Fractional Integrals ............... 227
Hüseyin Budak and Praveen Agarwal
x Contents
goyal.praveen2011@gmail.com
Closed-Form Discretization of
Fractional-Order Differential
and Integral Operators
Reyad El-Khazali and J. A. Tenreiro Machado
Abstract This paper introduces a closed-form discretization of fractional-order dif-
ferential or integral Laplace operators. The proposed method depends on extracting
the necessary phase requirements from the phase diagram. The magnitude frequency
response follows directly due to the symmetry of the poles and zeros of the finite
z-transfer function. Unlike the continued fraction expansion technique, or the infinite
impulse response of second-order IIR-type filters, the proposed technique general-
izes the Tustin operator to derive a first-, second-, third-, and fourth-order discrete-
time operators (DTO) that are stable and of minimum phase. The proposed method
depends only on the order of the Laplace operator. The resulted discrete-time opera-
tors enjoy flat-phase response over a wide range of discrete-time frequency spectrum.
The closed-form DTO enables one to identify the stability regions of fractional-
order discrete-time systems or even to design discrete-time fractional-order PIλDμ
controllers. The effectiveness of this work is demonstrated via several numerical
simulations.
Keywords Fractional calculus ·Transfer function ·Discrete-time operator ·
Discrete-time integro-differential operators ·Frequency response
1 Introduction
Fractional calculus is a generalization of the integer-order one. Most practical sys-
tems exhibit fractional-order dynamics, which could be of real or complex values.
Fractional-order systems enjoy the hereditary effect that is approximated by infinite-
dimensional models [8,20]. It is used in many fields such as in economy, physics,
R. El-Khazali (B
)
ECE Department, Khalifa University, Abu Dhabi, United Arab Emirates
e-mail: reyad.elkhazali@ku.ac.ae
J. A. T. Machado
Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Porto,
Portugal
e-mail: jtm@isep.ipp.pt
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_1
1
goyal.praveen2011@gmail.com
2 R. El-Khazali and J. A. T. Machado
biology, chemistry, medicine, social sciences, and engineering. To analyze fractional-
order systems, one has to look for finite-dimensional and realizable models that
approximate such systems [11,12,19,21–25].
The use of microprocessors nowadays are necessary for signal processing and sys-
tem analysis. Thus, a straightforward method is required to discretize a continuous-
time fractional-order system into a discrete-time one. This can be accomplished by
discretizing the fractional-order Laplacian operator sαand replacing it with a finite-
order DTO. In general, there are two methods that are used to discretize sα; i.e., a
direct and an indirect one. In the indirect discretization method, a rational continuous-
time operator (CTO) is first obtained and then discretized using techniques such as
the bilinear transformation, the Al-Alaoui operator, the Euler’s backward method, or
the stable Simpson’s method [1–3].The direct method, however, allows one to gen-
erate discrete-time operators that converts a continuous-time operator (CTO) into a
DTO [4,5,17].
The indirect discretization method is achieved in two steps; the first one is to
approximate the Laplacian operator sαby a rational transfer function in the s-domain,
which is then simplified using the continued fraction expansion (CFE), and the sec-
ond step is to discretize the expanded form using either the bilinear transformation,
Simpson’s method, Euler’s method, or a linear combination of them or other exist-
ing forms [6,24]. It is important to realize that the CFE method could yield an
unstable non-minimum phase discrete-time operator. An alternative approach to the
CFE was discussed in [19], where infinite impulse response (IIR) autoregressive
moving-average (ARMA) models are used to develop DTO operators, which may
result in developing higher order approximation. Notice that the Al-Alaoui operator
is obtained as a linear combination of the trapezoidal and the rectangular integration
rules [2,14,15,26]. The interpolation and inversion processes may induce, in some
cases, unstable fractional-order operators.
This work introduces a straightforward discretization direct method to discretize
continuous differential and/or integral operators. It can be considered as a dynamic
(or adaptive) discretization technique, where the poles and the zeros of the generated
z-transfer function are all located inside the unit disc and their values depend only on
the fractional-order α. The proposed method yields finite-order DTO that exhibits a
competitive frequency response to higher order operators developed in [2,6,16].
The paper is organized as follows. Section 2summarizes some preliminary con-
cepts and background. Section 3introduces the main results of first-, second-, third-,
and fourth-order operators, while Sect. 4summarizes the numerical simulation and
a comparison between different operators. Section5outlines the main conclusions.
2 Preliminary Concepts and Background
The general fractional-order differential (integral) operator is denoted by aD±α
t(aIα
t),
respectively [18], where aand trepresent the starting time and α∈Ris the order
of the operator. For example, if one wishes to implement a discrete-time fractional-
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 3
order controller, then it is necessary to look for a stable non-minimum phase DTO
operator of low order. The design and implementation of fractional-order discrete-
time controllers cannot accommodate higher order operators since this will increase
the complexity of the controlled system, and could yield unstable ones. Therefore, the
proposed technique provides a competitive DTO that benefits from the IIR structure
of such operators; i.e., a second-order DTO is competitive to that of a ninth-order
one introduced in [6,19,20].
As mentioned in Sect. 1, the indirect discretization method starts by develop-
ing a rational finite-order transfer functions, that is, s±α≈N(s,α)
D(s,α)[10,13,21], and
it is followed by using any existing discretization technique, or a linear combina-
tion of such methods. For example, the Al-Alaoui discrete-time integral operator is
simply a linear interpolation of the backward rectangular rule and the trapezoidal
rule, namely H(z)=aHRect (z)+(1−a)HTrap (z), where 0 <a<1[1–3]. A
similar approach was used to derive a hybrid digital integrator using a linear com-
bination of Trapezoidal and Simpson integrator [6,10]. Such interpolation reduces
the frequency warping over a limited frequency band, and their phase frequency
response is not constant. For comparison, Fig. 1displays the frequency response of the
Tustin operator, s=H(z)=2
T
1−z−1
1+z−1, Al-Alaoui operator, s=H(z)=8
7T
1−z−1
1+1
7z−1,
and Chen discrete-time operator [5]. Another discrete-time operator that approx-
imates an integer-order integrator was also introduced in [6] and given here for
completeness:
H(z)=6z2−1
T(3−a)(
z+p1)(
z+p2),(1a)
p1=3+a+2√3a
3−a,(1b)
p2=3+a−2√3a
3−a,(1c)
where Tis the sampling time and 0 <a<1 is a scaling factor. Equation (1) can
then be used to generate several quadratic forms that discretize s±1.
Figure 1shows the frequency response of the aforementioned three DTO opera-
tors that approximate s1for T=0.001. The magnitude response of Tustin operator
exhibits large errors at both ends of the frequency spectrum. The magnitude response
of the Al-Alaoui operator, however, is almost identical to that of the Tustin operator
at low frequency, but provides a better response at high frequency. Moreover, it yields
a linear phase response due to the asymmetric pole-zero location, while the hybrid
ninth-order operator reported in [6] yields a perfect phase behavior. However, one
cannot afford this size of an operator since a discrete-time fractional-order phase-
locked loop, for example, will be modeled by an 18th-order discrete-time z-transfer
function.
Since the goal is to look for a closed-form discrete-time model for s±α,the
direct approach is adopted here to develop a straightforward discretization method.
goyal.praveen2011@gmail.com
4 R. El-Khazali and J. A. T. Machado
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
150
200
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−60
−40
−20
0
20
40
60
Normalized Frequency (×π rad/sample)
Magnitude (dB)
Tustin
Al−Alaoui
Chen et. al.
Fig. 1 Frequency response of Tustin, Al-Alaoui, and the DTO of Eq. (1)fora=1
In all direct methods, the continuous frequency operator is replaced by a generat-
ing function, that is, s±α=ωz−1±α. To gain more insight, one may start with
the Grünwald–Letnikov (GL) definition of the fractional-order differential (integral)
operator [7,13,14,21,22]:
aD±α
tf(t)=lim
h→0
1
h±α
∞
j=0
C±α
jf((t−j)h).(2)
where
C±α
j=(−1)j±α
j=1−1±α
jC±α
j−1,j=1,...,n,(3a)
C±α
0=1.(3b)
Taking the Z-transform of (2) and using the short memory principle [14], the
following generating function may discretize s±α:
ωz−1±α=T∓α⎛
⎝
[L
T]
j=0
C±α
jz−j⎞
⎠,(4)
where T=his the sampling time, and L
T=nh−a
his an increasing memory size
L−nh −a.
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 5
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−20
−15
−10
−5
0
5
Normalized Frequency (×π rad/sample)
Magnitude (dB)
Fig. 2 Frequency response of a FIR-type discrete-time differentiator for s0.5with T=0.001 s,
and L=0.011
Equation (4) defines a transfer function of a finite impulse response (FIR) discrete-
time system of s±α. The memory size, L, determines the accuracy of the approxi-
mation. Hence, a compromise has to be made between the accuracy and the size of
the operator. Figure 2shows the frequency response of (4)forα=0.5, T=0.001,
and L=11. Clearly, the phase diagram is close to the expected angle of π
4over a
very narrow frequency band ω∈(0.06,0.08)rad/s, which may not be suitable for
realization techniques.
Obviously, in spite of its large size, the frequency response of the FIR discrete-time
form of Eq. (4) does not provide the expected constant phase response. Therefore,
an alternative discrete-time IIR-type rational z-transfer function, of lower size than
the FIR form, to discretize s±αwill be the choice to overcome such problem.
Since, the CFE approach does not always yield a minimum phase and stable
system, or a flat-phase response [2,6,7,11,13], a compromise has to be made
between the size of the expansion and the type of the generating functions used for
approximation. The following generating functions can be used to discretize s±αand
replace it with DTO operators [5,6,14,17]:
(a) Backward-Euler method: ωz−1±α=1−z−1
T±α
(b) Trapezoidal (Tustin) discretization rule: ωz−1±α=2
T
1−z−1
1+z−1±α
(c) Al-Alaoui Operator: ωz−1±α=8
7T
1−z−1
1+1
7z−1±α
goyal.praveen2011@gmail.com
6 R. El-Khazali and J. A. T. Machado
(d) A Hybrid interpolation of Simpson and Trapezoidal discrete-time integrators:
H(z)=aHS(z)+(1−a)HT(z),0<a<1,(5)
where HS(z)=T
3
1+4z−1+z−2
1−z−2and HT(z)=T
2
1+z−1
1−z−1.
The interpolation in (5) represents a generalization of the first three methods.
Since the magnitude frequency response of the integer-order integrator, s−1, lies
between the Simpson rule and that of the Trapezoidal discrete-time integrator [2,3],
the linear combination in (5)for0 <a<1 can be used to generate a typical IIR-type
discrete-time operator as follows [6]:
ωz−1±α=k01−z−2
1+bz−12α
,(6)
where α∈[0,1],k0=6z2
T(3−a)α
and b=z2=3+a−2√3a
3−a.
Several transfer functions of different sizes can be obtained to approximate
ωz−1±α. For example, when α=0.5 and T=0.001, Eq. (6), yields the fol-
lowing z-transfer functions, G(n,a)(z), that discretize s0.5, where nand arepresent
the order and the weighting factor of the approximation, respectively [6]:
G(2,0,5)z−1=127 +41.26z−1−112.6z−2
4+2.98z−1−z−2,(7a)
G(3,0,5)z−1=1501 −503.6z−1−1298z−2+446.5z−3
47.26 +4z−1−23.63z−2−z−3,(7b)
G(4,0,5)z−1=508.1−1501z−1−4.478z−2+1298z−3−382.9z−4
16 −40.54z−1−12z−2+20.27z−3+z−4.(7c)
Figure 3shows the frequency response of (7)forω∈(−π, π). The magnitude
frequency response of the second-order approximation yields a warping effect at
high frequency, while the phase diagram of the three forms exhibit a decreasing
phase value over most of the spectrum.
Remark 1 The approximation given by (7c), reported in [6], represents an unstable
non-minimum phase DTO since it has a pole and a zero outside the unit circle at
p=2.6298, and z=2.6328, respectively. Even though p≈z, that almost cancel
each other, implementing such an operator would cause system instability. Further-
more, according to [6], one must improve the phase performance of G(4,0,5)(z)by
cascading a causal lead compensator z0.5=z−0.5
z−1, which requires the implementation
of a fractional-order sampler.
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 7
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−150
−100
−50
0
50
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
20
30
40
50
Normalized Frequency (×π rad/sample)
Magnitude (dB)
2nd−order
3rd−order
4th−order
Fig. 3 Frequency response of s0.5using (7a), (7b), (7c)
3 New Fractional-Order Discrete-Time Operators
As discussed in Sect. 2, the discretization technique of generating functions using the
CFE yields high order and an unstable non-minimum phase discrete-time approx-
imation. The aim of this work is to avoid such subtleties by developing an adap-
tive closed-form DTO that effectively discretizes the fractional-order operators, s±α,
which only depend on its order ±α. Furthermore, one can also define the stability
region of the discrete form of s±α.
3.1 First-Order Operators
The following first-order operator based on a closed-form solution was first intro-
duced in [8,9]. It represents an approximation of a first-order discrete-time differ-
ential operator (DTDO), where its reciprocal also defines a discrete-time integral
operator (DTIO):
s±α≈H1K(z)=2
T±αz∓z1(α)
z∓p1(α),(8)
where
z1(α)=−p1(α)=1
tan (2−α)π
4,0<α<1,(9)
and where z1(α)=−p1(α)∈R.
goyal.praveen2011@gmail.com
8 R. El-Khazali and J. A. T. Machado
Obviously, for 0 <α<1, |z1(α)|=|p1(α)|<1 are located inside the unit
circle.
3.2 Second-Order Operator
The second-order discrete-time operator was also introduced in [8,9]. It yields a
normalized biquadratic discrete-time transfer function that approximates s±αand is
given by (Fig. 4):
s±α≈H2K(z)=2
T±α(z∓z1(α)) (z∓z2(α))
(z∓p1(α)) (z∓p2(α)) ,(10)
where
z1(α)=
η2−2+5η2
2+4
2η2
,η
2=tan απ
4,(11)
and ⎧
⎨
⎩
z2(α)=z1(α)−1
p1(α)=−z2(α)
p2(α)=−z1(α)
.(12)
Clearly, for large values of α, the first-order DTO yields a competitive frequency
response to that of the second-order DTO as shown in Fig.5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−15
−10
−5
0
5
10
15
Normalized Frequency (×π rad/sample)
Magnitude (dB)
2nd−order
1st−order
Fig. 4 Frequency response of discrete-time first- and second-order operators for s0.5
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
20
40
60
80
100
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−40
−30
−20
−10
0
10
20
30
40
Normalized Frequency (×π rad/sample)
Magnitude (dB)
Frequency response of 1st & 2nd DTO
1st−order DTO
2nd−order DTO
Fig. 5 Frequency response of (8)and(10)fors0.95
3.3 Third-Order Operator
The third-order operator is developed to improve the accuracy of the discrete-time
approximation over a wider frequency range. Similar to (10), the third-order operator
is given by
s±α≈H3K(z)=2
T±α(z∓z1(α)) (z∓z2(α)) (z∓z3(α))
(z∓p1(α)) (z∓p2(α)) (z∓p3(α)) ,(13)
where ⎧
⎪
⎪
⎨
⎪
⎪
⎩
p3(α)=−z1(α)
z2(α)=1−z1(α)
p2(α)=−z2(α)
z3(α)=−p1(α)
(14)
The pole-zero map of (14) is shown in Fig. 6, which represents a distribution of
alternating real poles and zeros.
goyal.praveen2011@gmail.com
10 R. El-Khazali and J. A. T. Machado
Fig. 6 Pole-zero map of
third-order DTO operator
Due to the symmetry of the poles and zeros and since zi(α)=−pi(α),i=
1,2,3, the phase requirement is assumed to meet the phase contribution of the
fractional-order operator at the discrete-time frequency =απ
2:
ϕz1+ϕz2+ϕz3−ϕp1+ϕp2+ϕp3=απ
2(15)
Substituting (14)into(15) yields
z1=max root s z2
1−z1+q(α),(16)
where
q(α) =2−α(1+η3)
1+η3(1−α)(17)
and
η3=tan απ
4.(18)
Hence z1is found, the rest of poles and zeros are determined from (17) and
(15). For example, for α=0.5Eq.(19)givesz1=0.8425, z2=0.1575, and z3=
−0.5, while p1=−z3,p2=−z2and p3=−z1. Therefore, the third-order DTO
that discretizes s0.5for T=2 is given by
s0.5≈H3K(z)=z3−0.5z2−0.3673z+0.06635
z3+0.5z2−0.3673z−0.06635 .(19)
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 11
Fig. 7 Pole-zero map of the
fourth-order DTO operator
Remark 2 When α=1, then from (17), q(1)=0, and Eq. (16) reduces to z2
1−z1=
0, which yields a nontrivial solution z1=1, and the third-order DTO operator given
by (13) and (14) for this case reduces to the well-known bilinear transformation
H3K=2
T
1−z−1
1+z−1.
3.4 Fourth-Order Operator
The fourth-order z-transfer function that discretizes the fractional-order operators is
similarly developed as the previous three operators and given by the following finite
z-transfer function:
s±α≈H4K(z)=2
T±α(z∓z1(α)) (z∓z2(α)) (z∓z3(α)) (z∓z4(α))
(z∓p1(α)) (z∓p2(α)) (z∓p3(α)) (z∓p4(α)) ,
(20)
where ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
p4(α)=−z1(α)
p2(α)=1−z1(α)
z3(α)=−p2(α)
z4(α)=−p1(α)
p3(α)=−z2(α)
(21)
The pole-zero map of (21) is shown in Fig. 7, which also represents a distribution
of alternating real poles and zeros of (20).
The phase contribution of (20) is given by
goyal.praveen2011@gmail.com
12 R. El-Khazali and J. A. T. Machado
ϕz1+ϕz2+ϕz3+ϕz4−ϕp1+ϕp2+ϕp3+ϕp4=απ
2(22)
Since there is a symmetry between the poles and zeros as depicted in Fig. 7, one
may focus on the phase contribution of the poles and zeros that lie on the positive
real axis. By other words, from the symmetry, and without loss of generality, one
may conclude from (22), that,
ϕz1+ϕz2−ϕp1+ϕp2=απ
4(23)
where
ϕzi=π−arctan 1
zi(α),ϕ
pi=π−arctan 1
pi(α),i=1,2.(24)
Assumption 2 Let p1(α)and z2(α)lie in the geometric mean of their adjacent
zeros and poles, respectively,
p1(α)=z1(α)z2(α),(25a)
z2(α)=p1(α)p2(α).(25b)
Substituting (24) and (25)into(23) yields the following nonlinear function in
z1(α):
f(z)=ηz4
1+2(1−η)z3
1−(η+3)z2
1+(2η−1)z1+(η+1)
+ηz
5
3
1(1−z1)1
3+z
1
3
1(1−z1)5
3−z
4
3
1(1−z1)2
3−z
2
3
1(1−z1)4
3
+z
5
3
1(1−z1)4
3−z
4
3
1(1−z1)5
3+z
2
3
1(1−z1)1
3−z
1
3
1(1−z1)2
3=0,
(26)
where η=tan απ
4.
Obviously, the nonlinearities in (26) are due to placing the inner pole/zero at
the geometric mean of its surrounding zeros/poles. Solving (26) numerically with
an accuracy |f(z)|<, for small >0 yields a desired solution 0 <z1<1. For
α=0.5, the fourth- order operator that discretizes s0.5is found to be
s0.5≈H4k(z)=2
Tαz4−0.5295z3−0.3835z2+0.05843z+0.01218
z4+0.5295z3−0.3835z2−0.05843z+0.01218 .(27)
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 13
Fig. 8 Frequency response of the second-, third-, and fourth-order operators for α=0.5, α=0.7
and α=0.9
goyal.praveen2011@gmail.com
14 R. El-Khazali and J. A. T. Machado
Fig. 8 (continued)
4 Numerical Simulation
Figure 8a–c shows the frequency response of the second-, third-, and the fourth-order
operators for α=0.5, α=0.7, and α=0.9, respectively. As noted, the second-order
operator is a good competitor to the third-order one, especially for α>0.7, while
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−10
0
10
20
30
40
50
Normalized Frequency (×π rad/sample)
Phase (degrees)
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−30
−20
−10
0
10
20
Normalized Frequency (×π rad/sample)
Magnitude (dB)
2nd−order (eq.(27))
9th−order (eq.(28))
Fig. 9 Frequency response of (27)and(28)fors0.5
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 15
the fourth-order operator exhibits a much better frequency response with a constant
phase at the middle frequency with some overshoot at both ends of the spectrum.
To appreciate the proposed DTO, the frequency response of the second- order
operator described by (10) is compared with other forms of DTO reported in [2,6].
The case when α=0.5forT=0.001 is taken as a benchmark. Equations (10)–(12)
then yield
s0.5≈44.7214 −22.0313z−1−8.4670z−2
1.0+0.4926z−1−0.1893z−2.(28)
The following ninth-order DTO that discretizes s0.5using the CFE and reported
in [6] is investigated against the one given by (28)
G9(z)=44.72 z9−0.5z8−2z7+0.875z6+1.313z5−0.4688z4
z9+0.5z8−2z7−0.875z6+1.313z5+0.4688z4···
−0.3125z3+0.07813z2+0.01953z−0.001953
−0.3125z3−0.07813z2+0.01953z−0.001953 .(29)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−10
0
10
20
30
40
50
60
Normalized Frequency (×π rad/sample)
Phase (degrees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−30
−20
−10
0
10
20
Normalized Frequency (×π rad/sample)
Magnitude (dB)
4th−order
9th−order
Fig. 10 Comparison between the approximations of (27)and(29)fors0.5
goyal.praveen2011@gmail.com
16 R. El-Khazali and J. A. T. Machado
Figure 9displays the frequency response of both (28) and (29). Both forms exhibit
similar magnitude and phase responses. However, the proposed second-order oper-
ator of (28) has a better magnitude response at low frequency than the ninth-order
one given by (29), while the phase response of (29) at low frequency is better than
that of (28). However, the significant improvement of (28) over the one in (29)is
evident by the order reduction. For example, if one wishes to discretize a system
with Laplace operators of two different orders, one would need an 18th-order model,
while a second-order DTO given by (10) is faster and requires less hardware (Fig. 10).
The appealing factor in the proposed techniques lies in the fact that in all cases
and for different fractional orders, the reciprocals of all proposed operators yield
stable minimum phase discrete-time integrators.
5 Conclusion
A closed-form discrete-time first-, second-, third-, and fourth-order operators (DTO)
are introduced to discretize the fractional-order Laplacian operator, s±α. Each opera-
tor is described by finite-dimensional rational z-transfer function. The discretization
method is straightforward and depends on the order of the operator. The proposed
method generates an adaptive, symmetrical real poles, and zeros that migrate to differ-
ent locations inside the unit disc. The corresponding z−transfer functions represent
stable non-minimum phase IIR-filters that exhibit constant phase and gain frequency
responses over a wide frequency spectrum. As αapproaches 1, the poles and zeros
of all four operators converge to ±1 and they reduce to the well-known discrete-time
bilinear transformation. It is worth noting that the first and the second-order opera-
tors will be sufficient to discretize s±αfor high fractional orders; (say 0.8≤α≤1).
The proposed DTO operators exhibit competitive frequency responses to those ones
obtained by different discretization methods.
References
1. Al-Alaoui, M.A.: Novel digital integrator and differentiator. IEE Electron. Lett. 29(4), 376–378
(1993)
2. Al-Alaoui, M.A.: Novel stable higher order s-to-z transforms. IEEE Trans. Circuit. Syst. I:
Fundam. Theory Appl. 48(11), 1326–1329 (2001)
3. Al-Alaoui, M.A.: Al-Alaoui operator and the α-approximation for discretization of analog
system. FACTA Universitatis (NIS) 19(1), 143–146 (2006)
4. Barbosa, R.S., Machado, J.A.T., Ferreira, I.M.: Pole-zero approximations of digital fraction-
alorder integrators and diferentiators using modeling techniques. In: 16th IFACWorld Congress.
Prague, Czech Republic (2005)
5. Chen, Y., Moore, K.: Discretization schemes for fractional-order differentiatorsand integrators.
IEEE Trans. Circuit. Syst.I: Fundam. Theory Appl. 49(3), 363–367 (2002)
6. Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion to discretize fraction-
alorder derivatives - an expository review. Nonlinear Dyn. 38(1), 155–170 (2004)
goyal.praveen2011@gmail.com
Closed-Form Discretization of Fractional-Order Differential and Integral Operators 17
7. Dorˇcák, L., Petráš, I., Terpák, J., Zborovjan, M.: Comparison of the method for discrete approx-
imation of the fractional order operator. In: Proceedings of the International Carpathian Control
Conference (ICCC’2003), pp. 851–856. High Tatras, Slovak Republic (2003)
8. El-Khazali, R.: Biquadratic approximation of fractional-order Laplacian operators. In: 2013
IEEE 56th International Midwest Symposium on Circuits and Systems(MWSCAS), pp. 69–72.
Columbus, OH (2013)
9. El-Khazali, R.: Discretization of fractional-order differentiators and integrators. In: 19th IFAC
World Congress, pp. 2016–2021. Cape Town, South Africa (2014)
10. Gupta, M., Yadav, R.: Design of improved fractional-order integrators using indirect discretiza-
tion method. Int. J. Comput. Appl. 59(14) (2012)
11. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equa-
tions, vol. 204. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)
12. Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Pro-
cess. 59(3), 386–426 (2011)
13. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)
14. Machado, J.T.: Analysis and design of fractional-order digital control systems. Syst. Anal.
Model. Simul. 27(2–3), 107–122 (1997)
15. Machado, J.T.: Fractional-order derivative approximations in discrete-time control systems.
Syst. Anal. Model. Simul. 34, 419–434 (1999)
16. Matignon, D.: Stability results for fractional differential equations with applications to control
processing. In: Computational Engineering in Systems Applications, vol. 2, pp. 963–968. Lille,
France (1996)
17. Nie, B., Li, W., Ma, H., Wang, D., Liang, X.: Research of direct discretization method of
fractional order differentiator/integrator based on rational function approximation. In: Tarn,
T.J., Chen, S.B., Fang, G. (eds.) RoboticWelding, Intelligence and Automation. Lecture Notes
in Electrical Engineering, pp. 479–485. Springer, Berlin, Heidelberg (2011)
18. Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation
and Integration to Arbitrary Order. Academic Press, New York (1974)
19. Ortigueira, M.D., Serralheiro, A.J.: Pseudo-fractional ARMA modelling using a double Levin-
son recursion. IET Control Theory Appl. 1(1), 173–178 (2007)
20. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.: Frequency-band complex noninteger differ-
entiator: characterization and synthesis. IEEE Trans. Circuit. Syst. I: Fundam. Theory Appl.
47(1), 25–39 (2000)
21. Podlubny, I.: The Laplace transform method for linear differential equations of the fractional
order. Tech. Rep. UEF-02-9, Slovak Academy of Sciences Institute of Experimental Physics,
Kosice, Slovakia (1994)
22. Podlubny, I.: Fractional differential equations, Volume 198: An Introduction to Fractional
Derivatives, Fractional Differential Equations, to Methods of Their Solution. Mathematics in
Science and Engineering. Academic Press, San Diego (1998)
23. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives and Some of Their
Applications. Nauka i Tekhnika, Minsk (1987)
24. Siami, M., Tavazoei, M.S., Haeri, M.: Stability preservation analysis in direct discretization of
fractional order transfer functions. Signal Process. 91(3), 508–512 (2011)
25. Vinagre, B., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order
operators used in control theory and applications. Fract. Calculus Appl. Anal. 3(3), 231–248
(2000)
26. Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fraction-
alorder differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003)
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics
of Vegetable Tissues and Edible Drinks
J. A. Tenreiro Machado and António M. Lopes
Abstract This chapter uses frequency response techniques to characterize vegetable
tissues and edible drinks. In the first phase, the impedance of the distinct samples is
measured and fractional-order models are applied to the resulting data. In a second
phase, hierarchical clustering and multidimensional scaling tools are adopted for
comparing and visualizing the similarities between the specimens.
Keywords Frequency response ·Fractional-order models ·Clustering ·
Visualization
1 Introduction
The frequency response technique with electrical signals, often referred to as electri-
cal impedance spectroscopy (EIS), measures the electrical impedance of a specimen
across a given range of frequencies [5,17,22,23,25]. This technique has the
advantage of being nondestructive, while avoiding complex and time-consuming
experimental or laboratory procedures. The EIS has been widely used for studying
vegetable tissues [4,36], animal, and human samples [1,13], beverages [30,39],
nonbiological materials [18,37], and devices [2,14].
This chapter addresses the application of the EIS for characterizing different
products, namely plant leaves, vegetables, wine, and milk [22–24,26]. In the first
phase, the impedance Z(jω) is measured and fractional calculus (FC) is applied to
model the samples with a reduced number of parameters. In a second phase, the
EIS experimental data are processed by means of hierarchical clustering (HC) and
J. A. T. Machado (B
)
Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto,
Rua Dr. António Bernardino de Almeida, 431, 4249–015 Porto, Portugal
e-mail: jtm@isep.ipp.pt
A. M. Lopes (B
)
Faculty of Engineering, UISPA–LAETA/INEGI, University of Porto, Rua Dr. Roberto Frias,
4200–465 Porto, Portugal
e-mail: aml@fe.up.pt
© Springer Nature Singapore Pte Ltd. 2019
P. Agar w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981-15-0430-3_2
19
goyal.praveen2011@gmail.com
20 J. A. T. Machado and A. M. Lopes
multidimensional scaling (MDS) algorithms for visualizing similarities between the
specimen.
The chapter is organized as follows. Section2introduces the tools and meth-
ods adopted in the follow-up. Section3describes the impedance spectra, Z(jω),
by means of FC models. Section4applies the MDS for clustering and visualizing
similarities between the specimens. Finally, Sect.5draws the main conclusions.
2 Tools and Methods
2.1 The Canberra Distance
The Canberra distance was proposed, and later modified, by Lance and Williams
[19,20]. Given 2 points in a K-dimensional space, X=(x1,...,xK)and Y=
(y1,...,yK), the Canberra distance between Xand Yis given by
dC(X,Y)=
K
k=1
|xk−yk|
|xk|+|yk|.(1)
Equation (1) is a metric widely used for quantifying data scattered around an
origin. The Canberra distance has several interesting properties, namely it is unitary
when the arguments are symmetric, biased for measures around the origin, and highly
sensitive for values close to zero.
2.2 Electrical Impedance Spectroscopy
In practical terms, the EIS method involves exciting a specimen with frequency-
variable electric sinusoidal signals and registering the system response. The voltage
v(t)and current i(t)across the specimen at steady state are sinusoidal functions of
time given by
v(t)=Vcos(ωt+θV)
i(t)=Icos(ωt+θI),(2)
where {V,I}are the amplitudes of the voltage and current, {θV,θ
I}denote their
phase shifts, ω=2πfrepresents the angular frequency, and fis the frequency.
The voltage and current can be represented in the frequency domain by
V(jω) =V·ejθV
I(jω) =I·ejθI,(3)
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 21
where j=√−1. The experimental complex impedance Ze(jω) is defined as the
ratio of phasors:
Ze(jω) =V(jω)
I(jω) =V
I·ej(θV−θI)=|Ze(jω)|·ejarg [Ze(jω)].(4)
Given an impedance spectrum Ze(jω), it is often necessary to find a mathematical
description, that is, a heuristic model, Zm(jω), that fits well into the experimental
data, and has a reduced number of parameters [28,29].
Different empirical models in the scope of the dielectric relaxation phenomenon
were proposed [12], namely the Debye, Cole-Cole, Cole-Davidson, and Havriliak-
Negami models [7–10,16,21,34,35]:
ZD(jω) =1
1+jωτ ,(5)
ZCC(jω) =1
1+(jωτ)α,(6)
ZCD(jω) =1
(1+jωτ)β,(7)
ZHN(jω) =1
[1+(jωτ)α]β,(8)
where 0 <α,β≤1, and τdenotes a relaxation time.
These empirical models are, in fact, particular cases of FC, and represent the fun-
damental bricks of any more complex fractional-order expressions, that may include
further poles and zeros.
In this chapter, the Bode diagrams of the electrical impedance Ze(jω) are approxi-
mated using fractional-order (FO) based models, while minimizing a fitness function,
J, based on the Canberra distance [6] between the experimental, Ze, and model, Zm,
impedances:
J=1
L
L
k=1|[Ze(jωk)]−[Zm(jωk)]|
|[Ze(jωk)]|+|[Zm(jωk)]|+|[Ze(jωk)]−[Zm(jωk)]|
|[Ze(jωk)]|+|[Zm(jωk)]|,
(9)
where Ldenotes the number of frequencies, ωk, used for measuring the electrical
impedance Ze(jω), and (·)and (·)represent the real and imaginary parts of a
complex number [22,24].
The function, J, leads to good results because it calculates the ratio between the
difference and the sum of two values. Therefore, it is possible to capture the relative
goyal.praveen2011@gmail.com
22 J. A. T. Machado and A. M. Lopes
error of the adjustment, avoiding saturation-like effects, that occurs when using the
standard Euclidean norm due to the simultaneous presence of large and small values.
2.3 Experimental Setup for EIS Measurements
The diagram of Fig. 1represents schematically the experimental arrangement adopted
for the measurements [22–24,26]. The specimens are connected in series with an
adaptation resistance, Rs=15 k, for signal measurement, while yielding a good
signal/noise ratio. A Hewlett Packard/Agilent 33220A function generator applies a
sinusoidal 5 V AC voltage to the circuit (i.e., a voltage divider). A Tektronix TDS
2002C two- channel oscilloscope measures the voltages Vab and Vcb. The impedance
Z(jω) is obtained for the frequency range 2π×10 ≤ω≤2π×105rad/s, at L=25
logarithmically spaced points, using the expression:
Z(jω) =Rs·Vab(jω)
Vcb(jω) −1.(10)
Several experimental tests demonstrated good stability in what concerns the oxi-
dation of the copper electrodes, while different electrode geometries revealed a neg-
ligible influence on the results. Moreover, experiments with various amplitudes of
the excitation signal showed good linearity, allowing data treatment using transfer
function concepts.
2.4 Hierarchichal Clustering
Clustering is a data analysis technique [15] that groups similar items. In HC, two
possible iterative strategies generate a hierarchy of clusters, namely the (i) agglom-
erative and the (ii) divisive clustering. With (i) each item starts in its own cluster and
the algorithm merges the two most similar clusters until there is one single cluster.
With (ii) all items start in a single cluster and the algorithm removes the outsiders
from the least cohesive cluster, until each item is in its own cluster. In both cases,
it is required a linkage criterion, that is a function of the distances between pairs of
items, for quantifying the dissimilarity between clusters. For 2 clusters, Rand S,the
distance d(xR,xS)between items xR∈Rand xS∈Sis based on metrics such as the
maximum, minimum, and average linkages given by [3]
dmax (R,S)=max
xR∈R,xS∈Sd(xR,xS),(11)
dmin (R,S)=min
xR∈R,xS∈Sd(xR,xS),(12)
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 23
Fig. 1 Experimental EIS setup for measuring impedance Z(jω)
dave (R,S)=1
R S
xR∈R,xS∈S
d(xR,xS).(13)
After using one of the algorithms, the results of HC are presented in a graphical
object such as a dendrogram or a hierarchical tree.
To assess the quality of the clustering, the cophenetic correlation (CC) coefficient
is used [33]. The CC gives a measure of how well the generated graphical object
preserves the original pairwise distances. If the clustering is successful, the links
between items in the graphical object have a strong correlation with those in the
original dataset. The closer the CC value to 1, the better the clustering result. The
quality assessment is plotted in a Shepard diagram that compares the original and
the cophenetic distances. A good clustering leads to a layout of points close to the
45-degree line.
goyal.praveen2011@gmail.com
24 J. A. T. Machado and A. M. Lopes
2.5 Multidimensional Scaling
MDS is a computational technique for clustering and visualizing data [31]. In the
first phase, given sitems and a measure of dissimilarity, a s×ssymmetric matrix,
C=[cij],(i,j)=1,...,s, of item-to-item dissimilarities is calculated. The matrix
Crepresents the input information for starting the MDS computational scheme. The
MDS rational is to assign points for representing items in a multidimensional space
and to try to reproduce the measured dissimilarities, cij. In a second phase, MDS
evaluates different configurations for maximizing some fitness function, arriving
at a set of point coordinates (and, therefore, to a symmetric matrix of distances
D=[dij]) with the reproduced dissimilarities that best approximates cij. A common
fitness function for measuring the difference between cij and dij is the raw stress:
S=dij −f(cij)2,(14)
where f(·)indicates some types of transformation.
The MDS interpretation is based on the patterns of points that can be visualized
in the generated map. Similar (dissimilar) objects are represented by points that are
close to (far from) each other. So, the information retrieval is not based on the point
coordinates, or the geometrical form of the clusters. Indeed, we can rotate, translate,
or magnify the map (for better visualization) because the distances remain identical.
The MDS axes have neither special meaning nor units.
The quality of the MDS mat can be assessed by means of the stress and Shepard
plots. The stress plot represents Sversus the number of dimensions mof the MDS
map. The plot S(m)is a monotonic decreasing chart and choosing the value of m
is a compromise between achieving low values of Sor m. Often the values m=2
or m=3 are adopted since they allow direct visualization. On the other hand, the
Shepard diagram compares dij and cij for a particular value m. A narrow scatter
around the 45-degree line represents a good fit between dij and cij.
3 Modeling Vegetable Tissues and Edible Drinks
3.1 EIS Analysis of Vegetable Tissues
A total of Nl=6 angiosperm leaves and Nv=4 vegetables are studied, as summa-
rized in Tables1and 2, respectively [22,23].
Each leaf (vegetable) is submerged in salted water, with except ion of its peti-
ole (base). Two copper electrodes of 0.5mm diameter connect the specimen to the
measurement circuit. One electrode is inserted into the leaf petiole (vegetable base),
aligned with its longitudinal axis, and the other one is placed in the water (see Fig.1,
setup A).
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 25
10210310 4105
|Z(jω)|dB
80
85
90
95
100
Ilex aquifolium
Ze(jω)
Zm(jω)
ω
10210 310 410 5
arg[Z(jω)]
-40
-30
-20
-10
0
Fig. 2 The Bode diagram of the experimental, Ze, and model, Zmimpedance of the Ilex aquifolium
(IA)
For the leaves, several numerical tests proved that the 4-parameter FO model
Zm(jω) =R+K
1+jω
pβ(15)
leads to a good approximation to the experimental data (see details in [22]). Figure2
depicts the Bode diagram of the experimental, Ze, and model, Zmimpedance of the
Ilex aquifolium (IA), illustrating the fit. Table1summarizes the values of the model
parameters that approximate the spectra of all leaves.
For the vegetables, the following 5-parameter FO model is needed to fit the exper-
imental data:
Zm(jω) =R+K
1+jω
pαβ.(16)
An example of the experimental and model Bode diagram is depicted in Fig.3
for the Cauliflower. The model parameters of the four specimens are summarized in
Table2(see details in [23]).
goyal.praveen2011@gmail.com
26 J. A. T. Machado and A. M. Lopes
Table 1 Parameters of the FO-based model for Nl=6leaves
iSpecies Tag R K p αJ
1Citrus limon CL 8.9 ×1035.6 ×1042×1030.59 0.035
2Ilex aquifolium IA 7.2 ×1038.4 ×1042.5 ×1030.72 0.288
3Ficus elastica FE 7.5 ×1037.8 ×1046×1020.55 0.338
4Hydrangea macrophylla HM 2×1036.6 ×1045×1020.48 0.493
5Acacia dealbata AD 7×1034.9 ×1051.2 ×1010.37 0.398
6Acer pseudoplatanus AC 1.5 ×1045.5 ×1051×1020.47 0.581
10 210 310 410 5
|Z(jω)|dB
80
90
100
110
120
Cauliflower
Ze(jω)
Zm(jω)
ω
10 210 310 410 5
arg[Z(jω)]
-40
-30
-20
-10
Fig. 3 The Bode diagram of the experimental, Ze, and model, Zmimpedances for the Cauliflower
Table 2 Parameters of the FO-based model for the Nv=4 vegetables
iDesignation Tag R K p α β J
1Cauliflower CF 1.8×1044.8×1055.5×1010.470 1.046 0.0026
2Broccoli BR 0.1×1044.0×1055.5×1010.859 0.897 0.0027
3Round cabbage RC 5.2×1040.6×1051.7×1030.444 1.470 0.0002
4Brussels sprout BS 5.9×1047.2×1051.7×1030.354 1.292 0.0008
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 27
Table 3 The set of Nw=16 wine samples analyzed
iTag Wine region Wine style
1W1Alentejo White
2W2Alentejo White
3W3Alentejo White
4W4Península de Setúbal White
5W5Tej o White
6W6Douro White
7W7Vinhos Verdes Green white
8W8Vinhos Verdes Green white
9R1Alentejo Red
10 R2Alentejo Red
11 R3Península de Setúbal Red
12 R4Península de Setúbal Red
13 R5Bairrada Red
14 R6Douro Red
15 R7Vinhos Verdes Green red
16 R8Vinhos Verdes Green red
These results demonstrate that FO empirical formulae constitute simple, yet reli-
able models to characterize vegetable structures.
3.2 EIS Analysis of Edible Drinks
In this Section, Nw=16 wine types and Nm=12 UHT milk varieties are studied.
The wine set includes samples from distinct Portuguese regions [11], and involves a
mix of ripe and green, both red and white, styles (Table3). The milk set comprises
samples from distinct brands, with different fat contents, and includes a mix of
normal, reduced, and fortified milk varieties (Table4).
The experimental setup for EIS measurements is depicted in setup B of Fig.1.
A parallelepipedic container with dimensions (l×w×h)=(120 ×100 ×55)mm
is filled with 200 ml of wine. Two 0.5mm diameter copper electrodes connect the
samples to the measurement circuit. The electrodes are immersed at 5 mm from the
bottom of the container, and are placed diametrically opposed to each other.
For both wine and milk, several numerical tests revealed that a good fit between
the experimental, Ze, and model, Zm, impedance occurs for the 6-parameter FO
model:
Z(jω) =K·1+jω
z1α1·1+jω
z2α2
(jω)β.(17)
goyal.praveen2011@gmail.com
28 J. A. T. Machado and A. M. Lopes
Table 4 The set of Nm=12 milk samples analyzed
iTag Milk type
1M1Skimmed
2M2Skimmed
3M3Skimmed
4M4Semi-skimmed
5M5Semi-skimmed
6M6Semi-skimmed
7M7Whole
8M8Whole
9M9Whole
10 M10 Organic semi-skimmed
11 M11 Reduced skimmed
12 M12 Fortified skimmed
Table 5 Impedance parameters of the Nw=16 wine samples
iTag Impedance parameters
K z1α1z2α2βJ
1W16700 1000 0.33 24 ×1040.88 0.29 0.2298
2W25000 1000 0.32 19 ×1040.87 0.32 0.2372
3W35800 1300 0.40 22 ×1040.88 0.31 0.2678
4W47000 1000 0.35 22 ×1040.86 0.31 0.2639
5W55500 1000 0.32 19 ×1040.87 0.33 0.2885
6W66000 900 0.32 20 ×1040.88 0.29 0.2888
7W77500 1600 0.40 15 ×1040.75 0.36 0.2667
8W87500 1600 0.40 17 ×1040.75 0.36 0.2339
9R15500 950 0.33 19 ×1040.87 0.32 0.2859
10 R26800 1100 0.33 23 ×1040.92 0.33 0.2936
11 R35000 1000 0.33 23 ×1040.86 0.31 0.2731
12 R46000 1000 0.34 22 ×1040.88 0.33 0.2941
13 R57500 1600 0.39 15 ×1040.75 0.35 0.2644
14 R65000 1000 0.32 20 ×1040.88 0.32 0.2977
15 R76500 1100 0.30 20 ×1040.89 0.35 0.3651
16 R87200 1700 0.41 19 ×1040.88 0.38 0.2985
Figures 4and 5depict the Bode diagrams of Zeand Zmfor the wine and milk sam-
ples W5and M2, respectively, illustrating the adequacy of expression (17). Tables5
and 6summarize the impedance parameters for all wine and milk specimens (see
details in [24,26]).
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 29
10210 310 410 5
|Z(jω)|dB
50
55
60
65
Wine sample W5
Ze(jω)
Zm(jω)
ω
10210 310 410 5
arg[Z(jω)]
-50
0
50
100
Fig. 4 The Bode diagram of the experimental, Ze, and model, Zm, impedances for the wine sample
W5
Table 6 Impedance parameters for the Nm=12 milk samples
iTag Impedance parameters
K z1α1z2α2βJ
1M133325 5340 0.68 5606 0.57 0.7 0.735
2M236563 8840 0.67 8233 0.65 0.7 0.693
3M345133 3844 0.72 15813 0.64 0.72 0.71
4M429400 9880 0.67 8906 0.59 0.67 0.726
5M535150 4290 0.72 8190 0.56 0.73 0.772
6M634380 9172 0.76 7717 0.54 0.68 0.69
7M759463 2600 0.72 16225 0.68 0.8 0.691
8M835750 7150 0.66 8450 0.61 0.69 0.763
9M931859 4572 0.68 5999 0.62 0.76 0.71
10 M10 32626 9667 0.76 6656 0.57 0.72 0.72
11 M11 27300 3500 0.72 65000 0.53 0.64 0.717
12 M12 34613 3500 0.85 65000 0.53 0.72 0.747
goyal.praveen2011@gmail.com
30 J. A. T. Machado and A. M. Lopes
10210 310 410 5
|Z(jω)|dB
30
40
50
60
70
80
Milk sample M2
Ze(jω)
Zm(jω)
ω
10210 310 410 5
arg[Z(jω)]
-100
-50
0
50
100
Fig. 5 The Bode diagram of the experimental, Ze, and model, Zm, impedances for the milk sample
M2
The results demonstrate that FO models yield a convincing description and reliable
characterization of the samples, and that the EIS technique leads to a simple and
straightforward procedure to characterize the specimen.
In conclusion, analyzing the results of Sects. 3.1 and 3.2 we verify the emergence
of FO effects that are not captured by classical integer-order models.
4 Clustering and Visualization of Vegetable Tissues
and Edible Drinks
4.1 HC of Vegetable Tissues and Edible Drinks
The HC processes a matrix C=[cij]based on the distance:
cij =1
L
L
k=1[Zei(jωk)]−[Zej(jωk)]
[Zei(jωk)]+[Zej(jωk)]+[Zei(jωk)]−[Zej(jωk)]
[Zei(jωk)]+[Zej(jωk)]
,
(18)
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 31
W3
W8
W6
W7
W12
W14
W5
W11
W9
W15
W16
W10
W13
W1
W2
W4
M1
M6
M10
M3
M5
M4
M2
M8
M7
M9
M11
M12
CL
FE
IA
RC
AD
BR
AC
CF
BS
HM
Fig. 6 Dendrogram generated by the HC for the s=38 samples and matrix C
where the indices i,j=1,...,s, so that sdenotes the number of samples, and Ze
represents the impedances measured by means of EIS.
One must note that trying other alternative measures is a common procedure.
In fact, one can choose distinct distances, and the corresponding charts, to obtain
the best visualization. Nevertheless, several tests demonstrated that the Canberra
distance leads to relevant results.
The successive (agglomerative) clustering and average-linkage method are used.
Figure6depicts the dendrogram generated by the HC, with input C=[cij]and s=
Nl+Nv+Nw+Nm=38. One can note the emergence of patterns for vegetables,
wine, and milk.
4.2 MDS of Vegetable Tissues and Edible Drinks
Figure7depicts the 2- and 3-dimensional maps of items obtained by the MDS, with
input C=[cij]and s=Nl+Nv+Nw+Nm=38, where three clusters composed
of vegetables, wine, and milk emerge. Moreover, the s=Nl+Nv=10 vegetable
tissues and s=Nw+Nm=28 edible drinks are compared apart from each other,
and the corresponding 3-dimensional MDS maps are depicted in Fig.8. The charts
reveal blurred and clear clusters, respectively, confirming that leaves and vegetables
are quite similar, while milk and wine have strong dissimilarities.
goyal.praveen2011@gmail.com
32 J. A. T. Machado and A. M. Lopes
Fig. 7 The 2- and
3-dimensional MDS maps
for the s=38 samples and
matrix C
x - component
-0.5 0 0.5 1 1.5
y - component
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
CL IA
FE
HM
AD
AC
CF
BR
RC
BS
Milk
Wine
Leaves
Vegetables
Wine
Milk
M1 - M12
W1 - W16
Vegetables
Leaves
(a)
1.5
x - component
1
AD
BR
0.5
CF
FE
IA
AC
HM
CL
BS
RC
0
-0.5
-1
W1 - W16
M1 - M12
0
y - component
-1
0
0.5
1
-0.5
1
z - component
Milk
Wine
Leaves
Vegetables
Vegetables
Leaves
Milk
Wine
(b)
In conclusion, the dendrogram and MDS charts are alternatives with different
characteristics, both leading to identical clusters, namely with a clear separation
between the 3 groups, but some mixing between leaves and vegetables. Nevertheless,
from the point of view of visualization, the 3-dimensional MDS is superior to the
dendrogram technique.
5 Conclusions
In this chapter, the EIS technique was used to determine the electrical impedance
spectra of different materials, and FO models were adopted to describe the experi-
mental data. It was shown that FO transfer functions describe adequately the data.
The potential use of simple, nonintrusive, and economical techniques in food pro-
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 33
Fig. 8 The 3-dimensional
MDS maps for the s=10
vegetable tissues and s=28
edible drinks with matrix C
y - component
2
0
BS
AC
-2
CF
1
BR
AD
0.5
RC
x - component
IA
FE
HM
0
CL
-0.5
-1
-0.4
-0.2
0
0.2
0.4
0.6
z - component
Leaves
Vegetables
(a)
1
0.5
x - component
M8
M1
M12
M9
M7
M2
M11
M4
M5
M6
0
M10
M3
W12
W16
-0.5
W14
W15
W5
W10
W13
W8
W9
W3
W7
W11
W6
W4
W1
W2
-1
-0.5
y - component
0
0
0.2
0.4
-0.2
0.5
z - component
Milk
Wine
(b)
duction, biology, and medicine reveals possible directions to be further explored [27,
32,38].
References
1. Åberg, P., Birgersson, U., Elsner, P., Mohr, P., Ollmar, S.: Electrical impedance spectroscopy
and the diagnostic accuracy for malignant melanoma. Exp. Dermatol. 20(8), 648–652 (2011)
2. Adachi, M., Sakamoto, M., Jiu, J., Ogata, Y., Isoda, S.: Determination of parameters of electron
transport in dye-sensitized solar cells using electrochemical impedance spectroscopy. J. Phys.
Chem. B 110(28), 13872–13880 (2006)
3. Aggarwal, C.C., Hinneburg, A., Keim, D.A.: On the Surprising Behavior of Distance Metrics
in High Dimensional Space. Springer (2001)
4. Ando, Y., Maeda, Y., Mizutani, K., Wakatsuki, N., Hagiwara, S., Nabetani, H.: Effect of air-
dehydration pretreatment before freezing on the electrical impedance characteristics and texture
of carrots. J. Food Eng. 169, 114–121 (2016)
goyal.praveen2011@gmail.com
34 J. A. T. Machado and A. M. Lopes
5. Biswas, K., Bohannan, G., Caponetto, R., Lopes, A., Machado, T.: Fractional Order Devices.
Springer (2017)
6. Cha, S.H.: Comprehensive survey on distance/similarity measures between probability density
functions. City 1(2), 1 (2007)
7. Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics I. Alternating current charac-
teristics. J. Chem. Phys. 9(4), 341–351 (1941)
8. Davidson, D., Cole, R.: Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J.
Chem. Phys. 19(12), 1484–1490 (1951)
9. Debye, P.J.W.: Polar Molecules. Chemical Catalog Company, Incorporated (1929)
10. Emmert, S., Wolf, M., Gulich, R., Krohns, S., Kastner, S., Lunkenheimer, P., Loidl, A.: Elec-
trode polarization effects in broadband dielectric spectroscopy. Eur. Phys. J. B 83(2), 157–165
(2011)
11. Fraga, H., Malheiro, A., Moutinho-Pereira, J., Jones, G., Alves, F., Pinto, J.G., Santos, J.: Very
high resolution bioclimatic zoning of Portuguese wine regions: present and future scenarios.
Reg. Environ. Change 14(1), 295–306 (2014)
12. Freeborn, T.J.: A survey of fractional-order circuit models for biology and biomedicine. IEEE
J. Emerg. Sel. Topics Circut. Syst. 3(3), 416–424 (2013)
13. Freeborn, T.J., Elwakil, A.S., Maundy, B.: Compact wide frequency range fractional-order
models of human body impedance against contact currents. Math. Problems Eng. 2016 (2016)
14. Glatthaar, M., Riede, M., Keegan, N., Sylvester-Hvid, K., Zimmermann, B., Niggemann, M.,
Hinsch, A., Gombert, A.: Efficiency limiting factors of organic bulk heterojunction solar cellsi-
dentified by electrical impedance spectroscopy. Solar Energy Mater. Solar Cells 91(5), 390–393
(2007)
15. Hartigan, J.A.: Clustering Algorithms. John Wiley & Sons, Inc. (1975)
16. Havriliak, S., Negami, S.: A complex plane analysis of α-dispersions in some polymer systems.
J. Polymer Sci. Part C: Polymer Symp. 14, 99–117. Wiley Online Library (1966)
17. Ionescu, C., Lopes, A., Copot, D., Machado, J., Bates, J.: The role of fractional calculus in
modelling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. (2017)
18. Irvine, J.T., Sinclair, D.C., West, A.R.: Electroceramics: characterization by impedance spec-
troscopy. Adv. Mater. 2(3), 132–138 (1990)
19. Lance, G.N., Williams, W.T.: Computer programs for hierarchical polythetic classification
(“similarity analyses”). Comput. J. 9(1), 60–64 (1966)
20. Lance, G.N., Williams, W.T.: Mixed-data classificatory programs I—agglomerative systems.
Aust. Comput. J. 1(1), 15–20 (1967)
21. Laufer, S., Ivorra, A., Reuter, V.E., Rubinsky, B., Solomon, S.B.: Electrical impedance charac-
terization of normal and cancerous human hepatic tissue. Physiol. Measur. 31(7), 995 (2010)
22. Lopes, A.M., Machado, J.T.: Fractional order models of leaves. J. Vib. Control 20(7), 998–1008
(2014)
23. Lopes, A.M., Machado, J.T.: Modeling vegetable fractals by means of fractional-order equa-
tions. J. Vib. Control 22(8), 2100–2108 (2016)
24. Lopes, A.M., Machado, J.T., Ramalho, E.: On the fractional-order modeling of wine. Eur. Food
Res. Technol. 6(243), 921–929 (2016)
25. Lopes, A.M., Machado, J.T., Ramalho, E.: Fractional-order model of wine. In: Chaotic, Frac-
tional, and Complex Dynamics: New Insights and Perspectives, pp. 191–203. Springer (2018)
26. Lopes, A.M., Machado, J.T., Ramalho, E., Silva, V.: Milk characterization using electrical
impedance spectroscopy and fractional models. Food Analyt. Methods 11(3), 901–912 (2018)
27. Martinsen, O.G., Grimnes, S., Schwan, H.P.: Interface phenomena and dielectric properties of
biological tissue. Encycl. Surface Colloid Sci. 20, 2643–2653 (2002)
28. Maundy, B., Elwakil, A., Allagui, A.: Extracting the parameters of the single-dispersion Cole
bioimpedance model using a magnitude-only method. Comput. Electron. Agric. 119, 153–157
(2015)
29. Nigmatullin, R., Nelson, S.: Recognition of the “fractional” kinetics in complex systems:
dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz. Signal Process.
86(10), 2744–2759 (2006)
goyal.praveen2011@gmail.com
On Fractional-Order Characteristics of Vegetable Tissues and Edible Drinks 35
30. Riul, A., de Sousa, H.C., Malmegrim, R.R., dos Santos, D.S., Carvalho, A.C., Fonseca, F.J.,
Oliveira, O.N., Mattoso, L.H.: Wine classification by taste sensors made from ultra-thin films
and using neural networks. Sens. Actuat. B: Chem. 98(1), 77–82 (2004)
31. Saeed, N., Nam, H., Haq, M.I.U., Muhammad Saqib, D.B.: A survey on multidimensional
scaling. ACM Comput. Surv. (CSUR) 51(3), 47 (2018)
32. Sammer, M., Laarhoven, B., Mejias, E., Yntema, D., Fuchs, E.C., Holler, G., Brasseur, G.,
Lankmayr, E.: Biomass measurement of living lumbriculus variegatus with impedance spec-
troscopy. J. Electr. Bioimpedance 5(1), 92–98 (2014)
33. Sokal, R.R., Rohlf, F.J.: The comparison of dendrograms by objective methods. Taxon, pp.
33–40 (1962)
34. Stanislavsky, A., Weron, K., Trzmiel, J.: Subordination model of anomalous diffusion leading
to the two-power-law relaxation responses. EPL (Europhys. Lett.) 91(4), 40003 (2010)
35. Vosika, Z., Lazarevic, M., Simic-Krstic, J., Koruga, D.: Modeling of bioimpedance for human
skin based on fractional distributed-order modified Cole model. FME Trans. 42(1), 74–81
(2014)
36. Watanabe, T., Orikasa, T., Shono, H., Koide, S., Ando, Y., Shiina, T., Tagawa, A.: The influence
of inhibit avoid water defect responses by heat pretreatment on hot air drying rate of spinach.
J. Food Eng. 168, 113–118 (2016)
37. West, A.R., Sinclair, D.C., Hirose, N.: Characterization of electrical materials, especially fer-
roelectrics, by impedance spectroscopy. J. Electroceramics 1(1), 65–71 (1997)
38. Wolf, M., Gulich, R., Lunkenheimer, P., Loidl, A.: Broadband dielectric spectroscopyon human
blood. Biochimica et Biophysica Acta (BBA)-General Subjects 1810(8), 727–740 (2011)
39. Zheng, S., Fang, Q., Cosic, I.: An investigation on dielectric properties of major constituents
of grape must using electrochemical impedance spectroscopy. Eur. Food Res. Technol. 229(6),
887–897 (2009)
goyal.praveen2011@gmail.com
Some Relations Between Bounded
Below Elliptic Operators and Stochastic
Analysis
Rémi Léandre
Abstract We apply Malliavin Calculus tools to the case of a bounded below elliptic
right-invariant pseudo-differential operators on a Lie group. We give examples of
bounded below pseudo-differential elliptic operators on Rdby using the theory of
Poisson process and the Garding inequality. In the two cases, there are no stochastic
processes because the considered semi-groups do not preserve positivity.
Keywords Malliavin calculus ·Pseudo-differential operators ·Generalized
Poisson processes ·Garding inequality
1 Introduction
Let Gbe a compact connected Lie group, with generic element gendowed with its
biinvariant Riemannian structure and with its normalized Haar measure dg.eis the
unit element of G.
Let fibe a basis of TeG. We can consider as right-invariant vector fields. This
means that if we consider the action Rg0h→(g→h(gg0)) on smooth function h
on G,wehave
Rg0(fih)=fi(Rg0h)(1)
We consider a right-invariant elliptic pseudo-differential bounded below operator
Lof order larger than 2kon G. It generates by elliptic theory a semi-group Pton
L2(dg)and even on Cb(G)the space of continuous functions on Gendowed with
the uniform norm.
R. Léandre (B
)
Laboratoire de Mathématiques, Université de Bourgogne-Franche-Comté, 25030 Besançon,
France
e-mail: remi.leandre@univ-fcomte.fr
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_3
37
goyal.praveen2011@gmail.com
38 R. Léandre
Theorem 1 If t >0,
Pth(g0)=G
pt(g0,g)h(g)dg(2)
where g→pt(g0,g)is smooth if h is continuous.
This theorem is classical in analysis, but it enters in our general program to
implement stochastic analysis tools in the theory of non-Markovian semi-group. See
the review [7,13] for that. See [10,11] for another presentation where the Malliavin
Matrix plays a key role. Here we don’t use the Malliavin matrix. See [12]forthe
case of right-invariant differential operators.
Jump processes are generated by pseudo-differential operators, which satisfy the
maximum principle. Unlike the Malliavin Calculus for jump processes [1,5,6],
there is no limitation here on the size of jumps.
This theorem can be applied to any positive power of a right invariant strictly
positive differential operator on G[14].
2 Pseudo-differential Operators
Let us recall what is a pseudo-differential operator on Rd[3,5,6,15]. Let be a
smooth function from Rd×Rdinto Ca(x,ξ). We suppose that for all x
|Dr
xDl
ξa(x,ξ)|≤C|ξ|m−l+C(3)
We suppose that for all x
|a(x,ξ)|≥C|ξ|m(4)
for |ξ|>Cfor a suitable m>0. Let ˆ
hbe the Fourier transform of the continuous
function h. We consider the operator Ldefines on smooth function hby
ˆ
Lh(x)=a(x,ξ)ˆ
h(ξ)dξ(5)
Lis said to be a pseudo-differential operator elliptic of order larger than mwith
symbol a. This property is invariant if we do a diffeomorphism on Rdwith bounded
derivatives at each order. This remark allows to define by using charts a pseudo-
differential operator elliptic of order larger than mon a compact manifold M.
On a compact Riemannian manifold, we can consider the Riemannian measure.
In local coordinates, the Riemannian metric is given by a smooth map
x→gi,j(x)(6)
in the set of strictly positive matrix and the Riemannian measure is given by
goyal.praveen2011@gmail.com
Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis 39
dx =det(g.,.)−1/2dx1⊗.. ⊗dxd(7)
We can normalize the Riemannian measure to be of total mass 1.
The fact that Lis symmetric on L2(M)means that
M
<h1(x), Lh2(x)>dx =M
<Lh1(x), h2(x)>dx (8)
The fact that Lis bounded below means that for some C>0:
M
<h(x), Lh(x)>dx ≥−CM
<h(x), h(x)>dx (9)
In such a case Lhas a self-adjoint extension. This generates a semi-group of bounded
operators Pton L2(M)satisfying the heat equation
∂
∂tPth=−LP
th(10)
for h∈L2(M)and t>0. Moreover, we suppose that P0h=h. It generates moreover
a semi-group on Cb(M)by ellipticity.
An example can be given on Rdif we use the Garding inequality [15]. Suppose
that we consider the Lebesgue measure on Rdand that for |ξ|>C0we have
Re(a(x,ξ)) > C|ξ|m(11)
for some C>0. In such a case if we suppose Lsymmetric, it is bounded below.
3 Proof of the Theorem
3.1 Algebraic Scheme of the Proof: Malliavin Integration
by Parts
We consider the family of operators on C∞(G×Rn):
˜
Ln
t=L+
n
i=1
fji∂
∂ui
αi
t+
n
i=1
∂2k
∂u2k
i
(12)
αi
tare smooth function from R+into R. By elliptic theory, ˜
Ln
tgenerates a semi-group
˜
Pn
ton Cb(G×Rn). This semi-group is time inhomogeneous.
goyal.praveen2011@gmail.com
40 R. Léandre
˜
Pn+1
t[h(g)hn(u)v](., ., 0)=t
0˜
Pn
t,s[fj+1αn+1
s˜
Pn
s[h(g)hn(u)]](., .) (13)
Moreover
˜
Pn+1
t[uh(.)hn(.)](., ., un+1)=˜
Pn+1
t[uh(.)hn(.)](., ., 0)+˜
Pn
t[h(.)hn(.)](., .)un+1
(14)
his a function of g,hna function of u1, ..., un. This comes from the fact that ∂
∂un+1
commutes with ˜
Ln+1
t.
Therefore, the two sides of (13) satisfy the same parabolic equation with second
member. We deduce that
˜
Pn+1
t[un+1
n
j=1
ujh(.)](., ., 0)=t
0
ds ˜
Pn
t,s[fjn+1αn+1
s˜
Pn
s[h
n
j=1
uj]](., .) (15)
This is an integration by parts formula. We would like to present this formula in a
more appropriate way for our object.
We consider the operator
Ln=L+
n
j=1
∂2k
∂u2k
j
(16)
It generates a semi-group Pn
t. In the sequel, we will skip the problem of sign coming
if kis even or not. Since n
j=1ujis a polynomial, the Volterra expansion associated
to ˜
Ps[hn
j=1uj]is finite and converge. We get
˜
Ps[h
n
j=1
uj](., .) =(−1)ls>s1>..>sl>0
Il
s1,..,slds1..dsl(17)
where
Il
s1,..,sl=Pn
s−s1[
n
i=1
fjiαi
s1
∂
∂ui[Pn
s1−s2[
n
i=1
fjiαi
s2
∂
∂ui[Pn
s3−s2[[
n
i=1
fjiαi
s2
∂
∂ui[...[Pn
sl[h
n
j=1
uj]..](., .) (18)
Moreover
Pn
s[h
n
j=1
uj](g0,.) =Pn
s[h(.g0)
n
j=1
uj](e,.) (19)
goyal.praveen2011@gmail.com
Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis 41
such that
fijPn
s[h
n
j=1
uj](g0,.) =
Pn
s[fijh(.g0)
n
j=1
uj](e,.) =Pn
s[fijh(.)
n
j=1
uj](g0,.) (20)
We remark that in (17) the series is finite and stops at nbecause we consider a
polynomial in viand because ∂
∂uicommute with Pt. If we consider Pt(h1(g)h2(v))
it is a product of the Pt(h1)Qt(h2(v)), where Qtis generated by n
j=1
∂2k
∂u2k
j
.We
deduce that in the term of the Volterra expansion of length lsmaller than n, we get
(Pt−s(flh(g))Qt−s(h1(v) where h1(v) is an homogeneous polynomial with coeffi-
cient independent of gof degree n−l.
We do the following recursion hypothesis on l:
Hypothesis 1 There exists a positive real rlsuch that if (α)=(i(α), .., i(α)). is a
multi-index of length smaller than l constituted of |(α)|with the same element
|Pt[f(α)h
n
i=n
ui](g,v
.)|≤Ct−rlh∞(1+
n
i=n|vi|)(21)
where .∞is the uniform norm of h.
It is true for l=1by(13) and the next part.
If it is true for l, it is still true for l+1, by using (15) and the Volterra expansion
above for f(α)hand taking αn+1
s=srl.
By choosing suitable αj
t, we have accordingly the framework of the Malliavin
Calculus for any basis of the Lie algebra fi, for any l
|Pt[
i
(fi)lh](g0)|≤C(α)h∞(22)
in order to conclude, because the operator i(fi)lis an elliptic operator whose
degree tends to infinity when l→∞.
3.2 Estimates: the Davies Gauge Transform
We do as in [12](16). The problem is that in ˜
Pn
t[hn
j=1uj](., .) the test function uj
is not bounded and that ˜
Pn
tacts only on Cb(G×Rn).Wedoasin[12] the Davies
gauge transform n
Ig(ui)where
goyal.praveen2011@gmail.com
42 R. Léandre
g(u)=(|u|)(23)
if uis big and gis smooth strictly positive.
This gauge transform acts on the original operator by the simple formula
(n
i=1g(ui))−1˜
Ln
1((n
i=1g(ui).). On the semi-group it acts as
(
n
i=1
g(.))−1˜
Pn
t[(
n
i=1
g(ui)h(.)hn(.)](., .) (24)
But
(g(ui))−1∂
∂ui
(g(ui).) =∂
∂ui+C(ui)(25)
where the potential C(ui)is smooth with bounded derivatives at each order. There-
fore, the transformed semi-group act on Cb(G×Rn). It remains to choose
hn(u.)=
n
j=1
uj
g(uj)(26)
in order to conclude. We deduce the bound:
|˜
Pn
t|[h
n
j=1|uj|](.;v.)≤C(h∞(1+
n
i=n|vi|)(27)
where |˜
Pn
t|is the absolute value of the semi-group ˜
Pn
t.
4 Study of an Example on the Linear Space
We give in this part a big category of examples on Rdof symmetric bounded below
pseudo-differential operators which takes its origin in the theory of Poisson process
[5,6].
We consider the space C∞(Rd)of smooth functions hwith bounded derivatives
at each order.
We introduce a smooth function from Rd×Rdinto R(x,y)→g(x,y)which
equals to 0 for |y|>C>0forasmallCand with bounded derivatives at each order.
This allows us to introduce the integro-differential operator on C∞(Rd):
Lh(x)=(−1)l+1Rd
(h(x+y)−h(x)
−
2l
i=1
1/i!<y⊗i,h(i)(x))g(x,y)|y|−(2l+d+α)dy (28)
for α∈]−1,0[.
goyal.praveen2011@gmail.com
Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis 43
We do the following hypothesis: for all x∈Rd,h(x,0)>C>0.
In such a case, we have shown [8,9] that Lis a pseudo-differential elliptic operator
with symbol
a(x,ξ)=(−1)l+1Rd
(exp[√−1<y,ξ>]−
2l
i=1
1/i!(√−1<y,ξ>)i)g(x,y)|y|−(2l+d+α)dy (29)
Lis elliptic and satisfies Garding assumption (11) with m→∞when l→∞.We
produce a large class of examples of such operators which are moreover symmetric
in L2(dx).
Let be Xj(x), j=1, .., dbe some vector fields without divergence, with bounded
derivatives of each order and which are uniformly in xin Rda basis of Rd.
Let φt(y)(x)be the dynamical system generated by the vector field X(y,x)=
d
j=1yjXj(x);φ0(y)(x)=xand
dφt(y)(x)=X(y,φt(y))dt (30)
We suppose g(x,y)=g(y)=g(−y). We introduce the operator
L1h(x)=(−1)l+1Rd
(h(φ1(y)(x)) −h(x)−
l
i=1
1/(2i!)(X(y,x))(2i)h(x))g(y)|y|−(2l+d+α)dy (31)
In the previous formula, the vector field X(y,x)is considered as a one-order
differential operator in x.
Lemma 2 Under the symmetry condition on g,L
1is symmetric and is defined on
C∞(Rd).
Proof The fact that L1is defined on C∞(Rd)comes from the fact that the asymptotic
expansion of y→h(φ1(y)(x)) near 0 is
h(x)+
2l
i=1
1/i!X(y,x)(i)h(x)(32)
and from the fact that g(y)=g(−y)such that only even integers remain in the sum
(31).
The fact that L1is symmetric comes from two fact: the vector field X(y,x)is
divergence free such that
goyal.praveen2011@gmail.com
44 R. Léandre
Rd
h1(x)X(y,x)(2i)h2(x)dx =Rd
h2(x)X(y,x)(2i)h1(x)dx (33)
by integrating by parts. Moreover, x→φ1(y)(x)preserves the Lebesgue measure
such that Rd
h1(x)h2(φ1(y)(x))dx =Rd
h1(φ1(−y)(x))h2(x)dx (34)
and the result arises from the equality g(y)=g(−y).♦
Theorem 3 L1is an operator of the type (28) which is symmetric bounded below.
Proof It remains only to show that L1is an operator of the type (28). For that we
remark that the map
y→φ1(y)(x)−x(35)
is a local diffeomorphism at every point yand a local diffeomorphism of a neigh-
borhood of 0 in Rdonto a neighborhood of 0 in Rd.♦
Remark Let us give some heuristic explanation which explains this part. Let us
consider a formal path measure dQ on a “space” of paths ytwith jumps starting
from 0 which represents the semi-group Ptassociated to the operator
Lh(x)=(−1)l+1Rd
(h(x+y)−h(x)−
l
i=1
1/(2i!)<y⊗2i,h(2i)(x)>)g(y)|y|−(2l+d+α)dy (36)
such that formally
Pth(x)=h(yt+x)“dQ(y.)” (37)
We consider the “formal stochastic differential with jumps” whose solution (start-
ing from x)y1,t(x)satisfies
y1,t(x)=φ1((yt))(y1,t−(x)) −y1,t−(x)(38)
where yt−=lims→t−ysand yt=yt−yt−. We should get
P1,th(x)=f(y1,t(x)“dQ(y.)” (39)
Moreover, a lot of compensation should appear in the formal equation giving y1,t.
We refer to [5] in the case where the path integrals are rigorously defined (In such a
case only one compensation appears!).
goyal.praveen2011@gmail.com
Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis 45
References
1. Bismut, J.M.: Calcul des variations stochastiques et processus de sauts. Z. Wahr. Verw. Gebiete
63, 147–235 (1983)
2. Chazarain, J., Piriou, A.: Introduction a la théorie des équations aux dérivées partielles linéaires.
Gauthier-Villars, Paris, France (1981)
3. Hoermander, L.: The Analysis of Linear PartialOperators III. Springer, Berlin, Germany (1984)
4. Hoermander, L.: The Analysis of Linear Partial Operators IV. Springer, Berlin, Germany (1984)
5. Ishikawa, Y.: Stochastic Calculus of variations for jump processes, Basel. de Gruyter, Schweiz
(2012)
6. Léandre, R.: Extension du théoreme de Hoermander a divers processus de sauts. Université de
Besançon, France (1984). PHD Thesis
7. Léandre, R.: Stochastic analysis for a non-markovian generator: an introduction. Russ. J. Math.
Phys. 22, 39–52 (2015)
8. Léandre, R.: Large deviation estimates for a non-Markovian generator of Lévy type of big
order. 4th Int. Conf. Math. Modern. Phys. Sciences, J. Phys.: conf. Ser. 633, 012085, 2015 (E.
Vagenas and al esds)
9. Léandre, R.: A Class of non-Markovian pseudo-differential operators of lévy type”. Pseudo-
differential operators: groups, geometry and applications. Birkhauser, 149–159 (M.W. Wong
and al eds) (2017)
10. Léandre, R.: Perturbation of the Malliavin Calculus of Bismut type of large order. In: Gazeau,
J.P. (ed. Physical and Mathematical Aspects of Symmetries, pp. 221–225. Springer (2017)
11. Léandre, R.: Malliavin Calculus of Bismut type for an operator of order four on a Lie group.
J. Pseudo-differential Oper. Appl. 8, 419–430 (2019)
12. Léandre, R.: Bismut’s way of the Malliavin Calculus of large order generators on a Lie group.
In: Tosun, M. (ed.) 6th International European Conference Mathematical Sciences and Appli-
cations 1926, 020026 A.I.P. Proceedings (2018)
13. Léandre, R.: Bismut’s way of the Malliavin Calculus for non markovian semi-groups: an
introduction. to appear in Analysis of pseudo-differential operators. M.W. Wong and al eds
14. Seeley, R.T.: Complex powers of an elliptic operator. In: Singular Integrals Proceedings Sym-
posia in Pure Mathematics. Providence, U.S.A. A.M.S., pp. 288–307 (1966)
15. Taylor, M.: Partial Differential Equations II. Springer, Qualitative Studies of Linear Equations.
Heidelberg, Germany (1997)
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants:
How to Differentiate the Pattern
Sequences from the Tested Ones?
Raoul R. Nigmatullin and Artem S. Vorobev
Abstract Based on the new method (defined below as the discrete geometrical
invariants—DGI(s)), one can show that it enables to find the statistical differences
between random sequences that can be presented in the form of 2D curves. We gen-
eralized and considered the Weierstrass–Mandelbrot function and found the desired
invariant of the fourth order that connects the WM-functions with different fractal
dimensions. Besides, we consider an example based on real experimental data. A
high correlation of the statistically significant parameters of the DGI obtained from
the measured data (associated with reflection optical spectra of olive oil) with the
sample temperature is shown. This new methodology opens wide practical appli-
cations in differentiation of the hidden interconnections between measured by the
environment and external factors.
Keywords Weerstrass–Mandelbrot function ·Discrete geometrical invariants ·
Equipment calibration ·Nano-noise “reading”
2010 AMS Math. Subject Classification. Primary 40A05, 40A25; Secondary
45G05.
1 Introduction and Formulation of the Problem
If we follow for the modern tendencies in the applied sciences, one can notice that
the efforts of many types of research were concentrated presumably on the analysis
of complex systems. It implies the usage of methodology of many natural sciences as
physics, chemistry, biology, economy, and improvement of the mathematical meth-
ods that should be more general and enables to describe the hierarchy of interactions,
R. R. Nigmatullin (B
)·A. S. Vorobev
Radioelectronics and Informative-Measurements Techniques Department, Kazan National
Research Technical University (KNRTU-KAI), K. Marx str. 10, 420111 Kazan, Tatarstan,
Russian Federation
e-mail: renigmat@gmail.com
A. S. Vorobev
e-mail: vartems14@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_4
47
goyal.praveen2011@gmail.com
48 R. R. Nigmatullin and A. S. Vorobev
intermittency between the structural organization levels of the considered complex
systems. One of the main obstacles is the “invisible” boundary determination that
divides the chaotic and deterministic behaviors of the complex systems. For better
understanding of its behavior, a potential researcher needs to increase the determin-
istic part and decrease the part related to its chaotic and unpredictable behavior. If it
is possible to solve this task then as a “reward” a researcher pulls apart the forecast-
ing boundaries for prediction of the complex system behavior in the time evolution
process. Many new features came from the fractal geometry with appearance of the
B. Mandelbrot book [1] and its successful interpretation [2] that helps to consider
and describe mathematically many new complex systems with self-similar/fractal
geometry. This specific understanding helped to attract the mathematical tool as the
fractional calculus [3,4] and find for it its proper place in many practical applications.
Now this powerful combination of the fractal geometry with the fractional calculus
gives a new impact in the development of many natural sciences unifying them in
one perfect instrument for knowledge and establishing new relationships that exist
in the Mother Nature.
In this paper, we want to attract the attention of experts and many researches
working in this “hot” spot as the fractal geometry and fractional calculus to “a dis-
covery” made by Prof. Yu. I. Babenko in his books [5,6]. Actually, he was able to
generalize the well-known Pythagoras theorem and find new mathematical relation-
ships between the lengths of many symmetrical sets/polyhedrons located in 2D and
3D spaces. After reading this instructive book, one of us (RRN) formulated the fol-
lowing problem: is it possible to find some deterministic mathematical relationships
between random sequences at least in 2D space and apply them for a more detailed
comparison of the measured data?
The obtained results showed that these DGI(s) really exist. Therefore, one can
state that at least any two arbitrary random sets located in 2D space can relate with
each other by means of their inter-correlations and integer moments. This gener-
alization opens quite new possibilities in the reduced identification of different 2D
curves (images) and comparison of various random curves with each other without
the knowledge of a “true” fitting function, which, for many complex systems studied,
it is absent. The preliminary results related to the application of the DGI in electro-
chemistry was published in paper [7]. In this paper, we present the complete invariant
of the 4th order and show its possibilities for comparison of the WM-curves with
different fractal dimensions and in finding of the hidden deterministic relationships
between the measured data (reflectance optical olive oils spectra) and temperature
changes during the experiment.
2 Basic Relationships and Description of the Algorithm
As it was reminded in the first section in the books [5,6], it was shown that the well-
known Pythagoras theorem can be generalized and propagated for a set of random
points having coordinates (xk,yk)(k=1,2, ..., n). Really, one can consider the
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 49
square of the distance connecting an arbitrary point M(x,y)with the kth point
(xk,yk)belonging to the given set:
l2
k=(x−xk)2−(y−yk)2,(2.1)
one requires that
1
n
n
k=1
l2
k=I2≡const .(2.2)
Inserting expression (2.1)into(2.2), one obtains
(x−x)2+(y−y)2=I2−R2,
xp=1
n
n
k=1
xk
p,yp=1
n
n
k=1
yk
p,R2=x2+y2,
V2def
=V2−V2,V=x,y.
(2.3)
As one can notice from (2.3) that the set of circles can exist if the desired invariant
I2≥R2, the equality sign corresponds to the circle with the zeroth radius. It is
convenient to consider the invariant circle with radius I2=2R2. From another point
of view, the requirement (2.2) corresponds to the reduction of the given set of points to
the continuous circle with 4 statistical parameters (xp,yp,p=1,2). However,
for practical purposes, this simplest requirement (2.2)isnot sufficient and therefore,
it has sense to consider other combinations.
2.1 The DGI of the Second Order (General Form)
In order to have reduction to the deterministic curve with sufficient number of sta-
tistical parameters we consider another combination that is a little complicated in
comparison with the definition of the Euclidean distance (2.1):
L2
k=C2(y−yk)2−2B(x−xk)·(y−yk)+A2(x−xk)2,k=1,2, ..., n.(2.4)
The quadratic form (2.4) contains 5 statistical parameters (xp,yp,p=1,2),
xyand 3 unknown parameters (A,B,C)figuring in (2.4). We subject this combi-
nation to the requirement:
1
n
n
k=1
L2
k=I2≡const .(2.5)
goyal.praveen2011@gmail.com
50 R. R. Nigmatullin and A. S. Vorobev
Inserting (2.4)into(2.5) after simple algebraic manipulations one can obtain
C2(y−y)2−2B(y−y)·(x−x)+A2(x−x)2+E2≡I2,
E2=C2y2−2Bxy+A2x2.(2.6)
As before, we put I2=2E2. In order to find three unknown parameters (A,B,C),
it is convenient to use the obvious parameterization for the variables (x,y)relatively
the angle ϕ:
y=y+Acos(ϕ−α),
x=x+Ccos(ϕ), 0≤ϕ≤2π.(2.7)
Excluding the parameter ϕfrom (2.7) and identifying expression (2.6) with rela-
tionship:
C2(y)2−2AC cos α(x)·(y)+A2(x)2=A2C2−B2,
E2=C2y2−2AC cos αxy+A2x2=A2C2sin2α,(2.8)
one obtains
cos α=B
AC ,E2=A2C2−B2.(2.9)
In order to decrease the number of unknown parameters, we find from (2.7)the
values Aand Cfrom the obvious conditions:
ymax =y+A,ymin =y−A,→A=1
2(ymax −ymin),
xmax =x+C,xmin =x−C,→C=1
2(xmax −xmin).
(2.10)
Parameter Bis found from relationships (2.8) and (2.9) as a positive root of the
quadratic equation written relatively B
B2−2xyB−[A2C2−x2A2−y2C2]=0,
B=xy+[(xy)2+A2C2−x2A2−y2C2]1/2.(2.11)
This single root is chosen from the comparison of two identity sequences (xk=yk)
that follows from the obvious requirement B=A2,(α=0). Therefore, one can say
that with the help of the rotated counterclockwise ellipse (2.7) we reduced 2nrandom
points figuring in (2.4) to 8 statistical parameters (xp,yp,p=1,2), xy,α,
A,C). If it is necessary to include the higher moments xpys,(p=0,1,2, ...;
s=0,1,2...) then other combinations of the type (2.4) should be considered.
Some generalizations for the invariant of the fourth order are considered below.
It is easy to notice that the invariant (2.7) of the second order is not sufficient for a
detailed comparison of two random sequences. Expression (2.9) is equivalent to the
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 51
conventional Pearson Correlation Coefficient (PCC) that is related to the correlation
of the second order xy. Actually, we found the geometrical interpretation of
the PCC and showed that the closed curve as an ellipse can be used for pictorial
interpretation of the quadratic correlations between two random sequences. There-
fore, it has a sense to consider the invariants of the higher orders for a more detailed
and reliable comparison of a couple of random sequences with each other. Below,
we want to consider the complete invariant of the fourth order. It is instructive also to
give the basis of the proposed theory that will be useful for quantitative comparison
of any two random sets located in 2D plane.
2.2 The General Theory of the Geometrical Invariants Based
on the Higher Order Curves and the GDI of the Fourth
Order
Unifying the ideas expressed in books [5,6], one can consider the following combi-
nation:
L(m)
k=
m
q=0,p=0
Aq,p(x−xk)q(y−yk)p.(2.12)
This combination can be considered as the most general form that can be used
for comparison of two random sets having coordinates (xk,yk)(k=1,2,3...,n).
If one requires that
1
n
n
k=1
L(m)
k=Inv, (2.13)
then this form can be used for comparison of two random sequences of an arbi-
trary order in terms of different combinations of the integer moments. If we insert
(2.12)into(2.13) and open the corresponding terms then one can obtain possible
combinations of the integer moments of the type:
Mq,p=(x)q(y)p≡1
n
n
k=1
(x−x−xk)q(y−y−yk)p,
A=1
n
n
k=1
Ak,Ak=Ak−A.
(2.14)
In this section, having in mind its practical application for comparison of the
different experimental data with each other we consider the complete invariant of
the fourth order that will be helpful for a more fine comparison of two sets.
goyal.praveen2011@gmail.com
52 R. R. Nigmatullin and A. S. Vorobev
2.3 The Complete Invariant of the Fourth-Order Admitting
the Separation of Variables
A possible combination allowing to express the desired invariant in the analytical
form can be written as
L(4)
k=A40(x−xk)4+A31 (x−xk)3(y−yk)−2A22(x−xk)2(y−yk)2+
+A13(x−xk)( y−yk)3+A04 (y−yk)4.
(2.15)
Inserting expression (2.15)into(2.13) and equating the linear terms relatively the
variables
X≡x−x,Y≡y−y,(2.16)
to zero, we obtain the following combinations:
4A40(x)3+3A31(x)2y+A13(y)3=4A22(y)2x,
4A04(y)3+3A13(y)2x+ A31(x)3=4A22 (x)2y.
(2.17)
In order to decrease the number of the parameters entering in (2.15) we introduce
the following ratios:
A31 =σxA22,A13 =σyA22 ,A40 =θxA22 ,A04 =θyA22.(2.18)
These ratios help to cancel on an arbitrary constant A22 (= 0)and present system
(2.17) in the form:
4θx(x)3+3σx(x)2y+σy(y)3=4(y)2x,
4θy(y)3+σx(x)3+3σy(y)2x=4(x)2y.
(2.19)
In order to find these four unknown parameters, it is necessary to find some
additional relationships between them. One can notice that for identity relationships
(xk,yk)(k=1,2,3...,n), the following relationships from (2.19) are valid:
4θx+3σx+σy=4,4θy+σx+3σy=4
or
θx=1−3
4σx−1
4σy,θy=1−3
4σy−1
4σx.
(2.20)
The systems (2.19), (2.20) allow finding the unknown variables(ratios) and rewrite
them by means of different correlations belonging of two compared sets.
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 53
σx=4
3·x(y)2−(y)3·x(y)2−(x)3+
+(x)3−(y)3·y(x)2−(y)3,
σy=4
3·y(x)2−(x)3·y(x)2−(y)3+
+(y)3−(x)3·x(y)2−(x)3,
=9·y(x)2−(x)3·x(y)2−(y)3+
+(x)3−(y)32
.
(2.21)
Two other unknown parameters θx,yare found from (2.20).
Finally, we obtain the following invariant of the fourth order:
K(X,Y)=K2(X,Y)+K4(X,Y)=I4,
K2(X,Y)=AxX2+B·X·Y+AyY2,
K4(X,Y)=θxX4+θyY4−2X2Y2+σxX3Y+σyXY3.
(2.22)
The following combinations shown below define the constants figuring in the DGI
(2.22):
Ax=6θx(x)2−2(y)2+3σxxy,
Ay=6θy(y)2−2(x)2+3σyxy,
B=−8xy+3σx(x)2+3σy(y)2.
(2.23)
The constant I4from (2.22) is defined by expression:
I4=θx(x)4+θy(y)4+
+σx(x)3(y)+σy(y)3(x)−2(x)2(y)2.
(2.24)
Finally, we obtain the eight parametric curve (2.22), which combines 6 correla-
tions and 8 moments up to the fourth order inclusive:
x,y,(x)2(y)2,x(y)2,y(x)2,
x(y)3,y(x)3,(x)4(y)4,(x)2,3,4,(y)2,3,4.
(2.25)
The curve K(X,Y)in (2.22) can be separated in the polar coordinate system. We
present the desired curve in the form:
goyal.praveen2011@gmail.com
54 R. R. Nigmatullin and A. S. Vorobev
x(ϕ)=x+r(ϕ)cos ϕ,
y(ϕ)=y+r(ϕ)sin ϕ,
r(ϕ)=q2
2(ϕ)+4I4q4(ϕ)−q2(ϕ)
2q4(ϕ)1/2
.
(2.26)
The functions q2,4(ϕ)figuring in (2.26) are determined by expressions:
q2(ϕ)=Axcos2(ϕ)+Bsin ϕcos ϕ+Aysin2(ϕ),
q4(ϕ)=θxcos4(ϕ)−2sin
2(ϕ)cos2(ϕ)+θysin4(ϕ)+σxsin ϕcos3(ϕ)+σysin3(ϕ)cos ϕ.
(2.27)
This curve determines statistical proximity/difference between 2D random
curves/sets located in the plane. What happens if two random curves are identi-
cal to each other (xj=yj)for all numbers of the discrete points j=1,2, ..., N?. In
this case as it can be shown (the details are given in the Mathematical Appendix) that
σx,y=4/3, θx,y=−1/3, Ax=Ay,B=−2Axand I4=0. Hence, from (2.26)it
follows that r(ϕ)=0. In this case, expression (2.22) is degenerated into a point with
coordinates x=ylocated on the line y=x.IntheMathematical Appendix,it
was found the form of the simplified curve (2.22) when two discrete sets xkand yk
(k=1,2, ..., n)are becoming close to each other, i.e., yk=xk±fk(fk=
fk−f)- small factor distorting the set yk).
Concluding this section, one can say that we propose the complete invariant of the
fourth order (2.22), which enables to compare two random sets (sequences) located
on 2D plane. In general, this result shows that two random sequences have at least
the compact deterministic curve of the fourth order (2.22) combining 8 parameters
I4,Ax,y,B,σx,y,θx,y. These parameters, in turn, depend on 14 statistical parameters
(2.25) that help to compare one random set with another one. Therefore, we made a
next step and generalized the conventional Pearson correlation coefficient (2.9) that
is valid only for the correlations of the second order. A potential researcher receives
a new statistical tool for more “fine” analysis and comparison of a couple of random
sets with each other. In subsequent chapters, we want to show how to apply this new
tool for comparison of random sequences of different nature.
Finishing this section, it is necessary to remind about another important possibil-
ity of a new approach that makes this DGI-tool more significant and general. The
previous results (expression (2.15) and below) were obtained for vectors (xk,yk)
(k=1,2,3...,n). Are the previous results conserved if one replaces the vectors
for matrices (Mi,j,Li,j)(i=1,2, ..., I;j=1,2, ..., J)? Attentive analysis of the
results obtained above shows that one can obtain a positive answer. Really, for this
case the mean values for x,yare rewritten in the following form:
x= 1
I·J
I,J
i,j
Mi,j,y= 1
I·J
I,J
i,j
Li,j,(2.28)
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 55
and different moments and inter-correlations keep formally their forms:
Qq,p=(x)q(y)p≡1
I·J
I,J
i,j=1x−x−xi,jqy−y−yi,jp
,
A= 1
I·J
I,J
i,j=1
Ai,j,Ai,j=Ai,j−A.
(2.29)
As it follows from last expression, the subsequent algebraic transformations
remained the same and, therefore the final result (2.26) and its simplified expression
(6.6) keep their structures, as well. This important generalization allows applying the
DGI-tool for analysis of different 2D-images and 3 D-projections, especially in cases
when it is necessary to compare the random trajectories generated by unpredictable
movements of molecules, viruses, and other “small” objects. This new possibility,
undoubtedly, merits the separate research.
3 New “Reading” of the Weierstrass–Mandelbrot Function
As it is known [2], the WM-function is defined by the following relationship:
S(z)=
N
n=−N
bn·F(z·ξn), F(z)=1−cos(z), b=1
ξv,
v=2−D,N1.
(3.1)
The dependencies of these parameters with respect to the chosen parameter D are
shown in Figs. 6,7, and 8. We omit the dependence xc(D)which keeps its constant
value equal to −0.02444 for all values of Dfrom [0.5,2.0]. As one can notice
from (3.1), one can generalize the conventional definition of the WM-function and
propagate it for more wide class of the functions varying the function F(z).Thesum
(3.1) has the obvious property:
S(zξ)=1
b+Up(z)−Dn(z),
Up(z)=bNf(zξN+1), Dn(z)=b−N−1f(zξ−N).
(3.2)
Expression (3.2) satisfies approximately the functional equation (3.3):
S(zξ)∼
=1
bS(z), S(z)=zvPr(ln(z)), v =ln(1/b)
ln(ξ)=2−D,
Pr(ln(z)±ln(ξ)) =Pr(ln(z)),
(3.3)
goyal.praveen2011@gmail.com
56 R. R. Nigmatullin and A. S. Vorobev
Fig. 1 Verification of the
relationships (3.2)forthe
WM-function for D=0.6.
The contribution of the
functions Up(x)and Dn(x)
are negligible and therefore
they are not shown
Fig. 2 Verification of two
log-periodic functions
Pr(x)=Pr(zξ)entering in
expression (3.3)for D=0.6
if contributions of the functions Up(z)and Dn(z)on the ends of the corresponding
intervals are negligible. We want to stress here that expressions (3.3) are more correct
in comparison with expression (2.16) given in the book [2] that was found in the
results of numerical calculations [8]. Now it has a sense to formulate a problem that
can be solved with the help of the DGI approach. Is it possible to relate the parameter
Dfrom (3.3) with parameters (20), (21), (23), and (24) forming the desired curve (26)
and test the relationships (3.2) and (3.3) numerically? For the aim we chose N=60
in (3.1) and select the interval for Das [0.5–2.0]. Then we compare successively the
curve corresponding to the D=0.5 with other curves from the interval [0.6, 2.0]
with step h=0.1. The verifications of relationships (3.1) and (3.2) for the limiting
cases D=0.5 and D=2.0aregiveninFigs.1,2,3and 4, correspondingly.
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 57
Fig. 3 Verification of the relationships (3.2) for the WM-function for D=1.9. The contribution of
the functions Up(x)and Dn(x)are shown inside the small figure on the right. Their contributions
are small
Fig. 4 Verification of two
log-periodic functions
Pr(x)=Pr(zξ)entering in
expression (3.3)for D=1.9
Figure 5shows that the deviation factor (6.6) for all selected ranges of Dthat do
not exceed the unit value. It allows to apply the simplified version of the DGI (6.5)
that contains only 4 parameters (xc,yc,ε/3Q0,B,I4).
Finishing this section, we want to stress one important point. As it has been men-
tioned at the end of the section two, the DGI-tool can be applied successfully for
comparison of different images having random fractal dimensions or their distribu-
tions. It helps to “read” quantitatively two sequences or 2Dprojections: one of them
can be considered as the pattern one and another image can be defined as tested. The
goyal.praveen2011@gmail.com
58 R. R. Nigmatullin and A. S. Vorobev
Fig. 5 This dependence
demonstrates the criterion of
applicability of the simplified
version of the DGI shown in
the Mathematical Appendix.
It is calculated in accordance
with expression (6.6)and
signifies that for all values of
Dfrom the interval [0.6–2.0]
the simplified version of the
DGI from (6.5) is applicable
Fig. 6 The behavior of the
mean value of the compared
curve relative to the
WM-function having
D=0.5
new methodology helps to analyze two images in terms of the reduced number of
parameters (8 or 4) and express this comparison in the form of the DGI curves (2.26)
or (6.5), correspondingly.
This instructive example shows that the DGI can serve an additional source of
information that connects the fractal parameters of the WM function with the deter-
ministic curve as the DGI of the fourth order. It helps also to reduce the initial set of
data points (equaled 100) to some small number (4!) of significant parameters that
helps to compare the initial curves in terms of the integer moments and their mutual
cross-correlations defined by expression (2.25). Figure 9demonstrates the form of
the DGI for two limiting cases (D=0.5,2.0), including also the intermediate case
D=1.5. The next section demonstrates the results of the DGI application to real
data.
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 59
Fig. 7 In this plot, we show the dependence of two other parameters as ε/3Q0(central plot) and
B(D)(small plot above) with respect to the parameter Dfrom the interval [0.6–2.0]. One can notice
that these parameters keep their monotone behavior with respect to D
Fig. 8 Finally, this plot shows the behavior of the invariant I4(D)defined by expression (6.4). This
is the biggest parameter keeping its monotone behavior
goyal.praveen2011@gmail.com
60 R. R. Nigmatullin and A. S. Vorobev
Fig. 9 In this plot, we show the parametric dependence of the function X(ϕ)and Y(ϕ)or three
values of D=0.6 (small segment in the middle), D=1.5 (two “hyperbolic” red branches) and
D=2.0 (green ellipse-like curve). It is hard to imagine that these different curves are generated
by a monotone set of parameters shown on the previous figures
4 Some Examples Based on Real Data
The question of the finding of additional relationships between input predominant
factor and the measured response/output was raised in many papers [9–11]. We faced
the same problem while studying the optical reflectance experiments associated with
changes in chemical properties of extra virgin olive oil with respect to some external
factor. One of the main input factors that can change the quality of the olive oil
studied is the influence of the surrounding temperature. The finding of the desired
olive oil parameter with the temperature is not new and was considered in some
papers [12–14]. The description of the experiment and the data processing algorithm
with the results obtained are described below.
4.1 Details of Experiment
The testing of the DGI method was carried out on the measured data, which we
obtained in the process of olive oil temperature variations by nonchemical method.
The results of the experiment were obtained in accordance with optical characteristics
of the used equipment. The task was in measuring the reflectance optical spectra
intensity with respect to the measurement of the internal olive oil temperature and
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 61
Fig. 10 a Schematic ensemble of the experimental setup and bphoto of the cuvette with the optical
fiber (located on the right-hand side) and the temperature probe (located above and placed inside
the cuvette)
in finding their expected correlations in the frame of new approach. Figure10a, b
show the image and sketch of the experimental setup, respectively. The food olive
oil sample was chosen as a complex fluid. It has the Trade mark—“Desantis” (class:
“olio extra virgine di olive”, Italy).
Optical reflectance spectra in the wavelength range from 195nm to 1117 nm were
obtained with the usage the commercial software “SpectraSuite” and the “MAT-
LAB” script that, in turn, were controlled by the optical spectrometer (Ocean Optics
HR4000CG-UV-NIR). The volume of 4 ml of the plastic (polystyrene) cuvette was
filled with 3.5 ml of the olive oil and then was illuminated with a light source (Edmund
MI150). The optical fiber (QR400-7-UV/BX) connected the optical source with
cuvette and then the reflected signal was transmitted to the spectrometer. Black card-
board was placed on the back of the cuvette to avoid undesirable optical reflection.
The temperature was monitored using a digital thermometer (Probe thermometer
TFA LT-101), its sensitive part was placed inside the olive oil (see Fig.10b). All
optical absorptions were registered at the fixed temperature (T) in the range from
9.3 to 21.8 ◦C in dark conditions. Each measurement took a time of 3 s, and the
time duration for the whole experiment occupied 3 ·301 =903 s, i.e. 15 min. The
measurement time for the thermometer consists of one second; therefore, the averag-
ing over 3 points was carried out. This temporal requirement creates a possibility to
“feel” possible changes in the structure of the olive oil in accordance with possible
temperature variations.
The “zone of interest” for our experiments was in the range up to 20 ◦C. This zone
determines some chemical processes that take place in extra virgin olive oil. One of
the temperature ranges where these chemical reactions take place corresponds to the
interval 5–15 ◦C[15].
goyal.praveen2011@gmail.com
62 R. R. Nigmatullin and A. S. Vorobev
4.2 The Data Treatment Procedure. Algorithm
In the frame of the new algorithm, we use the DGI approach to treat the measured
data. Without loss of generality, the proposed algorithm can be divided into the
following six steps:
(1) The intensity reflectance optical spectra were measured in the full wavelength
range (195.98–1117.05 nm) and for each measuring cycle containing 3647 wave-
length points were obtained. As a final result, we had 301 measurement cycles,
corresponding to the measured T(see Fig. 11). Taking into account the averag-
ing procedure over 3 points, i.e., 903 (duration of the whole experiment) →301
(cycles), we obtain the desired plot T(number of cycle) shown in Fig. 12.
(2) Reduction to 12 measured data points. One of the aims of our measurements
was to increase the speed of measurements (number of repetitions/per time).
Therefore, we tried to avoid the measurement repetitions for each data point
several times. However, keeping the desired speed it is possible to improve
the quality of data through the reduction procedure. In this case, all data were
reduced to 12 points, i.e., every 12 points from each measurement were reduced
to one averaged point (this was done for all 301 measurement cycles). We
can do this due to the fact that we used a “food” thermometer with an error
of ±0.5◦C(Err.). The admissible temperature interval for our measurements
was 12.5◦C (from 9.3 to 21.8◦C), the number of measurement cycles was
301 (N). Therefore, the number of reduced points (R.p.) can be calculated
as R.p=N/(Range/Err)≈12. Therefore, we obtained finally the number of
the reduced cycles equal to 301/12 ≈25. In accordance with this requirement,
we obtained the same number of temperature and optical measurement cycles.
This reduction procedure helps to average the unwanted data fluctuations also.
(3) The DGI (x,ym)tool was used for data obtained by the optical method, where
xis the first reduced cycle (x=y1=const),ymis the subsequent cycle (m=
2,3,...,M). Therefore, 24 calculations of the desired DGI were obtained from
the total M=25 ymreduced cycles. Each compared curve has four quantitative
parameters (fq,I4,ε/3Q,B)depending on T.
(4) In order to decrease the number of plots, it is necessary to establish the maximal
correlations between the calculated parameters (fq,I4,ε/3Q,B)with Tm.We
want to notice that all these parameters and temperature had the same number
of data points equal to 24. In order to obtain the reliable correlations of these
parameters with temperature, we used the value of the Complete Correlation Fac-
tor (CCF(Pr,T);Pr =(fq,I4,ε/3Q,B)) that is calculated with the usage of
all admissible set of the fractional moments [16]. In the result of this evalua-
tion, the following values were obtained: CCFfq=0.8623, CC FI4=0.8946,
CCF
ε/3Q=0.8413, CCFB=0.8563. The highest value of the CCF (in terms
of the correlation degree with temperature change) belongs to the parameter I4
(below we use for it the simplified abbreviation I). Its variations with temperature
is shown the Fig.13 (black points).
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 63
Fig. 11 The visually selected 6 curves (chosen from a set of 301 measurement cycles) of the
reflectance optical spectra covering the range [196, 1117] nm
(5) Using the Procedure of the Optimal Linear Smoothing (POLS), it becomes pos-
sible to obtain a monotone and smoothed curve I(T)[17]. It is depicted in Fig. 13
by green points (CCF for IPOLS =0.8964.)
(6) In order to test the reduction procedure, we realized the same calculation, how-
ever, for non-reduced data. In this case, we have 301 measurements cycles
including the same number of the temperature points with possible tempera-
ture deviations (T=±0.5%C). We realized the similar steps (4 and 5) and
after the application of the POLS we obtained two other curves that take into
account the limiting values of temperature fluctuations. These limiting curves
(maxI(+0.5%C), minI(−0.5%C)) are shown by red and blues points, accord-
ingly.
The obtained results demonstrate undoubtedly the effectiveness of the DGI
approach that can be applied as a new working tool for the quantitative finding
of the “hidden” relationships between the correlation parameters of the DGI with the
predominant input factor (T). In addition, these relationships can be ordered with
the help of the CCF and smoothed by the POLS. Besides, one can confirm a similar
temperature trend found by researchers from Bulgaria [15] that discovered a quasi-
linear temperature dependence of the reflectance spectrum intensity with respect to
temperature in the range [9,12–17]◦C.
goyal.praveen2011@gmail.com
64 R. R. Nigmatullin and A. S. Vorobev
Fig. 12 The temperature plot covering the given optical range and located in the interval [9.3, 21.8]
◦C
Fig. 13 This figure demonstrates the variations of the curves I(T). Black points characterize a
“true” curve obtained with the help of reduction procedure. The green, red, and blue curves show
the temperature variations of the curve I(T)obtained with the help of the POLS procedure. The red,
blue curves (maxI,minI)correspond to the limiting cases (obtained without reduction procedure)
5 Results and Discussion
In this paper, we showed possible applications of the DGI of the fourth order that
admits its separation and presentation in the parametric form (2.26), (2.27). Defi-
nitely, the frame of this paper does not allow demonstrating all possibilities of this
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 65
new instrument. These new possibilities will be a subject of the further research.
However, based on the obtained results one can say the following:
1. The DGIs will help to compare two curves (including 2D random sets) with
each other. This comparison is universal and it will be useful especially in cases when
analytical expression for description of model/real data is absent.
2. The DGI helps to realize “automatically” the reduction procedure of 2N
data points to 8 statistical parameters: (I4,Ax,y,B,σx,y,θx,y). These parameters
are tightly related with 14 integer moments and inter-correlations (2.25) that signify
about the statistical proximity and differences of two compared random sets.
3. The simplified DGIs help to find the relationships between fractal dimension
D(3.2) and the reduced set of the statistical parameters (xc,yc,I4,ε/3Q,B)inthe
case of statistical proximity of the compared initial curves. It will give an additional
possibility for better understanding the random fractal sets and relate their basic
parameters with the inter-correlations of two random sets compared.
4. The simplified DGIs can help in finding additional relationships between three
basic parameters (I4,ε/3Q,B) with respect to temperature T, which are in optical
measurements described above, it was used as the predominant input factor. The DGI
approach can take into account the temperature fluctuations and confirm indepen-
dently some specific peculiarities found by other researches [15].
6 Mathematical Appendix
Expression for the invariant of the fourth order in the case when two sets are close
to each other.
In this Appendix, we want to obtain an approximate expression for the general
invariant (2.22) when the first set xkis distorted by the function fkand the second set
ykis expressed in the form yk=xk±fk( fk=fk−f). The evaluation of
this curve is not trivial because we should open the limit 0/0 that appears in calcula-
tions of the parameters σx,yin expressions (2.21). We take into account that deviations
of the factor fkin the both sides relates its mean value equal to zero and there-
fore the average value ±Afk≈0 (where Arepresents some value). However,
the combination (±Afk)·(±Bfk)=(Afk)·(Bfk)= 0 because the
positive and negative compensations in this product do not take place. Taking into
account this remark, one can evaluate the expressions for and σx,yin expressions
(2.21). After some cumbersome and long calculations, one can obtain
σx=4
3x( f)22
−(x)2f·( f)3+( f)32,
σy=4
3x( f)22
−(x)2f·( f)3,
∼
=9x( f)22
−3(x)2f·( f)2+( f)32
.
(6.1)
goyal.praveen2011@gmail.com
66 R. R. Nigmatullin and A. S. Vorobev
As one can notice from (6.1) we kept the values proportional to O(( f)4,(f)6),
inclusively. If we introduce the values:
Q0=3x( f)22
−(x)2f( f)3,
ε=( f)32
,
(6.2)
then the parameters σx,y,θx,yare expressed in the compact form:
σx∼
=4
3+8ε
9Q0
,σy∼
=4
3−4ε
9Q0
θx∼
=−1
3−5ε
9Q0
,θy∼
=−1
3+ε
9Q0
.
(6.3)
The second terms in (6.3) are considered as corrections and have the order
O( f)2. Based on (6.3), one can evaluate other parameters keeping the same accu-
racy:
Ax=Ay∼
=−2( f)2−2
3
ε
Q0(x)2,B=−2Ax
I4=2
3
ε
Q0(x)2( f)2.
(6.4)
As one can notice from expressions (6.3) and (6.4), the simplified invariant curve
contains four parameters: (x)2,( f)2,(x)2( f)2,ε/Q0.
Finally, the invariant curve of the fourth order takes the following form:
x=x+X(ϕ), X(ϕ)=r(ϕ)cos(ϕ),
y=y+f+Y(ϕ), Y(ϕ)=r(ϕ)sin(ϕ),
r(ϕ)=P2
2(ϕ)+4I4P4(ϕ)−P2(ϕ)
2P4(ϕ)1/2
,
P2(ϕ)=3Bcos(ϕ)−sin(ϕ)2
,
P4(ϕ)=cos(ϕ)−sin(ϕ)4
+ε
3Q05cos(ϕ)4
+8cos(ϕ)3
sin(ϕ)+
+ε
3Q04sin(ϕ)3
cos(ϕ)−sin(ϕ)4.
(6.5)
As it follows from these expressions, when two sets coincide with each other
they are reduced to the point x=y=x. Finally, it is necessary to demonstrate the
criterion for application of the simplified invariant:
goyal.praveen2011@gmail.com
Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences … 67
R=stdev(X−Y)
stdev(Y)<1.(6.6)
The functions Xand Yare defined by expressions (2.16) and (6.5) and the oper-
ation “stdev” is defined by the conventional expression:
stdev(F)=1
2
N
j=1
(Fj−F)21/2
.(6.7)
The model examples considered in section (3.2) satisfy to criterion (6.6).
References
1. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman and Company, San Francisco
(1983)
2. Feder, J.: Fractals. Plenum Press, Ny and London (1988)
3. Samko, S.B., Kilbas, A.A., Marichev, I.: Fractional Integrals and Derivatives—Theory and
Applications. Gordon and Breach, New York (1993)
4. Uchaikin, V.V. : Method of the Fractional Derivatives. Artishock Publishing House, Ulyanovsk
(2008)
5. Babenko, Y.I.: PowerRelations in a Circumference and a Sphere. Norell Press Inc., USA (1997)
6. Babenko, Yu.I.: The power law invariants od the point sets, Professional, S-Petersburg, ISBN
978-5-91259-095-5, www. naukaspb.ru, (Russian Federation) (2014)
7. Nigmatullin, R.R., Budnikov, H.C., Sidelnikov, A.V., Maksyutova, E.I.: Application of the
discrete geometrical invariants to the quantitative monitoring of the electrochemical back-
ground, Res. J. Math. Comput. Sci. (RJMCS). eSciPub LLC, Houston, TX USA. Website:
http://escipub.com/
8. Berry, M.V., Lewis, Z.V.: On the weierstrass mandelbrot fractal function. Proc. R. Soc. London
A370, 459–484 (1980)
9. Butler, J.M., Johnson, J.E., Boone, W.R.: The heat is on: room temperature affects laboratory
equipment–an observational study. J. Assis.t Reprod Genet., 1389–1393. Published online 2013
Aug 7. https://doi.org/10.1007/s10815-013- 0064-4
10. Hoffman, G.R., Birtwistle, J.K.: Factors affecting the performance of a thin film magnetoresis-
tive vector magnetometer. J. Appl. Phys. 53, 8266 (1982). https://doi.org/10.1063/1.330303
11. Mostafa Mohamed Abd El-Raheem, Hoda Hamid Al-Ofi, Abdullah Alhuthali, Ateyyah Moshrif
AL-Baradi, Effect of preparation condition on the optical properties of transparent conduct-
ing oxide based on zinc oxide. Optics. 4(3), 17–24 (2015). https://doi.org/10.11648/j.optics.
20150403.11
12. Saleem, M., Ahmad, N., Ali, H., Bilal, M., Khan, S., Ullah, R., Ahmed, M., Mahmood, S.:
Investigating temperature effects on extra virgin olive oil using fluorescence spectroscopy. IOP
Publishing: Laser Phys. 27 (2017), pp. 1–10. https://doi.org/10.1088/1555-6611/aa8cd7
13. Giuffrè, A.M., Zappia, C., Capocasale, M.: Effects of High Temperatures and Duration Of
Heating on Olive Oil Properties for Food use and Biodiesel Production. Springer: Journal of
the American Oil Chemists’ Society, June 2017, Volume 94, Issue 6, pp. 819–830 (2017)
14. Clodoveo, M.L., Delcuratolo, D., Gomes, T., Colelli, G.: Effect of different temperatures and
storage atmospheres on Coratina olive oil quality, Elsevier. Food Chemistry 102(3), 571–576
(2007)
goyal.praveen2011@gmail.com
68 R. R. Nigmatullin and A. S. Vorobev
15. Bodurov, I., Vlaeva, I., Marudova, M., Yovcheva, T., Nikolova, K., Eftimov, T., Plachkova,
V.: Detection of adulteration in olive oils using optical and thermal methods. Bulg. Chem.
Commun. 45(Special Issue B), 81–85 (2013)
16. Nigmatullin, R.R., Ceglie, C., Maione, G., Striccoli, D.: Reduced fractional modeling of 3D
video streams: the FERMA approach. Nonlinear Dyn. (2014). https://doi.org/10.1007/s11071-
014-1792- 4
17. Nigmatullin, R.R.: New Noninvasive Methods for “Reading” of Random Sequences and Their
Applications in Nanotechnology. Springer: New Trends in Nanotechnology and Fractional
Calculus Applications, pp. 43–56
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear
Fractional Differential Equations with
Hilfer Derivative
Benaouda Hedia
Abstract This paper studies the existence of solutions for nonlocal semi-linear
fractional differential equations of Hilfer type in Banach space by using the non-
compact measure method in the weighted space of continuous functions. The main
result is illustrated with the aid of an example.
Keywords Semi-linear differential equations ·Nonlocal initial value problems ·
Hilfer fractional derivative ·Fixed point theorems ·Measure of non-compactness ·
Condensing map
AMS (MOS) Subject Classifications: 26A33 ·34K37 ·37L05 ·34B10.
1 Introduction
Differential equations of fractional order have recently proved to be valuable tools in
the modeling of many physical phenomena [8]. There has been a significant the-
oretical development in fractional differential equations in recent years; see the
monographs of Kilbas et al. [16], Zhou [19,20]. In [13], Hilfer proposed a general-
ized Riemann–Liouville fractional derivative, for short, Hilfer fractional derivative,
which is an interpolator between Riemann–Liouville and Caputo fractional deriva-
tives. This operator appeared in the theoretical simulation of dielectric relaxation in
glass-forming materials [14].
Recently, considerable attention has been given to the existence of solution of
initial and boundary value problems for fractional and semi-linear-fractional differ-
ential equations and inclusions involving Hilfer fractional derivative, see [12]. On
the other hand, when an existence result is proved for the fractional Cauchy problem
where the solutions are not unique, it is natural to discuss the topological structure
of the solution set [9,10].
B. Hedia (B
)
Loboratory of Mathematics and Informatics, University Ibn Khaldoun of Tiaret, PO BOX 78,
14000 Tiaret, Algeria
e-mail: nilpot_hedia@yahoo.fr
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_5
69
goyal.praveen2011@gmail.com
70 B. Hedia
Motivated by the papers cited above, in this paper, we consider a class of nonlocal
initial semi-linear fractional differential equation of Hilfer type described by the form
Dα,β
0+x(t)=Ax(t)+f(t,x(t)), t∈(0,b],(1.1)
I1−γ
0+x(t)=
m
i=1
λix(τi), α≤γ=α+β−αβ,τi∈(0,b],(1.2)
where the two-parameter family of fractional derivative Dα,βdenote the left-sided
Hilfer fractional derivative introduced in [13,14], 0 <α≤1, 0 ≤β≤1. The state
x(.) takes value in a Banach space Ewith norm .,Ais the infinitesimal generator
of semigroup of bounded linear operators (i.e., C0semigroup) T(t)t≥0that will be
specified later in Banach space E. The operator I1−γ
0+denotes the left-sided Riemann–
Liouville fractional integral, f:(0,b]× E→Ewill be specified in later sections.
τi,i=1,2,...,mare prefixed points satisfying 0 <τ1≤··· ≤ τm<band (γ)=
m
i=1λiτiwhere (γ)=+∞
0x1−γe−xdx.
Physically, condition (1.2) says that some initial measurements were made at the
times 0 and τi,i=1,...,m, and the observer uses this previous information in their
model. This type of situation can lead us to a better description of the phenomenon.
For example, [6], Deng considers the phenomenon of diffusion of a small amount of
gas in a tube and assumes that the diffusion is observed via the surface of the tube.
The nonlocal condition allows additional measurement which is more precise than
the measurement just at t=0.
Our main aim in this work is to extend the result given in [18], by using a fixed
point principle for condensing maps combined with Browder–Gupta approach [4]in
a general setting, namely when the function right-hand side has values in infinite-
dimensional Banach space.
This paper is organized in the following way. In Sect.2, we give some general
results and preliminaries and in Sect. 3we present our main results.
I wish you the best of success.
Tiaret
February 14, 2018
2 Preliminary Results
In this section, we introduce some notation and technical results which are used
throughout this paper [5].
Let J:= [0,b],b>0 and (E,·)be a Banach space. C(J,E)be the space of
E-valued continuous functions on Jendowed with the uniform norm topology
x∞=sup{x(t), t∈J}.
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 71
L1(J,E)the space of E-valued Bochner integrable functions on Jwith the norm
fL1=b
0
f(t)dt.
We consider the Banach space of continuous functions
C1−γ([0,b],E)={x∈C((0,b],E):lim
t→0+t1−γx(t)<+∞}.
A norm in this space is given by
xγ=sup
t∈[0,b]
t1−γx(t).
Obviously, C1−γ([0,b],E)is a Banach space. For a subset of the space C1−γ
([0,b],E), define γby
γ={xγ:x∈},(2.1)
where
xγ=t1−γx(t), if t∈(0,b];
limt→0+t1−γx(t), if t=0.(2.2)
It is clear that xγ∈C(J,E). We note the following Ascoli-Arzelà-type criteria.
Lemma 1 Aset⊂C1−γ([0,b],E)is relatively compact if and only if γis rel-
atively compact in C([0,b],E).
Proof See, for instance, [1].
Definition 1 Let Xand Ybe two topological vector spaces. We denote by P(Y)the
family of all nonempty subsets of Yand by
Pk(Y)={C∈P(Y):compact},
Pb(Y)={C∈P(Y):bounded}.
Let G:[0,b]→P(E)be a multifunction. It is called
(i) integrable, if it admits a Bochner integrable selection g:[0,b]→E,g(t)∈
G(t)for a.e. t∈[0,b];
(ii) integrably bounded, if there exists a function ζ∈L1([0,b]; R+)such that
G(t):=sup{g: g∈G(t)}≤ζ(t)a.e. t∈[0,b].
We give some concepts of fractional calculus. Let 0 <α<1. A function x:J→
Ehas a fractional integral if the following integral:
goyal.praveen2011@gmail.com
72 B. Hedia
Iαx(t)=1
(α)t
0
(t−s)α−1x(s)ds
is defined for t≥0. The Riemann–Liouville fractional derivative of xof order αis
defined as
Dαx(t)=1
(1−α)
d
dt t
0
(t−s)−αx(s)ds=d
dt I1−αx(t),
where (·)is the Gamma function, provided it is well defined for t≥0. The previous
integral is taken in Bochner sense.
The left-sided Hilfer fractional derivative of order 0 <α≤1 and 0 ≤β≤1is
defined by
Dα,β
0+x(t)=Iβ(1−α)d
dt I(1−β)(1−α)x(t).
for functions such that the expression on the right-hand side exists.
Lemma 2 ([7]) Let α,β∈R+. Then
1
0
tα−1(1−t)β−1dt =(α)(β)
(α+β),
and hence x
0
tα−1(x−t)β−1dt =xα+β−1(α)(β)
(α+β).
The integral in the first equation of Lemma 2is known as Beta function B(α,β).
Let us recall the following definitions and results that will be used in the sequel.
Definition 2 Let Ebe a real Banach space and (Y,≤)a partially ordered set. A
function β:P(E)→Yis called a measure of non-compactness in Eif
β() =β(co)
for every ⊂P(E), where codenotes the closed convex hull of .
Definition 3 ([15,17]) A measure of non-compactness βis called
(i) monotone if 0,
1∈P(E),0⊂1implies β(0)≤β(1);
(ii) nonsingular if β({a}∪) =β() for every a∈E,∈P(E);
(iii) invariant with respect to union with compact sets, if β({K}∪) =β() for
every K∈Pk(E)and ∈P(E);
If Yis a cone in a normed space, we say that the MNC is
(iv) regular if β() =0 is equivalent to the relative compactness of ;
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 73
(v) algebraically semi-additive, if β(0+1)≤β(0)+β(1)for each 0,
1∈
P(E).
One of most important examples of a measure of non-compactness possessing all
these properties is the Hausdorff measure of non-compactness defined by
χ() =inf{ε>0:has a finite ε−net}.
Definition 4 A continuous map F:X⊂E→Eis said to be condensing with
respect to a MNC β(β-condensing) if for every bounded set ⊂X, that is,
β(F()) < β(),
we have is relatively compact.
Lemma 3 ([3,15]) If {un}+∞
n=1⊂L1(J,E)satisfies un(t)≤κ(t)a.e. on J for
all n ≥1with some κ∈L1(J,R+). Then the function χ({un(t)}+∞
n=1)belongs to
L1(J,R+)and
χt
0
un(s)ds :n≥1≤2t
0
χ(un(s)ds :n≥1)ds.(2.3)
The application of the topological degree theory for condensing maps implies the
following fixed point principle.
Theorem 1 ([2,15]) Let V ⊂E be a bounded open neighborhood of zero and
:V→Eaβ-condensing map with respect to a monotone nonsingular MNC βin
E. If satisfies the boundary condition
x= λ(x)
for all x ∈∂V and 0<λ≤1, then the fixed point set Fix={x:x=(x)}is
nonempty and compact.
3 Main Result
Definition 5 The Wright function Mq(θ)defined by
Mq(θ)=
∞
n=1
(−θ)n−1
(n−1)!(1−qn)
is such that ∞
0
θδMq(θ)dθ=(1+δ)
(1+qδ),for δ≥0.
goyal.praveen2011@gmail.com
74 B. Hedia
Define the operators Kα,Sα,β
Kα(t)=tα−1Pα(t), Pα(t)=∞
0
αθMβ(θ)T(tαθ)dθ,
Sα,β=Iβ(1−α)
0+Kα(t).
The properties of these operators were explored by Zhou [19,20].
Suppose that there exists the bounded operator B:E→Egiven by
B=I−
m
i=1
λiSα,β(τi)−1
.(3.1)
Lemma 4 The operator B defined in (3.1) exists and is bounded if one of the fol-
lowing two conditions holds:
(i) The reals numbers λisatisfies M
m
i=1
|λi|<1.
(ii) T (t)is compact for each t >0and the homogeneous linear nonlocal problem
Dα,β
0+x(t)=Ax(t), t∈(0,b],1<α≤1,0<β≤1
I1−γ
0+x(t)=
m
i=1
λix(τi), α≤γ=α+β−αβ,τi∈(0,b]
has no nontrivial mild solutions.
Definition 6 A function x∈Cγ([0,b],E)is called mild solution of the problem
(1.1)–(1.2), if it satisfies the following equation Dα,β
0+x(t)=Ax(t)+f(t,x(t)),
t∈(0,b]and the condition (1.2).
Lemma 5 (See (5) Theorem 2.3) Let f (., u(.)) ∈C1−γ([a,b])for any u ∈C1−γ
[a,b]. A function u ∈C1−γ[a,b]is solution of the fractional initial value problem
Dα,βx(t)=f(t,u(t)), 0<α≤1,0<β≤1,
I1−γ
a+=ua,γ=α+β−αβ.
If and only if u satisfies the the following Volterra integral equation.
x(t)=tγ−1ua
(γ)+1
(α)t
0
(t−s)α−1(Ax(x)+f(s,x(s)))ds.(3.2)
According to the Lemma (4) and (5) we have the following lemma which will be
useful in the sequel:
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 75
Lemma 6 Let h be a continuous function, x is solution for the fractional integral
equation
x(t)=Sα,β|T|
m
i=1
λiB(g(τi)) +g(t)if and only if
Dα,β
0+x(t)=Ax(t)+h(t), t∈(0,b],
I1−γ
0+x(t)=
m
i=1
λix(τi), α≤γ=α+β−αβ,τi∈(0,b],
where
g(τi)=τi
0
Kβ(τi−s)h(s)ds,
g(t)=t
0
Kβ(t−s)h(s)ds
T:= 1
(γ)−m
i=1λiτi
.
We prove an Aronszajn-type result for this problem. We need to make the follow-
ing assumptions:
(H1) T(t)is continuous in the uniform operator topology for t>0, and {T(t)}t≥0f
is uniformly bounded, i.e., there exists M>1 such that sup
t∈[0,+∞)
|T(t)|<
M,M
m
i=1
<1.
(H2) The map f:[0,b]× E→Eis continuous.
(H3) There exists a function p∈C([0,b],R+)such that
f(t,x)≤ p(t)(1+t1−γx), for all t∈[0,b]and x∈E.
(H4) There exists a constant c>0 such that for each nonempty, bounded set ⊂
C1−γ([0,b],E)
χ(f(t,)) ≤cχ((t)), for all t∈[0,b],
where χis the Hausdorff measure of non-compactness in E.
(H5) There is a constant M>0 such that
goyal.praveen2011@gmail.com
76 B. Hedia
M
Mp∞b1−γ+β
(β+1)⎛
⎜
⎜
⎜
⎜
⎝
MBT
m
i=1
λi
(γ)+1⎞
⎟
⎟
⎟
⎟
⎠1+M
>1.(3.3)
To prove the existence of solutions to (1.1)–(1.2), we need the following auxiliary
lemmas.
Lemma 7 ([11]) Under assumption (H1),P
β(t)is continuous in the uniform oper-
ator topology for t >0.
Lemma 8 ([11]) Under assumption (H1), for any fixed t >0,{Kβ(t)}t>0and
{Sα,β(t)}t>0, are linear operators, and for any x ∈X
Kβ(t)x≤ Mtβ−1
(β)x,Sα,β(t)x≤ Mt(β−1)(α−1)
(α(1−β)+β)x
Lemma 9 ([11]) Under assumption (H1),{Kβ(t)}t>0, and {Sα,β(t)}t>0are strongly
continuous, which means that for any x ∈X and 0<t<t ≤b we have
Kβ(t)x−Kβ(t)x→0,Sα,β(t)x−Sα,β(t )x→0,
as t,t →0,
Theorem 2 Assume that (H1)–(H5) are satisfied. Then the set S(f,{τi}n
i=1)is
nonempty and compact.
We transform the problem (1.1)–(1.2) into a fixed point problem. Consider the
operator N:C1−γ([0,b],E)→C1−γ([0,b],E)defined by
N(x)(t)=Sα,β(t)T
n
i=1
λiBτi
0
Kβ(τi−s)f(s,y(s))ds
+t
0
Kβ(t−s)f(s,y(s))ds.
Clearly, from Lemma 1.12 [18], the operator Nis well defined and the fixed points
of Nare solutions to 1.1–1.2. Thus FixN =S(f,{τi}n
i=1). Next, we subdivide the
operator Ninto two operators Pand Qas follows:
(Px)(t)=Sα,β(t)T
n
i=1
λiBτi
0
Kβ(τi−s)f(s,y(s))ds.
and
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 77
(Qx)(t)=t
0
Kβ(t−s)f(s,y(s))ds.
Now, we show that S(f,{τi}n
i=1)=∅, the proof is devised into several steps.
Step 1.Pis continuous.
Let {xn}be a sequence such that xn→xin C1−α([0,b],E). Then
t1−γP(xn)(t)−P(x)(t)
≤M|T|
(γ)
m
i=1
λiBτi
0
Kβ(τi−s) f(s,xn(s)) −f(s,x(s))ds
≤M|T|B
(β)(γ)
m
i=1
λiτi
0
|(τi−s)β−1sγ−1f(·,xn(·)) −f(·,x(·))γds
≤M|T|B
(β)(γ)
m
i=1
λiτγ+β−1
iB(γ,β)f(·,xn(·)) −f(·,x(·))γ
≤⎡
⎢
⎢
⎢
⎢
⎣
M|T|B
m
i=1
λiτγ+β−1
i
(γ)(β)
⎤
⎥
⎥
⎥
⎥
⎦
B(γ,β)f(·,xn(·)) −f(·,x(·))γ
and
t1−γQ(xn)(t)−Q(x)(t)(3.4)
≤t1−γt
0
Kβ(t−s) f(s,xn(s)) −f(s,x(s))ds (3.5)
≤Mt1−γ
(β)t
0
(t−s)β−1sγ−1s1−γf(s,xn(s)) −f(s,x(s))ds (3.6)
≤Mt1−γ
(β)t
0
(t−s)β−1sγ−1f(·,xn(·)) −f(·,x(·))γds (3.7)
≤Mbβ
(β)B(γ,β)f(·,xn(·)) −f(·,x(·))γds.(3.8)
goyal.praveen2011@gmail.com
78 B. Hedia
Hence
N(xn)−N(x)α≤BT|
(γ)
m
i=1
λibβ+bβ
×MB(γ,β)
(β)f(·,xn(·)) −f(·,x(·))γ.
Using the hypothesis (H2),wehave
N(xn)−N(x)α→0,as n→+∞.
Step 2.Nmaps bounded sets into bounded sets in C1−γ([0,b],E).
Indeed, it is enough to show that there exists a positive constant such that for each
x∈Bη={x∈C1−α([0,b],E):xγ≤η}one has N(x)γ≤.
Let x∈Bη. Then for each t∈(0,b],by(H3)we have
t1−γNx(t)≤
MT
(γ)
m
i=1
λiBτi
0
Kβ(τi−s) f(s,x(s))ds
+t1−γt
0
|Kβ(t−s)| f(s,x(s))ds
≤
BTM
2p∞
m
i=1
λi
(β)(γ)τi
0
(τi−s)β−1(1+s1−γx(s))ds
+p∞Mt
1−γ
(β)t
0
(t−s)β−1(1+s1−γx(s))ds
≤
B p∞TM
2(1+xγ)
m
i=1
λiτβ
i
(β+1)(γ)
+p∞Mb1+β−γ(1+xγ)
(β+1)
≤(1+η)p∞
(β+1)
⎡
⎢
⎢
⎢
⎢
⎣
BTM
2
m
i=1
λiτβ
i
(γ)+Mb1−γ+β⎤
⎥
⎥
⎥
⎥
⎦
:= .
Step 3.Nmaps bounded sets into equicontinuous sets.
First, we prove {Px,x∈Bη}is equicontinuous. Let t1,t2∈(0,b],t1≤t2,letBηbe
a bounded set in C1−γ([0,b],E)as in Step 2, and let x∈Bη,wehave
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 79
t1−γ
2Px(t2)−t1−γ
1Px(t1))
≤
t1−γ
2Sα,β(t2)T
n
i=1
λiBτi
0
Kβ(τi−s)f(s,y(s))ds
−t1−γ
1Sα,β(t1)T
n
i=1
λiBτi
0
Kα(τi−s)f(s,y(s))ds
+
t1−γ
2Sα,β(t2)T
n
i=1
λiBτi
0
Kα(τi−s)f(s,y(s))ds
−t1−γ
1Sα,β(t1)T
n
i=1
λiBτi
0
Kα(τi−s)f(s,y(s))ds
.
from the fact that t1−γ
1Sα,β(t)is uniformly continuous on J, we deduce then {Px,x∈
Bη}is equicontinuous. Using condition (H2), one has
t1−γ
2Qx(t2)−t1−γ
1Qx(t1)|
≤
t1
t2
t1−γ
2(t2−s)β−1Pβ(t2−s)f(s,x(s))ds
+t1
0
t1−γ
2(t2−s)β−1Pβ(t2−s)f(s,x(s))ds
−t1
0
t1−γ
1(t1−s)β−1Pβ(t2−s)f(s,x(s))ds
+|t1
0
t1−γ
1(t1−s)β−1Pβ(t2−s)f(s,x(s))ds
−t1
0
t1−γ
1(t1−s)β−1Pβ(t1−s)f(s,x(s))ds
≤Mp∞(1+xγ)
(β)|t1
t2
t1−γ
2(t2−s)β−1ds+
Mp∞(1+xγ)
(β)
t1
0t1−γ
1(t1−s)β−1−t1−γ
2(t2−s)β−1 ds
+p∞(1+xγ)|
t1
0
t1−γ
1(t1−s)β−1!Pβ(t2−s)−Pβ(t1−s)"ds
I1+I2+I3,
where
goyal.praveen2011@gmail.com
80 B. Hedia
I1=Mp∞(1+xγ)
(β)t1
t2
t1−γ
2(t2−s)β−1ds
I2=Mp∞(1+xγ)
(β)
t1
0t1−γ
1(t1−s)β−1−t1−γ
2(t2−s)β−1 ds
I3=(1+xγ)p∞|
t1
0
t1−γ
1(t1−s)β−1!Pβ(t2−s)−Pβ(t1−s)")ds,
which yields limt2↔t1I1=0. Similarly, we can prove that limt2→t1I2=limt2→t1
I3=0
Thus {Qx,x∈Bη}is equicontinuous.
Step 4.Nis ν-condensing.
We consider the measure of non-compactness defined in the following way. For every
bounded subset ⊂C1−γ([0,b],E)
ν() =max
∈()
(#γ(), modC1−γ()). (3.9)
() is the collection of all countable subsets of and the maximum is taken in
the sense of the partial order in the cone R2
+.#γis the damped modulus of fiber
non-compactness
#γ() =sup
t∈[0,b]
e−Ltχ(γ(t)), (3.10)
where γ(t)={xγ(t):x∈}.mod
C1−γ() is the modulus of equicontinuity of the
set of functions given by the formula
modC1−γ() =lim
δ→0sup
x∈
max
|t1−t2|≤δxγ(t1)−xγ(t2).(3.11)
Let
q(L):= sup
t∈[0,b]t
0
(t−s)α−1sα−1e−L(t−s)ds.(3.12)
It is clear that
sup
t∈[0,b]t
0
(t−s)α−1sα−1e−L(t−s)ds −→
L→+∞ 0.
We can choose Lsuch that
¯q1:=
2cT M2
m
i=1
λiBq(L)
(β)(γ)<1
2(3.13)
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 81
and
¯q2:= 2cMb1−γq(L)
(β)<1
2.(3.14)
From Lemma 1, the measure νis well defined and gives a monotone, nonsingular,
semi-additive, and regular measure of non-compactness in C1−γ([0,b],E).
Let ⊂C1−γ([0,b],E)be a bounded subset such that
ν(N()) ≥ν(). (3.15)
We will show that (3.15) implies that is relatively compact. Let the maximum on
the left-hand side of the inequality (3.15) be achieved for the countable set {yn}+∞
n=1
with
yn(t)=S1fn(t)+S2fn(t), {xn}+∞
n=1⊂(3.16)
with
S1fn(t)
=Sα,β(t)T
n
i=1
λiBτi
0
Kα(τi−s)f(s,yn(s))ds,
S2fn(t)=t
0
Kα(t−s)f(s,yn(s))ds
and fn(t)=f(t,xn(t)). So that
#γ({yn}+∞
n=1)≤#γ({S1fn}+∞
n=1)+#γ({S2fn}+∞
n=1). (3.17)
We give now an upper estimate for#γ({yn}+∞
n=1).Byusing(H4)we have
χ({Kβ(t−s)fn(s)}+∞
n=1)
≤cM
(β)(t−s)α−1χ({xn(s)}+∞
n=1)
=cM
(β)(t−s)α−1sγ−1χ({xn
γ(s)}+∞
n=1)
≤cM
(β)(t−s)α−1sγ−1eLs sup
0≤s≤t
e−Lsχ({xn
γ(s)}+∞
n=1)
=cM
(β)(t−s)α−1sγ−1eLs#γ({xn}+∞
n=1)
(3.18)
for all t∈[0,b],s≤t. Then applying Lemma 3, we obtain
t1−γχ({S1fn(t)}+∞
n=1)≤
2cM2TB
(β)(γ)
m
i=1
λiti
0
(ti−s)α−1sγ−1eLs#γ({xn}+∞
n=1)ds
goyal.praveen2011@gmail.com
82 B. Hedia
and
t1−γχ({S2fn(t)}+∞
n=1)≤
2cMb1−γ
(β)t
0
(t−s)α−1sγ−1eLs#γ({xn}+∞
n=1)ds.
Taking (3.13) and (3.16) into account, we derive
#γ({yn}+∞
n=1)≤(¯q1+¯q2)#γ({xn}+∞
n=1). (3.19)
Combining the last inequality with (3.15), we have
#γ({xn}+∞
n=1)≤(¯q1+¯q2)#γ({xn}+∞
n=1).
Therefore
#γ({xn}+∞
n=1)=0.
Hence by (3.19), we get
#γ({yn}+∞
n=1)=0.
Furthermore, from Step 3, we know that modC1−γ(N()) =0 and (3.15) yields
modC1−γ() =0. Finally,
ν() =(0,0),
which proves the relative compactness of the set .
Step 5. A priori bounds.
Let x=λN(x)for some 0 <λ<1. This is implied by (H3), ( H5)follows
xγ= M.
Set
U={x∈: xγ<M}.
From the choice of U, there is no x∈∂Usuch that x=λN(x)for some λ∈[0,1]
yielding the desired a priori boundedness.
By Theorem (1), FixN =S(f,{τi}n
i=1)is nonempty compact subset of C1−α
([0,b],E).
4 Conclusion
In this paper, I have given a new result concerning the existence of solution of a class
of semi-linear Hilfer fractional differential equation with nonlocal conditions using
a measure of non-compactness combined with condensing map in Banach space.
goyal.praveen2011@gmail.com
Nonlocal Conditions for Semi-linear Fractional Differential Equations … 83
Acknowledgements The author would like to express his warmest thanks to all members of
ICFDA18 International Conference on Fractional Differentiation and its Applications 2018 for
his/her valuable comments and suggestions.
References
1. Agarwal, R.P., Hedia, B., Beddani, M.: Structure of solutions sets for imlpulsive fractional
differential equation. J. Fract. Calc. Appl 9(1), 15–34 (2018)
2. Andres, J., Górniewicz, L.: Topological Fixed Point Principles for Boundary Value Problems.
Kluwer, Dordrecht (2003)
3. Bothe, D.: Multivalued perturbations of m-accretive differential inclusions. Israel J. Math. 108,
109–138 (1998)
4. Browder, F.E., Gupta, G.P.: Topological degree and nonlinear mappings of analytic type in
Banach spaces. J. Math. Anal. Appl. 26, 390–402 (1969)
5. Deimling, K.: Nonlinear Functional Analysis. Springer (1985)
6. Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial
conditions. J. Math. Anal. Appl. 179, 630–637 (1993)
7. Diethelm, K.: Analysis of Fractional Differential Equations. Springer, Berlin (2010)
8. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional order differential equations
used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.)
Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction
Engineering and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999)
9. Djebali, S., Górniewicz, L., Ouahab, A.: Solutions Sets for Differential Equations and Inclu-
sions. De Gruyter, Berlin (2013)
10. Dragoni, R., Macki, J.W., Nistri, P., Zecca, P.: Solution Sets of DifferentialEquations in Abstract
Spaces, Pitman Research Notes in Mathematics Series 342. Longman, Harlow (1996)
11. Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional
derivative. Appl. Math. Comput. 257, 344–354 (2015)
12. Furati, K.M., Kassim, M.D., Tatar, N.E.: Existence and uniqueness for a problem involving
Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012)
13. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore, pp.
87–429 (2000)
14. Hilfer, R.: Experimental evidence for fractional time evolution in glass materials. Chem. Phys.
284, 399–408 (2002)
15. Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear
Differential Inclusions in Banach Spaces. De Gruyter, Berlin (2001)
16. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential
Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam (2006)
17. Toledano, J.M.A., Benavides, T.D., Azedo, G.L.: Measures of Noncompactness in Metric Fixed
Point Theory. Birkhauser, Basel (1997)
18. Wang, J.R., Zhang, Y.: Nonlocal initial value problems for differential equations with Hilfer
fractional derivative. Appl. Math. Comput. 266, 850–859 (2015)
19. Zhou, Y.: Basic Theory of Fractional DifferentialEquations. World Scientific, Singapore (2014)
20. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Academic
Press (2016)
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of
Energy 4.0: Ride Through Fault Aided
by Fractional PI Control and VRFB
Rui Melicio, Duarte Valério and V. M. F. Mendes
Abstract This chapter presents a simulation of a study to improve the ability of an
offshore wind system to recover from a fault due to a rectifier converter malfunction.
The system comprises: a semi-submersible platform; a variable-speed wind turbine;
a PMSG; a 5LC-MPC; a fractional PI controller using the Carlson approximation.
Recovery is improved by shielding the DC link of the converter during the fault using
as further equipment a redox vanadium flow battery, aiding the system operation as
desired in the scope of Energy 4.0. Contributions are given for: (i) the fault influence
on the behavior of voltages and currents in the capacitor bank of the DC link; (ii)
the drivetrain modeling of the floating platform by a three-mass modeling; (iii) the
vanadium flow battery integration in the system.
This work is funded by: European Union through the European Regional Development Fund,
included in the COMPETE 2020 (Operational Program Competitiveness and Internationalization)
through the ICT project (UID/GEO/04683/2019) with the reference POCI010145FEDER007690;
FCT, through IDMEC, under LAETA, Project UID/EMS/50022/2019; Portuguese Founda-
tion for Science and Technology (FCT) under Project UID/EEA/04131/2019, and grant
SFRH/BSAB/142920/2018.
R. Melicio (B
)
ICT, Instituto de Ciências da Terra, Universidade de Évora, Evora, Portugal
e-mail: ruimelicio@gmail.com
R. Melicio ·D. Valério
IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
R. Melicio ·V. M. F. Mendes
Departmento de Física, Escola de Ciências e Tecnologia, Universidade de Évora, Évora, Portugal
V. M. F. Mendes
Department of Electrical Engineering and Automation, Instituto Superior de Engenharia de
Lisboa, Lisbon, Portugal
CISE, Electromechatronic Systems Research Centre, Universidade da Beira Interior, Covilha,
Portugal
© Springer Nature Singapore Pte Ltd. 2019
P. Aga r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981-15-0430-3_6
85
goyal.praveen2011@gmail.com
86 R. Melicio et al.
Keywords Offshore wind system ·VRFB ·MPC five-level converter ·Ride
through capability ·Simulation ·Energy 4.0 ·Fractional control
1 Introduction
Wind energy conversion into electric energy through wind power systems either
onshore or offshore plays a significant role in a future shaped by the need for sustain-
able development concerning energy usage [1–40] and smart energy in the context
of Energy 4.0. The wind system quota has increased in capacity into the mixed gen-
eration of electric grids, improving the diversification of resources and contributing
to matching the needs in the usage of electric energy [6]. Fluctuation on the side
of the conversion into electric energy is expected to increase due to the uncertainty
inherent to the intermittency of exploitation of wind or solar energy sources [16].
Fluctuation on the side of the usage of electric energy is also expected to increase
in the future, for instance, due to the use of electric vehicles. Both fluctuations are
prone to lead to new challenges and threats to be faced in the scope of smart energy
in the context of Energy 4.0. Particularly, if not properly conduced, the exploitation
of wind energy sources for conversion into electric energy is a power system inter-
connection menacing the quality of energy and the transient stability of electric grid
[37] (Table1).
Grid codes establish interconnecting guidelines, i.e., instructions specifying tech-
nical and operative requirements to conduce power production and other parties
involved in the production, transportation, and usage of electric energy. Indeed, in
the context of sustainable power production in the scope of Energy 4.0, more restric-
tive grid codes are expected to be in force to conduce the operation of wind systems
to avoid abnormal behavior leading to menace, for instance, as loss of power quality
or of stability- appropriated margin into the electric grid. Wind systems must cope
with acceptable performance regulated in grid codes to integrate the electric grid. So,
after a failure, the recovery of normal operation in due time is of great importance
to avoid eventual coming off from the electric grid. Rethinking how a wind system
can satisfy grid codes and capture more value from the participation into the mixed
production of an electric grid while mitigating faults are challenges of research in
the way of Energy 4.0. Furthermore, abnormal behavior in the operation which is not
avoided in due time can lead to a fault going into a failure, needing human interven-
tion on the wind system. This is hampered for offshore wind systems by the access
of the place of exploitation, often impossible in days with severe weather in fall or
winter seasons [17,20]. So, improving the recovery of normal operation, particu-
larly Fault Ride Through capability [25], is of vital importance for offshore wind
systems. Research contributing to the operation continuity of wind systems in the
occurrence of an eventual failure is a promising line of research. A way of ensuring
operation continuity and capture more value often stated and justified as an advanta-
geous option is the use of energy storage. Energy storage through the technology of
VRFB has the advantage of a full depth of discharge without affecting performance
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 87
Table 1 Nomenclature
VRFB Vanadium redox flow battery
BMS Battery management system
IGBT Insulated gate bipolar transistor
MPC Multiple point clamped
PMSG Permanent magnet synchronous generator
5LC Five-level power converter
uWind speed value with disturbance
u0Average wind speed
nIndex of eigenswing excited
AnMagnitude of the eigenswing n
ωnEigenfrequency of the eigenswing n
ηWave elevation at the point x,y
ηaVector of harmonic wave amplitudes
ϑVector of harmonic wave frequencies
ζVector of harmonic wave phases (random)
φVector of harmonic wavenumbers
ψVector of harmonic wave directions
PtMechanical power with perturbation
Ptt Mechanical power without perturbation
mOrder of the harmonic in an eigenswing
anm Normalized magnitude of gnm
gnm Distribution of m-order harmonic in
eigenswing n
hnModulation of eigenswing n
ϕnm Phase of m-order harmonic in eigenswing n
usk Rectifier input or inverter output voltages
Ucj Voltage in the capacitor bank j
icj Current in the capacitor bank j
Udc Total DC voltage
CjCapacitance of the capacitor bank j
ifk Currents injected into the electric grid
LnInductance of the electric grid
RnResistance of the electric grid
ufk Voltage at the filter
ikInput or output current in MPC five-level
converter
ukVoltage at the electric grid, k∈{4,5,6}
μFractional order of the derivative or of the
integral
goyal.praveen2011@gmail.com
88 R. Melicio et al.
Fig. 1 VRFB reaction, charge/discharge process
and the useful life of the battery, customizable scalability high power, long duration
of about 15,000 cycles of charge/discharge, fast response, large capacity, and a fair
balance of energy efficiency and costs [12]. In comparison with other technologies,
VRFB technology is pointed out as an appropriate storage for electric grids, [2,3,
14], contributing to smooth the impact of uncertainty on the availability of renew-
able energy and allowing satisfaction of conditions imposed by grid codes [15]. A
schematic diagram of the electrochemical reactions charge/discharge processes of
the VRFB [2] is shown in Fig. 1.
In Fig. 1are shown the two tanks to store the electrolytes to be gradually pumped
into the stack of the electrochemical cells, where by chemical reactions the elec-
trolytes are charged or discharged. So, augmenting the volume of the tanks allows
scaling up the storage capacity. The membrane carries a selective transmission of
protons from the two sides of the VRFB. Each vanadium ion by a process of oxida-
tion drops at the positive terminal an electron during charge. The flow of electrons is
controlled by the BMS and directed to be collected by a vanadium ion at the cathode
by a process of reduction during charge, during discharge the process is reversed. A
VRFB is recognized by having not only capability to respond nearly instantaneously
to demand, standing in for the traditional means of meeting peak demand, but also the
ability to convey energy when required over significant periods of time, for instance,
a time of twelve hours as pointed out in [10]. So, the technology of VRFB at utility
scale is expected to be the way of the future for energy storage in the scope of Energy
4.0. The technology of VRFB is already in the way of application at utility scale.
In the Isle of Gigha, Scotland, a VRFB is in use due to the ability to balance vari-
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 89
able generation from renewable sources and due to the cost-effective time shifting.
In Dalian, China, a 200 MW/800 MWh VRFB utility scale is to be implemented
for peak-shaving and grid stabilization. Advantages pointed out are reliability, full
recyclability of the electrolyte, and more than 20-year useful life-time [9].
Many of the faults in wind systems can occur in power control and sensor elec-
tronic devices [39]. Faults are susceptible to disturb the operation and cause an inabil-
ity to perform within specified requirements, i.e., grid codes, or going into failure.
Hence, not only fault avoidance, but also tolerance to fault, i.e. the ability to continue
to perform within specified performance in the presence of a fault is regarded as
crucial. In this regard, a VRFB can be used during a short time span to replace the
delivery of energy coming from the generator interrupted, for instance, by a fault
in the rectifier converter. This replacement is intended to preserve the wind system
connection to the electric grid during a time span that ends with the recovery of the
full operation, avoiding a failure, implying the disconnection of the wind system
[35]. The VRFB can be incorporated into a wind system by a direct current circuit
a BMS required for: monitoring the state; processing and reporting secondary data;
protecting and controlling the environment of the VRFB [2]. However, a convenient
control strategy for the selection of voltage vectors to maintain the equilibrium of the
voltages in the capacitor bank must be carried out to further aid in avoiding failure
of the wind system. One part of the scope of the research in this chapter is the above
convenient control strategy for the selection of voltage vectors, aiding in avoiding a
failure of the wind system subject to a rectifier converter fault. This control strategy
is implemented by a convenient exploration of the use of the redundant vectors and
the dominant currents in the capacitor bank.
The chapter is concerned with a wind system equipped with a MPC-5LC and
the main contributions are: (i) the study of the fault influence on the behavior of
voltages and currents in the capacitor bank; (ii) the consideration of the dynamics
of the floating platform by a three-mass modeling drivetrain; (iii) the study of the
incorporation a VRFB assisting the recovery of operation. The rest of the chapter is
organized as follows: Sect. 2is concerned with the integration of wind systems in
the scope of Energy 4.0. Section 3presents the model. Section 4presents the control
method. Section 5presents a case study and discusses the consequence of the results.
Section 6presents the concluding remarks.
2 Energy in a Sustainable Way
The industry of energy is at the brink of a new industrial revolution, not only shift-
ing the present, but also the way of the future, in what regards information pro-
cessing, control and action of intervenient agents. Availability of energy has been
of paramount importance and a key influence on all Industrial Revolutions and is
expected to remain so in the future. But another important influence is the paradigm
for the organization of the energy business, which is expected to play a substantial
role.
goyal.praveen2011@gmail.com
90 R. Melicio et al.
The mix of sources of energy used in the power systems has changed, because
of the integration in an unprecedented scale of intermittent renewable sources of
energy, of nuclear phase-out, and of the appearance of utility-scale technology for
storage of electric energy. Also, new concepts for grids at the level of transmission
and distribution of electric energy are expected to happen together with the operation
of power systems in a sustainable way under Energy 4.0.
The increasing use of intermittent renewable energy, and in particular of wind
systems, requires control acting in due time, and needs to be balanced with flexible
generation, demand management, energy storage, or interconnection devices. Wind
systems must embrace Energy 4.0 concepts to cope with the future, implementing
convenient monitoring, transferring, and analyzing data in a smart grid and IoT
way. Systems interconnection and energy sustainability is the ambition of the future
energy business, using information not only from monitoring systems of the energy
industry, but also from other industrial systems [4,28]. Advanced smart technology
and better control systems for wind systems operation connected with electric grid
allow some flexibility on the requirements for the performance of electric generators.
But one of the most severe requirements in the case of offshore wind system could be
the black start capability, which requires the ability to recover from a total or partial
shutdown within a set timeframe, without any external supply. Under the Energy 4.0
framework, the integration of offshore wind systems in a smart grid is expected to
be aided by real-time monitoring and safety actions to mitigate the impact of faults
[34], improving efficiency and sustainability.
3 Modeling
The wind system under consideration is equipped with the following main compo-
nents: a semi-submersible platform of the category used in the WindFloat project
[29] anchored to the seabed by suspended cables; a platform where is placed the
variable-speed wind turbine and the equipment for power control by blade pitch
angle; a PMSG; the MPC-5LC; and an energy storage system assumed to be, but not
necessary a VRFB. What is necessary is that the energy storage system device has
enough energy to aid in avoiding failure due to the fault.
3.1 Wind and Marine Wave
The wind speed is modeled by a sum of harmonics ranging from 0.1 Hz to 10.0 Hz
as in [35] given by
u=u01+
n
Ansin (ωnt)(1)
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 91
The marine wave elevation is described as a phase/amplitude model as in [32] con-
sidered by a convenient sum of harmonic waves given by
η(x,y,z)=
i
ηa(i)cos [ϑit+ζi−υi(x,y)](2)
where
υi(x,y)=φi(xcos ψi+ysin ψi)(3)
In (3) is computed the inner product of the position vector in the horizontal plane
(x,y)by the wave vector, having the information concerned with the displacement
of the wave, pointing in the normal direction to the wave front.
3.2 Wind Turbine
The turbine mechanical power is modeled as a sum of a function of the magnitude
of three eigenswings as in [31] given by
Pt=Ptt 1+
3
n=1
An2
m=1
anmgnm (t)hn(t)(4)
where
gnm =sin t
0
mωn(t)dt+ϕnm(5)
The data considered for (4) and (5) are reported in [1].
3.3 Drivetrain Model
The aerodynamic loads to which wind turbines are subject have an important influ-
ence in the design of their structural components, due to the need to resist fatigue.
Fatigue-oriented design [13] is needed for the tower, the blades, and also the driv-
etrain. The model followed in this chapter (consisting of three masses coupled by
elastic elements) can be found to be reported in technical literature (e.g., [23,36]).
The reason why the drivetrain is modeled with several masses is its flexibility,
which is needed to increase reliability in the presence of fatigue [5,13]. In fact,
the vibrations of the structure, revealed by noise in the aerodynamic blades, can be
reduced by a better aerodynamic design, and so the efforts are now shifted to the
drivetrain itself.
goyal.praveen2011@gmail.com
92 R. Melicio et al.
3.4 Generator
The generator is a PMSG, and its model is that usually employed for a normal
synchronous electric machine. This has been reported in technical literature such
as [22]. An additional constraint is needed: the direct component of the electric
current in the stator is imposed to be zero. This constraint is intended to prevent the
demagnetization of the permanent magnet in a PMSG [33].
3.5 Five-Level Power Converter
Figure 2shows the details of the MPC 5LC rectifier. It shows that the MPC-5LC
rectifier and the inverter have 24 unidirectional commanded IGBTs. The IGBTs are
modeled as ideal components. Branch kof the converter consists of a group of eight
IGBTs connected to the same phase. IGBTs are identified by Sik with i∈{1,...,8},
corresponding to a branch k∈{1,2,3}(in the case of the rectifier) or k∈{4,5,6}
(in the case of the inverter) [32]. Figure 3shows the MPC-5LC, equipped with the
VRFB. For details, see [18].
The voltages in the rectifier (input, k∈{1,2,3}) and the inverter (output, k∈
{4,5,6})aregivenby
usk =1
3
p−1
j=1⎛
⎝2δjnk
3
a=1,a=k
δjna⎞
⎠Ucj (6)
On each capacitor bank Cj, the current icj can be found as
icj =
3
k=1
δnkik−
6
k=4
δnkik,k∈{1,...,6}(7)
The currents in each capacitor banks are, in (7), the input and output currents of
the MPC-5LC. The state equation the DC bar voltage Udc is
dUdc
dt=
p−1
j=1
1
Cj
icj,j∈{1,..., p−1}(8)
Note that (8) is valid even if the BMS has called the VRFB.
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 93
Fig. 2 MPC 5LC rectifier
3.6 Electric Grid
The current injected into the electric grid is modeled by the following state equation:
difk
dt=1
Lnufk −Rnifk −uk,k∈{4,5,6}(9)
This model consists of an ideal voltage source, in series with the short-circuit
impedance.
goyal.praveen2011@gmail.com
94 R. Melicio et al.
Fig. 3 Offshore wind system equipped with VRFB
3.7 VRFB
The VRFB is a DC source with a value of 7.14 kV, 2% plus of the DC link voltage.
Although this type of source can have significant discharge times [15], the capability
to respond nearly instantaneously is the most important for the purpose of the chapter.
4 Control Method
4.1 Fractional-Order Controllers
Fractional-order controllers are those designed by the application of fractional-order
derivatives, having advantageous robustness as the main reason for being useful in the
applications. Usual fractional-order controllers include fractional PIDs, introduced
in [27] and so-called because of their similitude with PID controllers, and CRONE
(Commande Robuste d’Ordre Non Entier) controllers [19,26]. The CRONE con-
trollers are designed according to a methodology conceived with control robustness
in mind. In the system under study, the controller for the variable-speed operation is
fractional PI controller [35], from the PID family, combining a fractional derivative
with a constant gain at low frequencies. The transfer function for the controller for
the variable-speed operation is given by
C(s)=2.6+0.61
s0.5(10)
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 95
Several tuning methods have been proposed for fractional PIDs, including tuning
rules such as in [11], or numerical methods as in [24]. Controller (16) was designed
according to the rules in [21]. The term s0.5is the Laplace transform of a half-
derivative. More about fractional derivatives is presented in [30,38]. An implemen-
tation of a term such as sαis usually carried out by means of an approximation. The
most usual approximations [38] are the CRONE approximation, due to the work of
Alain Oustaloup, the Matsuda approximation, and the Carlson approximation, the
one used for the system under study. It was introduced in [7], and is based on the
Newton–Raphson method for finding numerical solutions of equations, which can
be used to obtain numerical values for a1/nfinding the roots of f(x)=xn−a.The
method can also be used if ais not a positive real number, but rather the Laplace
transform variable sinstead. In this way, iterative approximations of s1/n,n∈N
can be found. Some calculations show that, beginning with the trivial (and far from
accurate) approximation
s1/n≈G1(s)=1,(11)
further approximations given by
s1/n≈Gk+1(s)=Gk(s)(n−1)Gn
k(s)+(n+1)s
(n+1)Gn
k(s)+(n−1)s,(12)
can be found, which are rather accurate in a frequency range that increases with
n, centered on frequency 1 rad/s. The approximation used to implement (10)was
obtained with two iterations:
s1
−2≈G3(s)=s4+36s3+126s2+84s+9
9s4+84s3+126s2+36s+1(13)
Transfer function (13) provides a good approximation a limited frequency range [ωl,
ωh]=[10−1,10]rad/s. Finally, (10) was implemented as C(s)≈2.6+0.6G3(s).
4.2 Power Converter Control
The modeling to be considered for the MPC-5LC control is of fractional order com-
plemented with a sliding mode control associated with PWM by space vector mod-
ulation. The output voltage vectors in the (α,β)space for the 5LC are shown in
Fig. 4.
In Fig. 4, the required selection for the output voltage vector in the (α,β)space
is carried out in function of the discrepancy between the current of the stator and the
reference current. A power converter is a time variable structure due to the IGBTs
switching blockage/conduction states [32]. The operation of time-variable structures
subject to uncertainties and external disturbances must be suitably complemented by
sliding mode control as is reported in the literature. This operation of the power con-
goyal.praveen2011@gmail.com
96 R. Melicio et al.
Fig. 4 Output voltage vectors for the MPC-5LC
verter as a time variable structure subject to uncertainties and external disturbances
is the one presented by the authors in [35].
5 Case Study
The case study is carried out using the computer application MATLAB/Simulink and
has a time horizon of 10 s. The electric grid voltage is of 5 kV at 50 Hz. A 10 kHz
switching frequency is assumed for the IGBTs. The capacitor bank reference voltage
U∗
dc is of 7 kV. The main data concerned with the wind system [36]issummarized
in Table 2.
The wind speed is shown in Fig. 5. The marine wave elevation is shown in Fig. 6
that shows a significant perturbation due to marine wave with a period of about 10 s
subjecting the wind system to a significant perturbation. The data for (1) and (4)are
u0=14.5m/s; A1=0.01; A2=0.08; A3=0.15; ω1(t)=ωt(t);ω2(t)=3ωt(t);
and ω3(t)=[g11(t)+g21 (t)]/2, g11(t),g21(t)given by (5).
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 97
Table 2 Wind system data
Turbine moment of inertia 5.5×106kgm2
Turbine rotor diameter 90 m
Hub height 45 m
Tip speed 17.64–81.04 m/s
Rotor angular velocity 6.9–31.6 rpm
PMSG rated power 2MW
PMSG inertia moment 400 ×103kg m2
Fig. 5 Wind speed
Fig. 6 Marine wave
elevation
goyal.praveen2011@gmail.com
98 R. Melicio et al.
Fig. 7 Power coefficient
Fig. 8 Vo lt a g e in t h e
capacitor bank
5.1 No Fault
This simulation is carried out with no-fault consideration for purpose of comparison,
i.e., is a normal operation with intermittent availability of wind energy, having the
wind speed shown in Fig. 5. The power range satisfaction due to the action of the
maximum power point tracking imposes the power coefficient shown in Fig. 7.The
voltages in the capacitor bank are shown in Fig. 8. The current in the capacitor
bank is shown in Fig. 9. These last two figures show that the control of the converter
surmounts the imbalance voltages in the capacitor bank, having appropriated currents
in the capacitors and almost a steady behavior for voltage in the DC link. So, normal
operation is pursued.
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 99
Fig. 9 Currents in the
capacitor bank
5.2 Failure
In what concerns the data, this simulation has the same wind speed and the marine
wave as the first one. It also has the same equipment for the wind system, but with
a fault. This fault imposes that a circuit breaker at the input of the rectifier is open
between 1.15 s and 1.65 s. So, during this period no energy flows to the capacitors,
i.e., the capacitors are not charged by the rectifier, implying a hovering menace of
fault going into failure on the wind system. This menace is in accordance with the
simulation results given for the voltages in the capacitors as shown in Fig. 10.In
this figure, it is shown that the voltage drop across capacitors is significant and the
level of voltage is not recovered in due time to avoid the disconnection. So, this
simulation is in accordance with the wind system having an inevitable fault going
into failure, i.e., the system goes into the necessary disconnection to avoid further
worst consequences.
The input voltages in the rectifier are shown in Fig. 11. Here it is once more
shown that the wind system is unable to recover voltage for feasible operation in the
rectifier. Again, this simulation results are in accordance with the wind system not
being able to avoid disconnection.
The currents in the capacitor bank are shown in Fig. 12. This figure shows that
after 1.15 s the currents in the capacitor bank have a smaller positive oscillation
than the negative one, meaning that the capacitors are discharging. The control of
the converter is trying to surmount the imbalance on voltages in the capacitor bank,
but there is not enough electric charge to sustain the drop on the voltages dropping
significantly near 1.65 s.
The main conclusion of this simulation is in accordance with the fact that the
system is unable to maintain the interconnection with the electric grid. So, ride
through fault is feasible for the wind system. The ability to perform within specified
goyal.praveen2011@gmail.com
100 R. Melicio et al.
Fig. 10 Vo lt a g e in t h e
capacitors of the capacitor
bank
Fig. 11 Input voltages in the
rectifier
performance requirements, i.e., in the way of fault tolerance in the way of the future
at Energy 4.0, is not feasible and disconnection is to happen to avoid further worst
consequences of equipment damage.
5.3 Ride Through Fault
In what concerns the data, this simulation has the same wind speed and the marine
wave as the first one. It also has the same equipment for the wind system, and a
fault considered as imposing that a circuit breaker at the input of the rectifier is open
between 1.15 s and 1.65 s. That is to say, the simulation has the same data as that of
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 101
Fig. 12 Currents in the
capacitor bank
Sect. 5.2, but with the addition of an energy storage system device assumed—but not
necessarily—to be given by the technology of VRFB. What is important to assume
is that the energy storage system device is designed to have enough energy to aid in
avoiding the fault going into failure during the time without energy flowing from the
rectifier converter. So, the simulation is concerned with the wind system calling due
to the fault the aid of the energy storage system device by the BMS to conveniently
charge the capacitor bank during the period of transient operation between 1.15 s
and 1.65 s. The behavior of the system during this period tends to be described by
the arrangement shown in Fig. 13.
In Fig. 13 is shown the VRFB call during the period of transient operation to
contribute with a convenient charge of the capacitors. So, recovering of normal
operation is feasible after fault clearing. The voltages in the capacitor bank are shown
in Fig. 14.
In Fig. 14 is shown that when the VRFB is called a transient occurs while the
returning of the VRFB to standby is done more smoothly. The voltage drop across
capacitors is not significant and the level of voltage is recovered in due time to avoid
the disconnection. So, this simulation is in accordance with the wind system having
a fault but not going into failure. The system can ride through the fault due to the halt
dropping of the level of voltages in the capacitor bank. The control surmounts imbal-
ance voltages on the capacitors, giving a recovering of the appropriated behavior of
the voltage in the DC link in a way of fault tolerance feasibility. The input voltages
in the rectifier, the currents in the rectifier, and the current in the capacitor bank are,
respectively, shown in Figs. 15,16 and 17.
In Fig. 15, it is shown that the wind system can recover the normal operation of
the rectifier converter, the system went into a state of the fault and goes into normal
operation after fault clearing. In Fig. 16, it is shown as expected that the input currents
in the rectifier are null during the period 1.15 s to 1.65 s. After this period, the currents
accommodate values given by the wind system control to achieve maximization of
goyal.praveen2011@gmail.com
102 R. Melicio et al.
Fig. 13 System during the transient
Fig. 14 Voltage in the capacitor bank
energy conversion in the power range of the wind system. In Fig. 17 is shown that
when the VRFB is called a transient occurs and the returning of the VRFB to a standby
state has smooth picks on currents in the capacitor bank. After the disconnection of
the VRFB, the rectifier can provide an electric charge to the capacitor bank to avoid
the imbalance voltages. Hence, the wind system can recover the feasible operation
after the fault clearing due to the assistance of the VRFB, conveniently charged. The
total harmonic distortion (THD) of the electric current injected into the electric grid
for the simulations with No fault and with Ride through fault are shown in Table 3.
These figures show, as expected, that the THD of the electric current into the electric
grid is not significantly greater and the difference is faded after fault clearing.
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 103
Fig. 15 Input voltages in the
rectifier
Fig. 16 Input currents in the
rectifier
Fig. 17 Current in the
capacitor bank
goyal.praveen2011@gmail.com
104 R. Melicio et al.
Table 3 Average THD of the current injected in the electric grid
Case study THD
No fault 0.51
Fault with VRFB 0.54
6 Conclusions
Grid codes are important tools to mitigate threats coming into the electric grid. But
faults in wind systems are likely to disable the ability to perform within grid codes.
Worst still, due to their location, offshore wind systems may require a considerable
time of disconnection due to the maintenance and repairs needed because of a fault
going into failure. If a wind system in fault does not have the Ride through fault
capability and is not disconnected in due time, then equipment damage and negative
impacts are to be expected in the electric grid. Consequently, not only fault avoidance,
but also tolerance to faults are of crucial importance for offshore wind systems, to
avoid the need of disconnecting from the grid, to reduce the economic consequence of
being not able of injecting energy into the grid, and to remain within the requirements
of grid codes.
A model is proposed for the simulation of a wind system having a threatening
on the continuity of the operation due to a fault in the rectifier converter. The fault
creates an interruption in the energy delivery from the PMSG to the electric grid. The
simulation of the fault is in accordance with an inability to maintain energy injection
into the grid, tapping the system in a state of no recovery, i.e., the system goes from
fault into failure and disconnection is expected. Then a VRFB controlled by a BMS
is suggested as an aid to introduce Ride through fault in a strategic hardware solution
with the MPC-5LC to satisfy grid codes in what regards continuity of the operation.
The strategic hardware solution of the VRFB with the MPC 5LC must be reinforced
with a convenient selection of voltage vectors to maintain the equilibrium of the
voltages to further aid in avoiding failure of the wind system when subject to the
rectifier fault. This strategic hardware solution and the reinforcement is in the line of
the objectives of Energy 4.0. The simulation carried out shows that normal operation
is recovered after rectifier fault clearing and imbalance voltages on the capacitor bank
are circumvented due to the action of the convenient selection of voltage vectors.
Also, the quality of energy injected in what regards the total harmonic distortion is
not significantly affected.
goyal.praveen2011@gmail.com
Offshore Wind System in the Way of Energy 4.0: Ride Through Fault Aided … 105
References
1. Akhmatov, V., Knudsen, H., Nielsen, A.H.: Advanced simulation of windmills in the electric
power supply. Int. J. Electr. Power Energy Syst. 22, 421–434 (2000)
2. Arribas, B.N., Melicio, R., Teixeira, J.G., Mendes, V.M.F.: Vanadium redox flow battery storage
system linked to the electric grid. Renew. Energy Power Qual. J. 1(14), 1025–1030 (2016)
3. Banham-Hall, D.D., Taylor, G.A., Smith, C.A.: Flow batteries for enhancing wind power inte-
gration. IEEE Trans. Power Syst. 27(3), 1690–1697 (2012)
4. Batista, N.C., Melicio, R., Mendes, V.M.F.: Services enabler architecture for smart grid and
smart living services providers under industry 4.0. Energy Build. 141, 16–27 (2017)
5. Brooke, J.: Wave Energy Conversion Systems, 1st edn. Elsevier Science, UK (2003)
6. Campanile, A., Scamardella, V.P.A.: Mooring design and selection for floating offshore wind
turbines on intermediate and deep water depths. Ocean Eng. 148, 349–360 (2018)
7. Carlson, G.E., Hajlijak, C.A.: Approximation of fractional capacitors (1/s)1/nby a regular
Newton process. IEEE Trans. Circuit Theory 11, 210–213 (1964)
8. Castro-Santos, L., Diaz-Casas, V.: Economic influence of location in floating offshore wind
farms. Ocean Eng. 107, 13–22 (2015)
9. CNESA China Energy Storage Alliance: China’s top 10 storage headlines of 2016
(2017). URL http://en.cnesa.org/featured-stories/2017/1/6/chinas-top-10-storage- headlines-
of-2016/. Accessed 1 June 2017
10. Community Energy Scotland: Gigha battery project (2017). URL http://www.
communityenergyscotland.org.uk/gigha-battery-project.asp/. Accessed 22 May 2017
11. D., D.V., da Costa, J.S.: Tuning of fractional PID controllers with ziegler-nichols type rules.
Signal Process. 86, 2771–2784 (2006)
12. Divya, K.C., Østergaard, J.: Battery energy storage technology for power systems—an
overview. Electr. Power Syst. Res. 79, 511–520 (2009)
13. El-Kafafy, M., Devriendt, C., Guillaume, P., Helsen, J.: Automatic tracking of the modal param-
eters of an offshore wind turbine drivetrain system. Energies 10(4), 1–15 (2017)
14. Gomes, L.L.R., Pousinho, H.M.I., Melcio, R., Mendes, V.M.F.: Stochastic coordination of joint
wind and photovoltaic systems with energy storage in day-ahead market. Energy 124, 310–320
(2017)
15. Guarnieri, M., Mattavelli, P., Petrone, G., Spagnuolo, G.: Vanadium redox flow batteries: poten-
tials and challenges of an emerging storage technology.IEEE Indu. Electron. Mag. 10(4), 20–31
(2016)
16. Haas, J., Olivares, M.A., Palma-Behnke, R.: Grid-wide subdaily hydrologic alteration under
massive wind power penetration in Chile. J. Environ. Manag. 154, 183–189 (2015)
17. Jichuan, K., Liping, S., Hai, S., Chulin, W.: Risk assessment of floating offshore wind turbine
based on correlation-FMEA. Ocean Eng. 129, 382–388 (2017)
18. Khomfoi, S., Tolbert, L.M.: Multilevel power converters. In: Rashid, M.H. (ed.) Power Elec-
tronics Handbook, 2nd edn, pp. 451–482. Academic Press, USA (2007)
19. Lanusse, P., Malti, R., Melchior, P.: CRONE control system design toolbox for the control
engineering community: tutorial and case study. Philos. Trans. Royal Soc. A: Math. Phys. Eng.
Sci. 371, 1–14 (2013)
20. Lin, L., Kai, W., Vassalos, D.: Detecting wake performance of floating offshore wind turbine.
Ocean Eng. 156, 263–276 (2018)
21. Maione, G., Lino, P.: New tuning rules for fractional PI-alfa controllers. Nonlinear Dyn. 49,
251–257 (2007)
22. Melicio, R., Mendes, V.M.F., ao, J.P.S.C.: A pitch control malfunction analysis for wind turbines
with permanent magnet synchronous generator and full-power converters: proportional integral
versus fractional-order controllers. Electric Power Components Syst. 38, 387–406 (2010)
23. Melicio, R., Valério, D., Mendes, V.: Fractional control of an offshore wind system. In: Pro-
ceedings of The International Conference on Fractional Differentiation and its Applications,
pp. 1–6. Amman (2018)
goyal.praveen2011@gmail.com
106 R. Melicio et al.
24. Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order
controllers for industry applications. Control Eng. Pract. 16, 798–812 (2008)
25. Nanou, I.S., Papathanassiou, G.N.P.S.A.: Assessment of communication-independent grid code
compatibilitysolutions for vschvdc connected offshore wind. Electric Power Syst. Res. 121,
28–51 (2015)
26. Oustaloup, A., Levron, F., Matthieu, B., Nanot, F.M.: Frequency-band complex noninteger
differentiatot: characterization and synthesis. IEEE Trans. Circuit. Syst. I: Funda. Theory Appl.
47, 25–39 (2000)
27. Podlubny, I.: An Introduction to Fractional Derivatives, Fractional Differential Equations, to
Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)
28. Robison, R.P., Sengupta, M., Rauch, D.: Intelligent energy industrial systems 4.0. IT Profes-
sional 17, 17–24 (2015)
29. Roddier, D., Cermelli, C., Aubault, A., Weinstein, A.: Windfloat: a floating foundation for
offshore wind turbines. J. Renew. Sustain. Energy 2(3), 033,104 (2010)
30. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and
Applications. Gordon and Breach, Yverdon (1993)
31. Seixas, M., Melicio, R., Mendes, V.F.M.: Offshore wind energy system with DC transmission
discrete mass: modeling and simulation. Electr. Power Compon. Syst. 44, 2271–2284 (2016)
32. Seixas, M., Melicio, R., Mendes, V.F.M., Couto, C.: Simulation of OWES with five-level
converter linked to the grid: harmonic assessment. In: Proceedings of the 9th IEEE International
Conference on Compatibility and Power Electronics, pp. 126–131. Lisbon, Portugal (2015)
33. Seixas, M., Melicio, R., Mendes, V.M.F.: Offshore wind turbine simulation: multibody drive
train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control. Energy
69, 357–369 (2014)
34. Seixas, M., Melicio, R., Mendes, V.M.F.: Simulation of rectifier voltage malfunction on
OWECS, four-level converter, HVDC light link: smart grid context tool. Energy Convers.
Manag. 97, 140–153 (2015)
35. Seixas, M., Melicio, R., Mendes, V.M.F., Couto, C.: Blade pitch control malfunction simulation
in a wind energy conversion system with MPC five-level converter. Renew. Energy 89, 339–350
(2016)
36. Seixas, M., Mendes, V., Melicio, R.: Ride through fault on the rectifier controller of an offshore
wind system aided by VRFB. In: Proceedings of the IEEE International Symposium on Power
Electronics, Electrical Drives and Motion, pp. 925–930. Amalfi (2018)
37. Gryning, S.M.P., Wu, Q., Blanke, M., Niemann, H.H., Andersen, K.P.H.: Wind turbine inverter
robust loop-shaping control subject to grid interaction effects. IEEE Trans. Sustain. Energy 7,
41–50 (2016)
38. Valrio, D., da Costa, J.S.: An Introduction to Fractional Control. IET, Stevenage (2013)
39. Yang Z.C.Y.: A survey of fault diagnosis for onshore grid-connected converter in wind energy
conversion systems. IEEE Renew. Sustain. Energy Rev. 66, 345–359 (2016)
40. Zhao, H., Wu, Q., Wang, J., Lui, Z., Shahidehpour, M., Xue, Y.: Combined active and reactive
power control of wind farms based on model predictive control. IEEE Trans. Energy Convers.
16, 86–803 (2013)
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with
Convergence Analysis for
Time-Fractional Partial IDEs
Constrained by Neumann Conditions
Omar Abu Arqub, Mohammed Al-Smadi and Shaher Momani
Abstract Some scientific pieces of research are governed by classes of partial
integro-differential equations (PIDEs) of fractional order that are leading to novel
challenges in simulation and optimization. In this chapter, a soft numerical algorithm
is proposed and analyzed to fitted analytical solutions of PIDEs with appropriate
initial and Neumann conditions in Sobolev space. Meanwhile, the solutions are rep-
resented in series form with strictly computable components. By truncating n-term
approximation of the analytical solution, the solution methodology is discussed for
both linear and nonlinear problems based on the nonhomogeneous term. Analysis
of convergence and smoothness are given under certain assumptions to show the
theoretical structures of the method. Dynamic features of the approximate solutions
are studied through an illustrated example. The yield of numerical results indicates
the accuracy, clarity, and effectiveness of the proposed algorithm as well as provide
a proper methodology in handling such fractional issues.
Keywords Partial integro-differential equations ·Reproducing kernel algorithm ·
Fredholm and Volterra operators ·Fractional derivatives
1 Introduction
In the past years, fractional calculus theory has gained a considerable attention in
diverse fields of science and engineering according to the enormous range of real-
O. A. Arqub (B
)·S. Momani
Department of Mathematics, Faculty of Science, The University of Jordan,
Amman, 11942, Jordan
e-mail: o.abuarqub@ju.edu.jo
M. Al-Smadi
Department of Applied Science, Ajloun College, Al-Balqa Applied University,
Ajloun 26816, Jordan
S. Momani
Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman
University, Ajman, UAE
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_7
107
goyal.praveen2011@gmail.com
108 O. A. Arqub et al.
world applications and the critical role that it plays to describe the complex dynam-
ical behaviors to such models including fluid dynamic model, traffic flow model,
convection-diffusion model, heat flux model, and so forth [1–4]. As well as, this
topic aids to simplify the controlling design without any shortage of hereditary. The
derivatives of fractional order are powerful to interpret several physical problems,
for instance, electrical circuits, damping laws, and controlled damper. Further, frac-
tional partial differential equations (FPDEs) are constructed due to attention among
the scientists and engineers to exact explanation of nonlinear phenomena which
appear, for example, in fluid mechanics wherever continuum assumption does not
well, and therefore fractional model can be deemed to be the best operator [5–10].
Developing numeric-analytic techniques to the solutions of PIDEs of fractional order
is an essential task. Since it is challenging to get closed-form solutions to fractional
PIDEs in many situations. So a lot of efforts have been made to introduce and improve
analytical techniques that help us to obtain the analytic solution of those fractional
differential equations [11–17].
The reproducing kernel (RK) technique is a well-known systematic approach
in obtaining a feasible solution of both linear and nonlinear differential or integral
operators involving ordinary differential, partial differential, integral and integro-
differential, fractional differential, fuzzy differential, and delay differential equa-
tions [18–30]. The RKA is a superb, suitable, and useful tool to provide accurate
and appropriate algorithms for numeric simulations to natural phenomena arising
in physics, chemistry, biology, ecology, and engineering such as diffusive transport,
viscoelastic materials, fluid rheology, intelligent transportation systems, electromag-
netic theory, and probability [31–46]. Inspired by the areas mentioned above, the RK
method has been successfully implemented and utilized directly in solving nonlinear
initial or boundary value problems without the need for unphysically tied hypotheses,
linearization, transformations, discretization, or even perturbation.
In this chapter, we display a soft numerical algorithm, called reproducing kernel,
for handling classes of time-fractional PIDEs restricted by initial and Neumann
functions as follows:
∂α
ξαϕ(η,ξ)+μ1∂2
η2ϕ(η,ξ)+μ2∂ηϕ(η,ξ)+μ3ϕ(η,ξ)+P[ϕ(η,ξ)]+Q[ϕ(η,ξ)]=F(η,ξ),
P[ϕ(η,ξ)]=λ11
0
k1(η,ξ,ρ)μ4∂2
η2ϕ(η,ρ)+μ5∂ηϕ(η,ρ)+μ6ϕ(η,ρ)dρ,
Q[ϕ(η,ξ)]=λ2η
0
k2(η,ξ,ρ)μ7∂2
η2ϕ(η,ρ)+μ8∂ηϕ(η,ρ)+μ9ϕ(η,ρ)dρ,
(1)
with the initial and Neumann conditions
ϕ(η,0)=ω(η),
∂ηϕ(0,ξ)=ν1(ξ), ∂ηϕ(1,ξ)=ν2(ξ), (2)
where 0 <α≤1,0≤η,ξ,ρ≤1,μi,i=1,2, ..., 9 are real finite constant, λ1and
λ2are constant parameters, k1(η,ξ,ρ)and k2(η,ξ,ρ)are arbitrary continuous ker-
nel functions on the cube [0,1]3,ω(η), ν1(ξ),ν2(ξ)and F(η,ξ)are continuous
functions over the required domain, and ϕ(η,ρ)is an analytical solution to be found
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 109
numerically. Hereby, we assume that Eq. (1) with conditions (2) has a unique smooth
solution. Further, ∂α
ξαdenotes the time-fractional Caputo derivative of order α, which
is given by
∂α
ξαϕ(η,ξ)=1
(1−α)
ξ
0
(ξ−τ)−α∂τϕ(η,τ)dτ.(3)
The rest of this chapter is organized as follows. In Sect. 2, the Hilbert spaces
required are extended, as well the reproducing kernel functions are provided. The
issue is formulated and the computational RK algorithm is presented in Sect. 3.
Meanwhile, some necessary theoretical results and convergent validity are studied
in the same section. In Sect. 4, numerical results are discussed to show the reliability
and efficiency of the proposed algorithm. Finally, this chapter ends with conclusions.
2 Preliminary Spaces
For clarity of presentation, some necessary definitions and preliminary facts are pre-
sented. Henceforth, ||u||2
=u(η),u(η),u∈, η∈[0,1],and is a Hilbert
space, and L2[0,1]=r|1
0r2(ρ)dρ<∞.
Definition 1 [17]Ifis a Hilbert space defined on a nonempty set , then F:
×→Ris called a reproducing kernel function (RKF) of the space when
both two conditions are met:
1. For each η∈, we have F(ρ,η)∈.
2. For each ψ∈and each η∈, we have ψ(ρ),F(ρ,η)=ψ(η).(Reprod−
ucing Property)
The space that possesses a reproducing kernel is said to be reproducing kernel
Hilbert space. The RKF Fof the space completely determines the Hilbert space
.
Next, the required Hilbert spaces rH1
2[0,1]and dH2
2[0,1]will be defined, which
are possessing RKFs rR{1}
η(ξ)and dR{2}
η(ξ),respectively.
Definition 2 [17]Letr(η)be in the space L2[0,1].The space rH1
2[0,1]given by
rH1
2[0,1]={r=r(η):ris absolutely continuous over [0,1]}. The inner product
is equipped by
r1(η), r2(η)rH1
2=r1(0)r2(0)+
1
0
r
1(ρ)r
1(ρ)dρ.(4)
Remark 1 The space rH1
2[0,1]is complete reproducing kernel and the RKF is
obtained as follows:
rR{1}
η(ξ)=1+mim{η,ξ}.(5)
goyal.praveen2011@gmail.com
110 O. A. Arqub et al.
Anyhow, if [0,1]is the desired domain in direction of τ, then the complete
reproducing kernel space rH1
2[0,1]can be defined. Further, the inner product is
equipped by
r1(τ), r2(τ)rH1
2=r1(0)r2(0)+
1
0
r
1(ρ)r
1(ρ)dρ.
The kernel function is rR{1}
τ(ξ)=1+mim{τ,ξ}.
Definition 3 Let r(η)be in the space L2[0,1].The space dH2
2[0,1]is given by
dH2
2[0,1]={r=r(η):r,rare absolutely continuous over [0,1], and r(0)=0}.
The inner product is equipped by
r1(η), r2(η)dH2
2=
2
i=0
r(i)
1(0)r(i)
2(0)+
1
0
r
1(ρ)r
2(ρ)dρ.(6)
Remark 2 The space dH2
2[0,1]is complete reproducing kernel and the RKF is
obtained as follows:
dR{2}
η(ξ)=1
12 hη(ξ), η≤ξ,
hξ(η), η>ξ,(7)
where hη(ξ)=12ηξ +6ηξ2−2ξ3.
Definition 4 Let r(η)be in the space L2[0,1].The space dH3
2[0,1]is given by
dH3
2[0,1]={r=r(η):r,r,r are absolutely continuous over [0,1], and r(0)=
r(1)=0}. The inner product is equipped by
r1(η), r2(η)dH3
2=
1
i=0
r(i)
1(0)r(i)
2(0)+r1(1)r2(1)+
1
0
r
1(ρ)r
2(ρ)dρ.(8)
Remark 3 The space dH3
2[0,1]is complete reproducing kernel, and the RKF is
obtained as follows:
dR{3}
η(ξ)=1
120 (1−ξ)3η3gη(ξ), η≤ξ,
(1−η)3ξ3gξ(η), η>ξ,(9)
where gη(ξ)=6η2ξ2+3ηξ(ξ−5η)+(10η2−5ηξ +ξ2).
Next, to extend the novel inner product spaces and to fit its reproducing kernel
functions, we construct a reproducing kernel space W(), =[0,1]⊗[0,1],in
which every function satisfies the constraints homogeneous initial and Neumann
conditions of the above-mentioned time-fractional PIDEs.
Definition 5 Let be ∂3
η3∂3
ξ3ϕin the space L2(). The Hilbert space W() is defined
as W() ={ϕ=ϕ(η,ξ):∂2
η2∂2
ξ2ϕare complete continuous functions in , and
ϕ(η,0)=∂ηϕ(0,ξ)=∂ηϕ(1,ξ)=0}. The metric system structure lies in
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 111
ϕ1(η,ξ), ϕ2(η,ξ)W=
1
i=0∂i
ξiϕ1(η,0),∂i
ξiϕ2(η,0)dH3
2
+
1
01
i=0
∂2
ξ2∂i
ηiϕ1(0,ξ)∂2
ξ2∂i
ηiϕ2(0,ξ)+∂2
ξ2ϕ1(1,ξ)∂2
ξ2ϕ2(1,ξ)dξ(10)
+
1
0
1
0
∂3
ξ3∂2
η2ϕ1(η,ξ)∂3
η3∂2
ξ2ϕ2(η,ξ)dηdξ.
Theorem 1 The space W () is a complete reproducing kernel, and its RKF is
defined by
RW
(x,t)(η,ξ)=dR{3}
x(η)dR{2}
t(ξ),(11)
where the functions dR{2}
t(ξ)and dR{3}
x(η)are RKFs of dH2
2[0,1]and dH3
2[0,1],
respectively.
Proof By utilizing the features of r1(η), r2(η)dH2
2and r1(η), r2(η)dH3
2,it follows
that
ϕ(η,ξ),dR{3}
x(η)dR{2}
t(ξ)W=
1
i=0∂i
ξiϕ(η,0),∂i
ξidR{3}
x(η)dR{2}
t(0)dH3
2
+
1
0
⎡
⎢
⎢
⎣
1
i=0
∂2
ξ2∂i
ηiϕ(0,ξ)∂2
ξ2∂i
ηidR{3}
x(0)dR{2}
t(ξ)
+∂2
ξ2ϕ(1,ξ)∂2
ξ2dR{3}
x(1)dR{2}
t(ξ)⎤
⎥
⎥
⎦dξ
+
1
0
1
0
∂3
ξ3∂2
η2ϕ(η,ξ)∂3
η3∂2
ξ2dR{3}
x(η)dR{2}
t(ξ)dηdξ
=
1
i=0∂i
ξiϕ(η,0),dR{3}
x(η)∂i
ξidR{2}
t(0)dH3
2
+
1
0
⎡
⎢
⎢
⎣
1
i=0
∂2
ξ2∂i
ηiϕ(0,ξ)∂2
ξ2dR{2}
t(ξ)∂i
ηidR{3}
x(0)
+∂2
ξ2ϕ(1,ξ)dR{3}
x(1)∂2
ξ2dR{2}
t(ξ)
⎤
⎥
⎥
⎦dξ
+
1
01
0
∂3
ξ3∂2
η2ϕ(η,ξ)∂3
η3dR{3}
x(η)∂2
ξ2dR{2}
t(ξ)dηdξ
=
1
i=0∂i
ξiϕ(η,0),dR{3}
x(η)dH3
2
∂i
ξidR{2}
t(0)
+
1
0
∂2
ξ2dR{2}
t(ξ)∂2
ξ2⎡
⎢
⎢
⎣
1
i=0
∂i
ηiϕ(0,ξ)∂i
ηidR{3}
x(0)+ϕ(1,ξ)dR{3}
x(1)
+1
0∂3
ξ3ϕ(η,ξ)∂3
η3dR{3}
x(η)
⎤
⎥
⎥
⎦dξ
=
1
i=0
∂i
ξiϕ(x,0)∂i
ξidR{2}
t(0)+
1
0
∂2
ξ2dR{2}
t(ξ)∂2
ξ2ϕ(η,ξ),dR{3}
x(η)dH3
2
dξ
=
1
i=0
∂i
ξiϕ(x,0)∂i
ξidR{2}
t(0)+
1
0
∂2
ξ2dR{2}
t(ξ)∂2
ξ2ϕ(x,ξ)dξ
=ϕ(x,ξ),dR{2}
t(ξ)dH3
2
=ϕ(x,t).
Hence, ϕ(η,ξ), R(x,t)(η,ξ)W=ϕ(x,t).
goyal.praveen2011@gmail.com
112 O. A. Arqub et al.
Definition 6 Let be ∂η∂ξϕin the space L2(). The space H() is given by
H() ={ϕ=ϕ(η,ξ):ϕis continuous functions on }, and the metric space can
be obtained by
ϕ1(η,ξ), ϕ2(η,ξ)H=ϕ1(η,0),ϕ2(η,0)dH2
2(12)
+
1
0
∂ξϕ1(0,ξ)∂ξϕ2(0,ξ)dξ+
1
0
1
0
∂2
ηξϕ1(η,ξ)∂2
ηξϕ2(η,ξ)dηdξ.
Theorem 2 The space H []is complete reproducing kernel, and the RKF is
RH
(x,t)(η,ξ)=rR{1}
τ(ξ)rR{1}
η(ξ),(13)
where the functions rR{1}
η(ξ)and rR{1}
τ(ξ)are RKFs of rH1
2[0,1]and rH1
2[0,1],
respectively.
3 Numerical RK Algorithm
In this section, the statement of PIDEs (1) and (2) is being redrafted in Hilbert
space W(). After homogenizing the inhomogeneous restriction conditions using
appropriate transformation, the differential operator T:W() →H() of frac-
tional order αcan be defined,which is invertible, linear, and bounded, such that
Tϕ(η,ξ)=∂α
ξαϕ(η,ξ)+μ1∂2
η2ϕ(η,ξ)+μ2∂ηϕ(η,ξ)+μ3ϕ(η,ξ)+P[ϕ(η,ξ)]+Q[ϕ(η,ξ)].
Therefore, the original FPIED statement will be converted equivalently into the
following form:
Tϕ(η,ξ)=F(η,ξ),(14)
with respect to the homogeneous initial and Neumann conditions
ϕ(η,0)=0,
∂ηϕ(0,ξ)=0,∂ηϕ(1,ξ)=0.(15)
The orthogonal function systems of W() can be constructed by choosing a count-
able dense set {(ηi,ξi)}∞
i=1of , and defining ωi(η,ξ)=RH
(ηi,ξi)(η,ξ)and ψi(η,ξ)=
T∗ωi(η,ξ), in which T∗is the adjoint operator of Tsuch tah T∗:H()→W().
While, the orthonormal basis ¯
ψi(η,ξ)∞
i=1of the space W()can be obtained using
the procedures of G-Schmidt normalization to {ψi(η,ξ)}∞
i=1such that
ψi(η,ξ)=
i
k=1
βikψk(η,ξ).(16)
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 113
Theorem 3 The system {ψi(η,ξ)}∞
i=1is complete orthogonal basis of the space
W() as follows:
ψi(η,ξ)=T(x,t)RW
(x,t)(η,ξ)(x,t)=(ηi,ξi),
where T(x,t)indicates that the operator T applies to the function of (x,t).
Proof Clearly that
ψi(η,ξ)=T∗ωi(η,ξ)=T∗ωi(x,t),RW
(η,ξ)(x,t)W
=ωi(x,t),T(x,t)RW
(η,ξ)(x,t)H
=T(η,ξ)RW
(η,ξ)(x,t)(x,t)=(ηi,ξi)(17)
=T(x,t)RW
(x,t)(η,ξ)(x,t)=(ηi,ξi)∈W().
Consequently, for each fixed ϕ∈W(),letϕ(η,ξ), ψi(η,ξ)W=0, i=1,2, ....
Then, ϕ(η,ξ), ψi(η,ξ)W=ϕ(η,ξ), T∗ωi(η,ξ)W=Tϕ(η,ξ), ϕi(x)H=
Tϕ(ηi,ξi)=0. Since {(ηi,ξi)}∞
i=1is dense on , therefore Lϕ(η,ξ)=0. It follows
that ϕ(η,ξ)=0 by applying T−1ϕ.
Remark 4 The sequence RW
(xi,ti)(η,ξ)∞
i=1is linear independent basis on W().
Theorem 4 Suppose that Am=m
k=1βik F(ηk,ξk). Let ϕ(η,ξ)∈W()be the the
analytical solution of Eqs. (14)and (15), then it has the following form:
ϕ(η,ξ)=
∞
m=1
Am¯
ψm(η,ξ).(18)
Proof By utilizing the features of ϕ1(η,ξ), ϕ2(η,ξ)W,itfollowsthat
ϕ(η,ξ)=
∞
m=1ϕ(η,ξ), ¯
ψm(η,ξ)W¯
ψm(η,ξ)
=
∞
m=1
m
k=1
βmk ϕ(η,ξ), ψk(η,ξ)W¯
ψm(η,ξ)
=
∞
m=1
m
k=1
βmk ϕ(η,ξ), T∗ωk(η,ξ)W¯
ψm(η,ξ)
=
∞
m=1
m
k=1
βmk Tϕ(η,ξ), ωk(η,ξ)H¯
ψm(η,ξ)
=
∞
m=1
m
k=1
βmk F(η,ξ), ωk(η,ξ)H¯
ψm(η,ξ)
=
∞
m=1
m
k=1
βmk F(ηk,ξk)¯
ψm(η,ξ)
=
∞
m=1
Am¯
ψm(η,ξ).
goyal.praveen2011@gmail.com
114 O. A. Arqub et al.
Remark 5 The n-term approximate solution of the analytical solution described in
Eq. (18) can be given by
ϕn(η,ξ)=
n
i=1
i
k=1
βik F(ηk,ξk)¯
ψi(η,ξ).(19)
According to the proposed algorithm, the required domain can be divided into
finite r×sgrid points with respect to ηspace-direction, η=1
r, and to ξtime-
direction, ξ=1
sover [0,1], respectively, r,s∈N,r,s>0. Anyhow, the ordered
pair (ηl,ξm)of can be given simultaneously by
(ηl,ξm)=(lη,mξ),l=0,1, ..., r,m=0,1, ..., s.
The following computational RK algorithm is given to summarize the procedures
of the proposed method in solving those time-fractional PIDEs.
Algorithm 1 To obtain approximate solution ϕn(η,ξ)of the analytical solution
ϕ(η,ξ)of BVPs (14)and (15)on , the following steps can be carried out:
Step A: Divide the required domain into grid points such as n=rs,r,s∈N,
r,s>0;
Step B: Put ψn(ηn,ξn)=TRH
(x,t)(η,ξ)(x,t)=(ηn,ξn);
Step C: Find orthonormal coefficients βnk,then let ψn(ηn,ξn)=n
k=1βnkψn
(ηn,ξn),n=1,2, ...,rs;
Step D: Ues the initial data ϕ0(η1,ξ1);then do the following subroutine:
For n=1, n++;
Let An=n
k=1βnk F(ηk,ξk);
Set ϕn(ηn,ξn)=n
k=1n
j=1Akψk(ηk,ξk);
Step E: If n<rs, then let n=n+1 and go to Step D; Otherwise, Stop.
Theorem 5 Let ||ϕ(η,ξ)||Wbe bounded on , then the n-term approximate solution
ϕn(η,ξ)in Eq. (19)converges to the analytical solution ϕ(η,ξ)of Eqs. (14)and (15)
in W () that is given as ϕ(η,ξ)=
∞
i=1
i
k=1
βik F(ηk,ξk)¯
ψi(η,ξ).
Proof Let δn=||ϕ(η,ξ)−ϕn(η,ξ)||Wthe nature error at (η,ξ)∈,
then δ2
n−1=
∞
i=n
i
k=1
βik F(ηk,ξk)¯
ψi(η,ξ)
2
W
=
∞
i=ni
k=1βik F(ηk,ξk)2
and δ2
n=
∞
i=n+1
i
k=1
βik F(ηk,ξk)¯
ψi(η,ξ)
2
W
=
∞
i=ni
k=1βik F(ηk,ξk)2
. Thus, δn−1≥δn.
Consequently {δn}∞
n=1is decreasing with respect to the norm of W().If
∞
i=1
Ai¯
ψi
(η,ξ)is convergent, then, ||ϕ(η,ξ)−ϕn(η,ξ)||W→0 as soon as n→∞.
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 115
4 Numerically Gained Results
In this section, some examples are quantitatively discussed at certain grid points on
to demonstrate the ability and performance of the proposed method in solving
those fractional PIDEs. For computation, all symbolic and numerical calculations
are performed using the Mathematica 9.0.
Example 1 We consider the linear time-fractional PIDE in the following form:
∂α
ξαϕ(η,ξ)+∂2
η2ϕ(η,ξ)−∂ηϕ(η,ξ)+xϕ(η,ξ)+P[ϕ(η,ξ)]−Q[ϕ(η,ξ)]=F(η,ξ),
P[ϕ(η,ξ)]=1
0
(ξ−ρ)eη+ρ∂2
η2ϕ(η,ρ)+∂ηϕ(η,ρ)+ϕ(η,ρ)dρ,
Q[ϕ(η,ξ)]=η
0
ρα+1eη−ξ∂2
η2ϕ(η,ρ)+3∂ηϕ(η,ρ)+4ϕ(η,ρ)dρ,
(20)
with the initial and Neumann conditions
ϕ(η,0)=0,
∂ηϕ(0,ξ)=e−1ξ2α,∂ηϕ(1,ξ)=2ξα+1(ξα−1+1), (21)
where 0 ≤η,ξ≤1, 0 <α≤1 and F(η,ξ)are given functions such that the selected
solution will be satisfied in the both left- and right-hand sides of BVPs. Equations (20)
and (21) over the domain . Here, the exact solution is ϕ(η,ξ)=ηeη−1ξ2α+η2ξα+1.
Example 2 Consider the following nonlinear time-fractional PIDE:
∂α
ξαϕ(η,ξ)−∂2
η2ϕ(η,ξ)+eηϕ3(η,ξ)+sin (η)ϕ2(η,ξ)+∂ηϕ(η,ξ)−P[ϕ(η,ξ)]−Q[ϕ(η,ξ)]=F(η,ξ),
P[ϕ(η,ξ)]=1
0
(ξ+η)eηρα∂2
η2ϕ(η,ρ)+∂ηϕ(η,ρ)+ϕ(η,ρ)dρ,
Q[ϕ(η,ξ)]=η
0
sin (ηξ)∂2
η2ϕ(η,ρ)−∂ηϕ(η,ρ)−ϕ(η,ρ)dρ,
(22)
with the initial and Neumann conditions
ϕ(η,0)=0,
∂ηϕ(0,ξ)=ξ3α+1,∂ηϕ(1,ξ)=ln (0.5)ξ3α+1,(23)
where 0 ≤η,ξ≤1, 0 <α≤1 and F(η,ξ)is given function such that the selected
solution will be satisfied in the both left- and right-hand sides of BVPs (20) and
(21) over the domain . Here, the exact solution is ϕ(η,ξ)=(1−η)ln (1+η)
cos2(3πη)ξ3α+1.
To demonstrate the effectiveness of the RK solutions, the examples above are
tested across the domain . Anyhow, results from numerical analysis are an approx-
imation, in general, which can be made as accurate as desired. Because a computer has
a finite word length, only a fixed number of digits are stored and used during computa-
tions. Following, absolute errors of approximate solution ϕn(η,ξ)for both Examples
goyal.praveen2011@gmail.com
116 O. A. Arqub et al.
Table 1 Absolute errors of Example 1 over
x t α=0.25 α=0.5α=0.75 α=1
0 0.25 1.5202955220 ×10−78.5851595338 ×10−79.3131586223 ×10−71.6088074877 ×10−8
0.5 3.1310929434 ×10−62.9629678339 ×10−74.4292618364 ×10−79.7271247494 ×10−7
0.75 4.0499972764 ×10−61.6829336033 ×10−68.1459842130 ×10−65.9375835113 ×10−7
1 8.9639505999 ×10−69.9022004243 ×10−62.3384483768 ×10−68.1051406404 ×10−6
0.25 0.25 8.2373427553 ×10−76.5983172765 ×10−79.3054980205 ×10−77.6310676740 ×10−7
0.5 2.7232898676 ×10−61.0693336383 ×10−73.9973040691 ×10−77.7728964454 ×10−7
0.75 3.2714639390 ×10−63.4265168332 ×10−68.9924037150 ×10−68.1729535402 ×10−6
1 1.8522344410 ×10−68.6625479440 ×10−69.7462801838 ×10−61.1749765902 ×10−6
0.5 0.25 4.5870682134 ×10−73.1634826495 ×10−72.7065700155 ×10−75.9471694210 ×10−8
0.5 9.9056258776 ×10−64.4738559389 ×10−64.6166202647 ×10−76.8824054303 ×10−7
0.75 3.0618500642 ×10−66.7163511298 ×10−66.9702799521 ×10−63.8457050803 ×10−7
1 2.7560569249 ×10−64.2505438452 ×10−68.1256012945 ×10−68.2173135294 ×10−6
0.75 0.25 1.7835134763 ×10−77.3344192438 ×10−73.6082013498 ×10−76.2822299556 ×10−8
0.5 3.2605016445 ×10−63.4355177778 ×10−75.8928801842 ×10−78.8955534782 ×10−7
0.75 1.9684148538 ×10−68.5361825235 ×10−65.8141127051 ×10−63.1743929263 ×10−7
1 5.3327573861 ×10−66.2303305719 ×10−62.0146526217 ×10−68.0470697269 ×10−6
1 0.25 4.8687548990 ×10−71.4518234026 ×10−77.3231124692 ×10−76.7200224276 ×10−7
0.5 5.3763575915 ×10−68.8579046722 ×10−62.5430060005 ×10−77.4596696195 ×10−7
0.75 5.9469042577 ×10−67.7876144332 ×10−66.8853386269 ×10−63.6563750502 ×10−6
1 1.3313627925 ×10−68.9735110839 ×10−62.2569043006 ×10−64.8476533804 ×10−6
Table 2 Absolute errors of Example 2 over
x t α=0.25 α=0.5α=0.75 α=1
0 0.25 7.9776844684 ×10−73.4840685648 ×10−72.0885121979 ×10−74.1424682191 ×10−8
0.5 8.3783958819 ×10−74.1528820307 ×10−77.1702389557 ×10−86.1668778742 ×10−8
0.75 3.3967362498 ×10−76.6991474431 ×10−78.4884903069 ×10−72.0811339500 ×10−7
1 5.3104194940 ×10−65.2032473529 ×10−71.8581559833 ×10−75.1027116115 ×10−8
0.25 0.25 7.7999626080 ×10−72.4640451071 ×10−72.8311662609 ×10−78.6689276035 ×10−8
0.5 8.3989967791 ×10−75.7091389550 ×10−74.8072774473 ×10−75.2358140673 ×10−7
0.75 6.9639358686 ×10−72.8087915523 ×10−77.0935196083 ×10−76.6709448740 ×10−8
1 5.1420418936 ×10−64.0159489813 ×10−72.2581468314 ×10−73.2212477612 ×10−7
0.5 0.25 6.9540160215 ×10−75.6446045032 ×10−77.8194176413 ×10−81.3560757675 ×10−8
0.5 1.8873258794 ×10−78.7256088711 ×10−77.3103863842 ×10−76.1796585570 ×10−7
0.75 9.3946809462 ×10−75.5063241201 ×10−73.0104629974 ×10−71.7518180465 ×10−8
1 9.4341574440 ×10−69.3968995759 ×10−78.0664793597 ×10−76.1269428722 ×10−8
0.75 0.25 4.7458237332 ×10−72.7794343641 ×10−71.2112309375 ×10−74.8362338856 ×10−8
0.5 3.4525853552 ×10−75.0629795416 ×10−74.0379454165 ×10−79.4477540589 ×10−7
0.75 6.6148187244 ×10−72.9964614478 ×10−72.2900033128 ×10−78.9941236417 ×10−8
1 9.8926059525 ×10−62.6698929941 ×10−76.7181321202 ×10−81.1564777914 ×10−8
1 0.25 2.8313095737 ×10−76.6972801239 ×10−77.4614502464 ×10−73.0846376150 ×10−8
0.5 6.5203528576 ×10−75.8144559231 ×10−74.3056435055 ×10−77.5776950397 ×10−8
0.75 3.2788169448 ×10−76.3469459479 ×10−76.2841762561 ×10−77.8177679269 ×10−7
1 4.9070667887 ×10−64.9990426724 ×10−79.2211292620 ×10−76.7229592233 ×10−7
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 117
1and 2for different values (η,ξ)in with step-size 0.25 are listed in Tables 1and 2,
respectively. From these tables, it can be observed that the error estimate confirm the
accuracy of numeric results related to fill time ξm=mξ,m=0,1, ...,r,ξ=1
r
and distance ηl=lη,l=0,1, ...,s,η=1
s. Hence, more accurate numeric solu-
tions can be found by utilizing more grid points (η,ξ).
5 Concluding Remarks
In this chapter, the RKM has been applied to obtain approximate solutions for both
linear and nonlinear PIDEs of fractional order. The fractional derivative has been
described in the Caputo sense. Two examples have been tested to show the efficiency
of the proposed method. By comparing our results with the exact solution for integer
and non-integer orders derivative, one can observe that the proposed method yields
accurate approximations. This adaptive can be used as an alternative technique in
solving several nonlinear partial fractional problems arising in diverse engineering,
chemistry, biology, and physical sciences.
References
1. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press,
London, UK (2010)
2. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press (2005)
3. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA, USA (1999)
4. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and
Applications. Gordon and Breach, New York (1993)
5. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional DifferentialEqua-
tions. Elsevier, Amsterdam, Netherlands (2006)
6. Arshed, S.: B-spline solution of fractional integro partial differential equation with a weakly
singular kernel. Numer. Methods Part. Differe. Equ. (2017). In Press. https://doi.org/10.1002/
num.22153.
7. Rostami, Y., Maleknejad, K.: Numerical solution of partial integro-differential equations by
using projection method. Mediterranean J. Math. 14, 113 (2017). https://doi.org/10.1007/
s00009-017- 0904-z
8. Huang, L., Li, X.F., Zhao, Y., Duan, X.Y.: Approximate solution of fractional integro-
differential equations by Taylor expansion method. Comput. Math. Appl. 62, 1127–1134 (2011)
9. Mohammed, D.S.: Numerical solution of fractional integro-differential equations by least
squares method and shifted Chebyshev polynomial. Math. Problems Eng. 2014, Article ID
431965, 5 p. (2014). https://doi.org/10.1155/2014/431965
10. Momani, S., Qaralleh, R.: An efficient method for solving systems of fractional integro-
differential equations. Comput. Math. Appl. 52, 459–470 (2006)
11. Tohidi, E., Ezadkhah, M.M., Shateyi, S.: Numerical solution of nonlinear fractional Volterra
integro-differential equations via Bernoulli polynomials. Abstract Appl. Anal. 2014, Article
ID 162896, 7 p. (2014). https://doi.org/10.1155/2014/162896
12. Wang, Y., Zhu, L.: Solving nonlinear Volterra integro-differential equations of fractional order
by using Euler wavelet method. Adv. Differ. Equ. 2017, 27 (2017). https://doi.org/10.1186/
s13662-017- 1085-6
goyal.praveen2011@gmail.com
118 O. A. Arqub et al.
13. Abu Arqub, O., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions
for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293,
385–399 (2015)
14. El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A.: A novel expansion iterative
method for solving linear partial differential equations of fractional order. Appl. Math. Comput.
257, 119–133 (2015)
15. El-Ajou, A., Abu Arqub, O., Momani, S.: Approximate analytical solution of the nonlinear
fractional KdV-Burgers equation: A new iterative algorithm. J. Comput. Phys. 293, 81–95
(2015)
16. Ray, S.S.: New exact solutions of nonlinear fractional acoustic wave equations in ultrasound.
Comput. Math. Appl. 71, 859–868 (2016)
17. Ortigueira, M.D., Machado, J.A.T.: Fractional signal processing and applications. Signal Pro-
cess 83, 2285–2286 (2003)
18. Zaremba, S.: L’equation biharminique et une class remarquable defonctionsfoundamentals
harmoniques. Bulletin International de l’Academie des Sciences de Cracovie 39, 147–196
(1907)
19. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
20. Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Sci-
ence, New York, NY, USA (2009)
21. Al-Smadi, M.: Simplified iterative reproducing kernel method for handling time-fractional
BVPs with error estimation. Ain Shams Eng. J. 9(4), 2517–2525 (2018)
22. Daniel, A.: Reproducing Kernel Spaces and Applications. Springer, Basel, Switzerland (2003)
23. Weinert, H.L.: Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Process-
ing, Hutchinson Ross (1982)
24. Lin, Y., Cui, M., Yang, L.: Representation of the exact solution for a kind of nonlinear partial
differential equations. Appl. Math. Lett. 19, 808–813 (2006)
25. Zhoua, Y., Cui, M., Lin, Y.: Numerical algorithm for parabolic problems with non-classical
conditions. J. Comput. Appl. Math. 230, 770–780 (2009)
26. Abu Arqub, O.: Fitted reproducing kernel Hilbert space method for the solutions of some
certain classes of time-fractional partial differential equations subject to initial and Neumann
boundary conditions. Comput. Math. Appl. 73, 1243–1261 (2017)
27. Abu Arqub, O., Shawagfeh, N.: Application of reproducing kernelalgorithm for solving Dirich-
let time-fractional diffusion-Gordon types equations in porous media. J. Porous Media (2017).
In Press
28. Abu Arqub, O., Rashaideh, H.: The RKHS method for numerical treatment for integrodiffer-
ential algebraic systems of temporal two-point BVPs. Neural Comput. Appl., 1–12 (2017).
https://doi.org/10.1007/s00521-017- 2845-7
29. Abu Arqub, O.: The reproducing kernel algorithm for handling differential algebraic systems
of ordinary differential equations. Math. Methods Appl. Sc. 39, 4549–4562 (2016)
30. Abu Arqub, O., Al-Smadi, M., Shawagfeh, N.: Solving Fredholm integro-differential equations
using reproducing kernel Hilbert space method. Appl. Math. Comput. 219, 8938–8948 (2013)
31. Abu Arqub, O., Al-Smadi, M.: Numerical algorithm for solving two-point, second-order peri-
odic boundary value problems for mixed integro-differential equations. Appl. Math. Comput.
243, 911–922 (2014)
32. Momani, S., Abu Arqub, O., Hayat, T., Al-Sulami, H.: A computational method for solving
periodic boundary value problems for integro-differential equations of Fredholm-Voltera type.
Appl. Math. Comput. 240, 229–239 (2014)
33. Abu Arqub, O., Al-Smadi, M., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential
equations using reproducing kernel Hilbert space method. Soft Comput. 20, 3283–3302 (2016)
34. Abu Arqub, O., Al-Smadi, M., Momani, S., Hayat, T. : Application of reproducing kernel
algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput.,
1–16 (2016). https://doi.org/10.1007/s00500-016- 2262-3
35. Abu Arqub, O.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-
Volterra integrodifferential equations. Neural Comput. Appl., 1–20 (2015). https://doi.org/
10.1007/s00521-015-2110- x
goyal.praveen2011@gmail.com
Soft Numerical Algorithm with Convergence Analysis for Time-Fractional … 119
36. Abu Arqub, O.: Approximate solutions of DASs with nonclassical boundary conditions using
novel reproducing kernel algorithm. Fundamenta Informaticae 146, 231–254 (2016)
37. Abu Arqub, O., Integral-initial, Z., Al-Smadi, M.: Numerical solutions of time-fractional partial
integrodifferential equations of Robin functions types in Hilbert space with error bounds and
error estimates. Nonlinear Dyn. 94(3), 1819–1834 (2018)
38. Abu Arqub, O.: Numerical solutions for the Robin time-fractional partial differential equations
of heat and fluid flows based on the reproducing kernel algorithm. Int. J. Numer. Methods Heat
Fluid Flow (2017). https://doi.org/10.1108/HFF- 07-2016-0278
39. Geng, F.Z., Qian, S.P.: Reproducing kernel method for singularly perturbed turning point
problems having twin boundary layers. Appl. Math. Lett. 26, 998–1004 (2013)
40. Jiang, W., Chen, Z.: A collocation method based on reproducing kernel for a modified anoma-
lous subdiffusion equation. Numer. Methods Part. Differ. Equ. 30, 289–300 (2014)
41. Geng, F.Z., Qian, S.P., Li, S.: A numerical method for singularly perturbed turning point
problems with an interior layer. J. Comput. Appl. Math. 255, 97–105 (2014)
42. Al-Smadi, M., Abu Arqub, O., Shawagfeh, N., Momani, S.: Numerical investigations for sys-
tems of second-order periodic boundary value problems using reproducing kernel method.
Appl. Math. Comput. 291, 137–148 (2016)
43. Jiang, W., Chen, Z.: Solving a system of linear Volterra integral equations using the new
reproducing kernel method. Appl. Math. Comput. 219, 10225–10230 (2013)
44. Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary
value problems with a delay. Appl. Math. Model. 39, 5592–5597 (2015)
45. Al-Smadi, M., Abu Arqub, O.: Computational algorithm for solving fredholm time-fractional
partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math.
Comput. 342, 280–294 (2019)
46. Abu Arqub, O., Al-Smadi, M.: Atangana-Baleanu fractional approach to the solutions of
Bagley-Torvik and Painlevé equations in Hilbert space. Chaos Solitons Fractals 117, 161–167
(2018)
goyal.praveen2011@gmail.com
Approximation of Fractional-Order
Operators
Reyad El-Khazali, Iqbal M. Batiha and Shaher Momani
Abstract In order to deal with some difficult problems in fractional-order sys-
tems, like computing analytical time responses such as unit impulse and step
responses; some rational approximations for the fractional-order operators are pre-
sented with satisfying results in simulation and realization. In this chapter, several
comparisons in the time response and Bode results between four well-known meth-
ods; Oustaloup’s method, Matsuda’s method, AbdelAty’s method, and El-Khazali’s
method are made for the rational approximation of fractional-order operator (frac-
tional Laplace operator). The various methods along with their advantages and limi-
tations are described in this chapter. Simulation results are shown for different orders
of the fractional operator. It has been shown in several numerical examples that the El-
Khazali’s method is very successful in comparison with Oustaloup’s, Matsuda’s, and
AbdelAty’s methods.
Keywords Fractional-Order models ·Oustaloup’s approximation ·Matsuda’s
approximation ·AbdelAty’s approximation ·El-Khazali’s approximation
R. El-Khazali (B
)
ECSE Department, Khalifa University, Abu Dhabi 127788, UAE
e-mail: reyad.elkhazali@ku.ac.ae
I. M. Batiha ·S. Momani
Department of Mathematics, The University of Jordan, Amman 11942, Jordan
e-mail: iqbalbatiha22@yahoo.com
S. Momani
e-mail: s.momani@ajman.ac.ae
S. Momani
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science,
King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_8
121
goyal.praveen2011@gmail.com
122 R. El-Khazali et al.
1 Introduction
Many attempts have been made by many researchers to obtain different forms
of finite-order rational approximation to the fractional-order Laplacian integro-
differential operators. Such attempts allow one to develop realizable models of dif-
ferent systems and processes using passive or active electronic devices to mimic the
behavior of such operators. For example, a diffusion process in the electrochemical
process, which exhibits fractional-order dynamics, can successfully be modeled by
finite-order electrical circuits using different approximation algorithms [1,2].
Fractional-order systems provide more freedom in control theory. For example,
fractional-order controllers proved to show superior performance over their integer-
order counterparts. It widens the scope of applications in systems and, in some
cases, simplifies the design of controllers. The Laplace transform of the input–output
relationship provides a powerful tool to investigate the frequency response of linear
systems. It is extended to systems that exhibit fractional-order dynamics [1–5]. The
frequency response of an integro-differential Laplacian operator, s±αcan be defined
as
(jω)±α=ω±αcos απ
2±jsin απ
2 (1)
It is not possible to estimate the exact time response of a fractional-order transfer
function since the analytical inverse Laplace transform does not exist [2], one may
compute the system time responses, namely the impulse and the step responses,
which may be described by fractional-order transfer function, by constructing rational
approximations of the fractional-order operators, s±α. Such approximation can be
used to generate equivalent integer-order transfer functions that describe the original
system within a limited frequency band [1].
There are several popular approximation methods that are used to approximate
s±α, such as the Continued Fractional Expansion (CFE), least square method,
Oustaloup’s, Carlson’s, Matsuda’s, Chareff’s, AbdelAty’s et.al., and El-Khazali’s
approximation methods [1–5]. In the latest paper by the authors in [6], the numerical
time-domain solution of fractional-order systems is obtained using two methods; the
first one uses the Fourier series representation of a square wave, and the second one
uses the inverse Fourier transform. The two methods use the exact numerical data of
the system frequency response to obtain accurate representations of the fractional-
order dynamics.
In this chapter, some basic concepts about the fractional-order models, the fre-
quency domain analysis, and some rational approximations of fractional-order oper-
ators are presented in the first four sections. Section 5, however, includes the main
comparison results in both time and frequency domains using four different approxi-
mation methods of the Laplacian operators, s±α, namely, the Oustaloup’s, Matsuda’s,
AbdelAty’s et.al, and El-Khazali’s approximations [1–5].Two numerical examples
are given to provide detailed comparison and to highlight the advantages and disad-
vantages of each method.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 123
Furthermore, providing modular approximating to s±α,0 <α≤1, by finite-order
rational-transfer functions simplifies the realization of larger class of fractional-order
controllers such as fractional-order integrators, Iλ, fractional-order differentiator,
Dδ, a combination of PIλ,PD
δ, to design PIλDδcontrollers (FOPID). The proper-
ties of these controllers and the effect of their fractional orders on system transient
response are briefly highlighted for completeness and compared with their integer-
order counterparts.
2 Fractional-Order Models
Figure 1shows a general classification of LTI systems. It is well known that sys-
tems, which exhibit hereditary effect, are described by fractional-order differential
equations. The fundamental definitions of fractional-order calculus are used to char-
acterize such systems [7]. The integer-order dynamics, however, is considered as a
subset of larger class of fractional-order systems. Furthermore, it was shown in [7–9]
that fractional-order controllers outperform their integer-order counterparts due to
their flexibility in accommodating more parameters, and due to the constant-phase
frequency response, which provides more robustness to the controlled plants.
The class of linear time-invariant systems that will be considered here is described
by the following fractional-order differential equations [7]:
anDαny(t)+···+a0Dα0y(t)=bmDβmu(t)+···+b0Dβ0u(t)(2)
where u(t)and y(t)are the control and output variables, and Dαdefines a fractional-
order differential operator (according to Caputo definition) of different fractional
orders αk;k=1,2,3,...,n,and βl;l=1,2,3,...,m,are arbitrary constants and
n,m∈N.
Fig. 1 Classification of LTI
systems
goyal.praveen2011@gmail.com
124 R. El-Khazali et al.
The system of (2) is said to be of commensurate order if all its orders are integer
multiples of a base of a fractional order, q, such that αk;βk=kq;q∈R+. The system
can then be expressed as [4]:
n
k=0
akDkqy(t)=
m
k=0
bkDkqu(t)(3)
Obviously, if one sets q=1
n, where n>1; then (3) defines a typical fractional-order
system of commensurate order. A fractional-order linear time-invariant (FOLTI)
system is mathematically equivalent to an infinite-dimensional LTI filter. In order
to model such systems, one has to realize finite-order approximate models that best
describe the original fractional-order systems. Obviously, the size of approximation
depends on the type of the numerical algorithms used to implement such models.
Discrete-time fractional-order systems are usually approximated by higher order
FIR or IIR filters of integer orders [10]. The size of the FIR filters needed to model
such systems are usually much larger than those of IIR filters. Therefore, most
researchers focus on developing a set of rational-transfer functions of much lower
orders. The presence of the feedback action imbedded in the rational transfer func-
tions compensate for the need to realize larger size of either continuous or discrete
FIR filters for the FOLTI systems [11,12]. Figure 1shows a classification of FOLTI
that is of interest in this chapter.
The frequency response of systems is usually carried out by using transfer func-
tions. The transfer function of a linear time-invariant system is defined as the ratio of
the Laplace transform of the output (system output response) to the Laplace trans-
form of the input (system input) under the assumption that all initial conditions are
zero [11], i.e.,
G(s)=L(output)
L(input)
zero initial conditions =Y(s)
U(s)(4)
A typical form of a fractional-order LTI (FoLTI) system can be described by the
following transfer function [11]:
G(s)=Y(s)
U(s)=bmsβm+bm−1sβm−1+···+b1sβ1+b0sβ0
ansαn+an−1sαn−1+···+a1sα1+a0sα0(5)
The next sections outline a complete comparison between the different types of
approximations used to evaluate the time and frequency response of the approximated
integro-differential operators. In addition, the finite-order rational approximations
will be used to model and design fractional-order PID (FOPID) controllers, and
compared with the ideal controllers.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 125
3 Frequency Response of FOLTI Systems
The frequency response of FOLT I systems (5) may be obtained by replacing, s,in
(5) by a generating function, s=(ω(z−1)) [11] to yield
G(z)=bm(ω(z−1))βm+bm−1(ω(z−1))βm−1+···+b1(ω(z−1))β1+b0(ω(z−1))β0
an(ω(z−1))αn+an−1(ω(z−1))αn−1+···+a1(ω(z−1))α1+a0(ω(z−1))α0
(6)
where s=(ω(z−1)) denotes the discrete equivalence of the Laplace operator s,
expressed as a function of the complex variable zor the shift operator z−1. Sys-
tem (5) defines an irrational continuous transfer function in the Laplace domain
or/and an infinite-dimensional discrete-time transfer function in the z-domain [5]. In
both cases, FOLTI systems have unlimited memory size, while integer-order ones
are described by finite-dimensional models [1].
4 Rational Approximation of Fractal-Order Operators
The rational approximation of fractional-order Laplacian operators simplifies the
realization of fractal elements of real orders, which can also be characterized as
Constant-Phase Elements (CPE) [13]. Such elements provide a good description of
these operators over a limited frequency range. In this study, our interest is limited to
fractional-order operators of real order. The electronic circuit realizations of different
fractional-order operators are left for future consideration [13].
The following section describes briefly the different approximation methods of
the fractional-order integro-differential Laplacian operators, s±α, that are used in
many applications [1,3–5,14].
4.1 Oustaloup’s Approximation
Oustaloup’s approximation is a popular one and generates rational-transfer functions
of odd order only. The bandwidth over which the approximation is considered can
be customized to yield a good fitting to the fractional-order elements s±αwithin
a predefined frequency band. Thus, for geometrically distributed frequencies over
the frequency range of interest (ωb,ωh), the following rational function is used for
approximating sα[1]:
sα∼
=K
N
k=−N
s+ω
k
s+ωk=Bnsn+Bn−1sn−1+···+B1s+B0
Ansn+An−1sn−1+···+A1s+A0
(7)
goyal.praveen2011@gmail.com
126 R. El-Khazali et al.
where the poles, zeros, and gain are evaluated from
ωk=ωbωh
ωbk+N+0.5(1+α)
2N+1(8a)
ω
k=ωbωh
ωbk+N+0.5(1−α)
2N+1(8b)
K=ωh
ωb−α
2
N
k=−N
ωk
ω
k
(8c)
Due to the geometrical distribution of frequencies, the unity gain geometric fre-
quency ωuis calculated from
ωu=√ωb·ωh(9)
where the approximation depends on the order of the filer Nand the upper and the
lower frequency range (ωb,ωh). Observe that the order of the transfer function (7)
will always be n=2N+1, i.e., only odd-order approximations are possible through
the Oustaloup’s method. In the special case where the limited frequencies ωband
ωhare symmetrical around the center frequency ωu=1rad/sec, (i.e., ωb=1/ωh),
then the coefficients of (7) are correlated to each other as follows [12]:
An−i=Bi,i=0,1,2,...,N.(10)
One can define the fractional-order derivative using the a fractional-order integral
as follows:
Dα
af(t)=DmJm−α
af(t)(11)
where mis a positive integer such that m−1<α≤m.
Consequently, for the case when the fractional orders α≥1, one may rewrite
sαas
sα=snsγ(12)
where n=α−γdenotes the integer part of αand sγis obtained by Oustaloup’s
approximation using (7).
For digital implementations, the obtained approximation may be discretized to
using suitable discretization methods [5].
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 127
4.2 Matsuda’s Approximation
This method provides continuous approximation by calculating gain at logarithmi-
cally spaced frequencies. If the value of a function F(jω)is known over a set of
frequencies ω0,ω1,ω2,...,ωN, then the following set of functions is recursively
defined by [15]:
d0(ω)=| F(jω)|(13)
dk+1(ω)=ω−ωk
dk(ω)−dk(ω),k=0,1,2,...,N(14)
Then, an (N+1)(N+1)superior upper triangular matrix is formed as follows:
D=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
d0(ω0)d0(ω1)d0(ω2)... d0(ωN)
d1(ω1)d1(ω2)... d1(ωN)
d2(ω2)... d2(ωN)
....
.
.
dN(ωN)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(15)
The desired approximation is then given by the following continued fraction:
F(s)=a0+s−ω0
a1+s−ω1
a2+s−ω2
a3+···
=a0+s−ω0
a1+···
s−ω1
a2+···
s−ω2
a3+··· (16)
such that the set of coefficients is defined as
ak=⎧
⎪
⎨
⎪
⎩
|F(jω0)|,if k =0
ωk−ωk−1
dk−1(ωk)−dk−1(ωk−1),if k =1,2,3,...,N
(17)
4.3 AbdelAty’s Approximation
The approximation is based on using a weighted sum of 1st-order high-pass fil-
ters. The parameters of the filters are obtained using a flower pollination algorithm
(FPA) for each fractional order [16], which can be synthesized using Forster II RC
realization [4] and given by
sα=1
(α)(1−α)
n
i=1
ki
s
τα
i(1+τis)(18)
where τiand kiare two constants found using the FPA optimization algorithm.
goyal.praveen2011@gmail.com
128 R. El-Khazali et al.
Notice that the high-pass structure of each filter section of (20) creates larger
phase errors for fractional orders less than 0.5 than other approximation methods,
since the phase value of each section is π/2 at low frequency.
4.4 El-Khazali’s Integro-Differential Approximation
Fractional-order integro-differential Laplacian operators, s±α, can be approximated
using a biquadratic approximation algorithm introduced in [17], where sαdefines a
differential operator, while s−αdefines a fractional-order integrator. The orders of
both the numerator and denominator are equal, where the reciprocal of one approx-
imation yields the other one. Thus, realizing fractal elements (i.e., fractional-order
capacitors or inductors) is straightforward and only depends on the order of differ-
entiation or integration. For a single module, it enjoys a flat phase response at its
center frequency, but with a narrower bandwidth than its counterparts. It consists of
cascaded several 2nd-order biquadratic transfer functions of the form [13,17]:
s
ωgα=
n
i=1
Hi(s/ωi)=
n
i=1
Nis
(ωi/ωg)
Dis
(ωi/ωg)(19)
where ωi,i=1,2,...,n, is the center frequency of each biquadratic module and
ωg=n
n
i=1ωiis their geometric mean.
If one selects the first center frequency, ω1, of the first section, then to obtain a
constant-phase element, the subsequent center frequencies of each section can be
calculated from the following recursive formula [17]:
ωi=ω2(i−1)
xω1;i=2,3,...,n(20)
where ωxis the maximum real solution of the following polynomial:
a0a2ηγ4+a1(a2−a0)γ3+(a2
1−a2
2−a2
0)ηγ2+a1(a2−a0)γ+a0a2η=0
(21)
and where η=tan(απ/4). Each biquadratic module in (21) is given by
s
ωgα=His
ωi=Ni(s
ωi)
Di(s
ωi)∼
=a0(s
ωi)2+a1(s
ωi)+a2
a2(s
ωi)2+a1(s
ωi)+a0
,i=1,2,3,... (22)
where
a0=αα+2α+1
a2=αα−2α+1
a1=(a2−a0)tan(2+α)π
4=−6αtan(2+α)π
4
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(23)
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 129
Observe that (24) is the only approximation that yields Hi(s
ωi)=(s
ωi)as α→1; i.e.,
His
ωi=a0(s
ωi)2+a0(s
ωi)
a0(s
ωi)+a0=s
ωi(24)
Moreover, the reciprocal of (21) approximates a fractional-order integrator [17],
or simply:
(s/ωg)(−α)=
n
i=1ˆ
Hi(s/ωg)=
n
i=1
Dis
(ωi/ωg)
Nis
(ωi/ωg).(25)
5 Comparison Results
In this section, we introduce a comparison simulation result between the four dif-
ferent approximation algorithms discussed in section (4) in both time and frequency
domains. Two numerical examples are investigated to highlight the main differences
of these methods; the first one approximates a fractional-order Laplacian differen-
tiator s0.4, and the second example is the approximation of a closed-loop transfer
function of a FOLTI system.
Example 1 The integer-order of a rational transfer function that approximates s0.4
using Oustaloup’s, Matsuda’s, AbdelAty’s, and El-Khazali’s approximation, respec-
tively, are given by
– Oustaloup’s approximation
s0.4=6.31s3+77.14s2+41.74s+1
s3+41.74s2+77.14s+6.31 ,0.1≤ω≤10 (26)
– Matsuda’s approximation
s0.4=10.01s3+163.6s2+83.01s+1
s3+83.01s2+163.6s+10.01 (27)
– AbdelAty’s approximation
s0.4=
412.48s8+206415397.69s7+11956108280000.56s6+94706047639064780s5
+104739477770482730000s4+1.616e22s3+3.399e23s2+8.429e23s
s8+1443327.34s7+190377994705.8s6+3320443435965385.5s5
+8064411171562614000s4+2.737e21s3+1.285e23s2+7.801e23s+3.550e23
(28)
goyal.praveen2011@gmail.com
130 R. El-Khazali et al.
Fig. 2 Bode plot of the approximations (26), (27), (28), and (29)
– El-Khazali’s approximation
s0.4=2.493s2+4.924s+0.8931
0.8931s2+4.924s+2.493 (29)
The Bode diagrams and the step response of (26)–(29) are shown in Figs. 2and 3,
respectively. Figure 2shows the frequency response of four different approximations.
They all show similar frequency response, except for AbdelAty’s approximation
since the phase error at low frequency is larger than the rest of approximations. This
error would increase for lower values of α. Figure 3, however, shows an almost
identical response of the fractional-order derivative of the unit-step function using
the four approximations given by (26)–(29). However, the 8th-order approximation
of AbdelAty does not give the right time response.
Example 2 Consider the following fractional-order transfer function reported in [6]:
H(s)=1
2s2.1+0.8s0.8+1(30)
If one wishes to sufficiently simplify (30) by a lowest integer-order transfer func-
tion using the four approximations discussed in Sect. 4, one needs to replace s0.1and
s0.8by their integer-order equivalence to be able to develop integer-order approxima-
tionto(30). Using Oustaloup’s 5th-order approximations for s0.1and s0.8yields [1]:
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 131
Fig. 3 Fractional-order derivative of order 0.4 of the unit-step function using (26), (27), (28), and
(29)
s0.1=1.585s5+68.37s4+403.3s3+367.9s2+51.87s+1
s5+51.87s4+367.9s3+403.3s2+68.37s+1.585 (31)
s0.8=39.81s5+901.4s4+2790s3+1336s2+98.83s+1
s5+98.83s4+1336s3+2790s2+901.4s+39.81 (32)
Substituting (31) and (32)into(30) yields a 10th-order transfer function of the form:
HOus(s)=
s10 +150.7s9+6830s8+1.088e05s7+6.769e05s6+1.619e06s5
+1.551e06s4+5.711e05s3+8.211e04s2+4151s+63.1
3.17s12 +450s11 +1.859e04s10 +2.745e05s9+1.593e06s8+3.871e06s7+4.771e06s6
+3.953e06s5+2.293e06s4+6.858e05s3+8.961e04s2+4331s+64.36
(33)
Repeating the same procedure using the following Matsuda’s 5th-order approxima-
tions of s0.1and s0.8:
s0.1=1.891s5+175.3s4+1329s3+1197s2+126.3s+1
s5+126.3s4+1197s3+1329s2+175.3s+1.891 (34)
s0.8=313.3s5+1.188e04s4+4.261e04s3+1.826e04s2+809.3s+1
s5+809.3s4+1.826e04s3+4.261e04s2+1.188e04s+313.3(35)
goyal.praveen2011@gmail.com
132 R. El-Khazali et al.
yields the following 10th-order rational-transfer function of (30) of the form:
HMat(s)=
s10 +935.6s9+1.216e05s8+3.317e06s7+2.831e07s6+7.689e07s5
+7.406e07s4+2.366e07s3+2.579e06s2+7.739e04s+592.3
3.78s12 +3411s11 +3.557e05s10 +8.757e06s9+6.709e07s8+1.806e08s7+2.219e08s6
+1.815e08s5+1.041e08s4+2.725e07s3+2.722e06s2+7.875e04s+593.8
(36)
In similar manner, the 8th-order AbdelAty’s approximations of s0.1and s0.8are given
by
s0.1=
4.51s8+3722617.69s7+390344733034.4s6+6445483209685290s5
+17462315946623164000s4+7.739e21s3+5.437e23s2+5.054e24s
s8+1053025.15s7+133752991203.54s6+2649165710288196s5
+8597754455256356000s4+4.569e21s3+3.872e23s2+4.503e24s+1.386e24
(37)
s0.8=
228721.34s8+56105355367.14s7+1671424211082485s6+7107665167043685000s5
+4.418e21s4+4.005e23s3+5.176e24s2+7.867e24s
s8+3327245.95s7+518122071483.89s6+10400295711020872s5
+30071114886647075000s4+1.273e22s3+7.848e23s2+6.793e24s+6.216e24
(38)
Consequently, the 16th-order approximation of (30)is
HAbd (s)=
s16 +4380271.1s15 +4155548728273.07s14 +1003674131746307100s13 +8.911e22s12
+2.824e27s11 +3.606e31s10 +1.732e35s9+3.401e38s8+2.529e41s7+7.661e43s6
+8.707e45s5+3.923e47s4+6.210e48s3+3.408e49s2+3.741e49s+8.618e48
9.02s18 +37456993.85s17 +30226279998733.27s16 +6561788513256727000s15
+5.251e23s14 +1.514e28s13 +1.758e32s12 +7.688e35s11 +1.376s10 +9.315e41s9
+2.574e44s8+2.655e46s7+1.090e48s6+1.558e49s5+7.889e49s4+9.057e49s3
+6.816e49s2+4.613e49s+8.618e48
(39)
In addition, the biquadratic approximation method of El-Khazali yields the following
rational transfer function for s0.1and s0.8, respectively.
s0.1=1.994s2+5.082s+1.594
1.594s2+5.082s+1.994 (40)
s0.8=3.436s2+4.404s+0.236
0.2365s2+4.404s+3.436 (41)
Substituting (40) and (41)into(30) gives a 6th-order approximation of the form:
Hk(s)=0.377s4+8.222s3+28.333s2+26.247s+6.852
0.943s6+19.968s5+63.984s4+76.78s3+62.978s2+34.234s+7.229
(42)
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 133
The frequency and the unit-step responses of all four approximations given by
(33), (36), (39), and (42) are shown in Figs. 4and 5, respectively. Obviously,
El-Khazali’s approximation yields a competitive frequency response to those of
Oustaloup’s and Matsuda’s approximations, but gives larger steady-state errors than
its counterparts as depicted in Fig. 5. This is due to using a 2nd-order approxima-
tion to the fractional-order derivatives in (40) and (41). Increasing their order of
approximation to a 4th-one, for example, would improve the rational approximation
of s0.1and s0.8and consequently, it would improve the steady-state step response.
Furthermore, the parameter values of El-Khazali’s approximation are smaller than
its competitive counterparts and that would imply less expensive circuit design.
The effectiveness of the approximation methods in designing different types of
controllers is further investigated in the next section.
−100
−80
−60
−40
−20
0
20
Magnitude (dB)
10−2 10−1 100101102
−225
−180
−135
−90
−45
0
Phase (deg)
Bode Diagram
Frequency (rad/sec)
Oustaloup
Matsuda
AbelAty
ElKazali
Fig. 4 Bode diagrams of (33), (36), (39)and(42)
goyal.praveen2011@gmail.com
134 R. El-Khazali et al.
020 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Step Response
Time (sec)
Amplitude
Oustaloup
Matsuda
AbelAty
Elkazali
Fig. 5 Step response of (33), (36), (39)and(42)
6 Approximation of Fractional-Order Controllers
The commonly used proportional–integral–derivative (PID) controller, which con-
sists of three parameters has been successfully used in industrial applications for
several decades. The popularity of the PID controller lies in the simplicity of the
design procedures and in the effectiveness of its system performance [18]. Some
applications of PID controller show undesirable system performance, which may be
enhanced by using fractional-order PID (FOPID) controllers [7–9,19–21].
A FOPID controller was introduced in [7]. it is a generalization of the conventional
integer-order PID controller and denoted by PIλDδ, where λand δare the fractional
orders of the integral and the derivative components, respectively. Thus, adding two
more parameters to tune the original three parameters of the classical integer-order
PID controllers, which add more degrees of freedom [7]. The PID controller processes
the present, past, and future values of the error signal, e(t). A digital implementation
of such controllers grew rapidly and overcomes the difficulties embedded in the
realizations of the continuous versions.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 135
Fig. 6 Block diagram of a controlled system using integer-order PID controller
Figure 6shows a typical block diagram of an integer-order PID controller. The
proportional controller amplifies the present value of the error signal. The integral
part refers to the accumulation of the past errors, while its derivative part predicts
the future values of the error; i.e., acts on the anticipated value of the error signal.
One may use the weighted sum of these three actions to make a final adjustment to
tune its parameters [7].
6.1 Proportional-Order Controllers
Proportional controllers are widely used and are simple to design. They simply form
a direct scaling of the error signal to alter the transient and steady-state system
responses, i.e.,
y(t)=Kpe(t)(43)
where e(t)=r(t)−y(t)is the error signal between a reference signal, r(t), and the
output signal, y(t). Thus,
Y(s)=KpE(s)(44)
The steady-state error in a proportional controller, C(s)=Kp, is inversely pro-
portional to the controller gain [22,23]. It is well known that increasing it makes
the system response faster and minimizes the steady-state errors, but increases the
system overshoot [15].
goyal.praveen2011@gmail.com
136 R. El-Khazali et al.
6.2 Fractional-Order Integral Controllers (I λ-Controllers)
The fractional-order integral controllers (Iλ- controllers) produce the control signal
y(t), which is proportional to the fractional integral of the error signal e(t). It can be
described by the following relation [23]:
y(t)=Ti.Iλe(t)≡Ti.D−λe(t)(45)
which is described by the following transfer function:
Y(s)
E(s)=Ti
sλ≡C(s,λ)(46)
where C(s,λ)denotes a fractional-order controller, (Iλ-controller) of order λ.
Obviously, increasing λreduces the steady-state error. The fractional-order inte-
grator, s−λ, eliminates the offset error when λ>1 without increasing the integral
constant Ti. It is possible to reduce the offset error when λ<1 by adding a pole and
zero at the origin. Therefore, s−λin this case can be expressed as [22,24]:
1
sλ=1
sλ.s
s=1
s.s(1−λ),0<λ<1 (47)
Equation (47) shows that s−λcan be expressed as a product of a pure integral (1/s)
and a fractional-order differentiator s(1−λ).
Now, we briefly summarize the effects of extending the integral control actions
to the fractional case. Let us first explore s−λwith λ=0.4 and Ti=1. The transfer
function of the I0.4-controller is then given by
C(s)=1
s0.4(48)
Using the approximations discussed in example (1), the I0.4-controller, (48), can be
expressed by the following transfer functions:
COus(s)=s3+41.74s2+77.14s+6.31
6.31s3+77.14s2+41.74s+1(49)
CMat(s)=s3+83.01s2+163.6s+10.01
10.01s3+163.6s2+83.01s+1(50)
CAbd (s)=
s8+1443327.34s7+190377994705.8s6+3320443435965385.5s5
+8064411171562614000s4+2.737e21s3+1.285e23s2+7.801e23s+3.550e23
412.48s8+206415397.69s7+11956108280000.56s6+94706047639064780s5
+104739477770482730000s4+1.616e22s3+3.399e23s2+8.429e23s
(51)
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 137
10−2 10−1 100101102
−20
−10
0
10
20
30
40
Magnitude (dB)
Bode Plot: Magnitude Response
10−2 10−1 100101102
−100
−80
−60
−40
−20
0
Frequency (rad/s)
Angle (deg)
Bode Plot: Phase Response
I0.4−Khazali
I0.4−Oustaloup
I0.4−AbdelAty
I0.4−Matsuda
Ideal−controller
Fig. 7 Bode diagram of an integer I- and a fractional-order I0.4-controller approximated by
Oustaloup’s, Matsuda’s, AbdelAty’s, and El-Khazali’s approximations for Ti=1
and
Ck(s)=0.8931s2+4.924s+2.493
2.493s2+4.924s+0.8931 (52)
Figure 7shows a numerical simulation of the ideal I0.4-controller’s four of its
approximations using Oustaloup’s, Matsuda’s, AbdelAty’s, and El-Khazali’s meth-
ods.
6.3 Fractional-Order Differential Controllers
(Dδ-Controllers)
The fractional-order differential controllers of order δ, denoted by Dδ-controllers,
produce a control signal, y(t), that is proportional to the fractional-order derivative
of the error signal e(t). It described by the following expression [23]:
y(t)=Td.Dδe(t)(53)
goyal.praveen2011@gmail.com
138 R. El-Khazali et al.
Hence, the corresponding transfer function of (53) is given by
Y(s)
E(s)=Td.sδ(54)
where Tdis a constant of the differential controller, C(s)=Td.sδ.
The fractional differentiator, sδ,actsasaδ-predictor that predicts the future value
of the error signal. It reduces the rate of change of error, which improves the control
performance. Notice that sδcannot be implemented alone since it does not eliminate
the steady-state errors. Thus, it must be augmented at least with a proportional con-
troller, P, to form a PD controller, known as lead controller, to shape the frequency
response of the controlled system. Notice that the δth-differential controller amplifies
noise signals when δincreases and has no effect on the steady-state error [9,23].
For comparison, let us consider the case when δ=0.4 and Td=1, then, the
D0.4-controller is given by C(s)=s0.4. Figure 8shows the frequency response of
the rational approximation of C(s)=s0.4using (7), (16), (18), and (19). Obviously,
the low-frequency deviation of (18) cannot be avoided due to the presence of the
10−2 10−1 100101102
−40
−30
−20
−10
0
10
20
Magnitude (dB)
Bode Plot: Magnitude Response
10−2 10−1 100101102
0
20
40
60
80
100
Frequency (rad/s)
Angle (deg)
Bode Plot: Phase Response
D0.4−Khazali
D0.4−Oustaloup
D0.4−AbdelAty
D0.4−Matsuda
Ideal−controller
Fig. 8 Bode diagram of a classical D-and D0.4-controller approximated by Oustaloup’s, Matsuda’s,
AbdelAty’s, and El-Khazali’s approximations with Td=1
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 139
differential effect of the high-pass filters approximations. However, the biquadratic
approximation of (19) does behave like a proportional controller for both the low-
and high-frequency bands, thus suppressing signal noise.
6.4 Fractional-Order PI Controllers (PIλ-Controller)
The fractional-order PI controller (PIλ-Controller), or lag controllers, combines both
the proportional and the fractional-order integral action and is defined as [23]
y(t)=Kpe(t)+Ti.Iλe(t)(55)
which yields a lag controller of order λof the form:
Y(s)
E(s)=C(s)=Kp+Ti
sλ(56)
As λincreases, this controller has the following features [23]:
•It reduces the steady-state error.
•It decreases the rise time.
•It filters out the noise at high frequencies.
•It increases bandwidth of the system.
•It increases the order and type of the system.
Consider the case when λ=0.4, Kp=1, and Ti=1, the following transfer
functions approximate PI0.4=1+1/s0.4using (7), (16), (18), and (19), respectively:
COus(s)=731s3+11888s2+11888s+731
631s3+7714s2+4174s+100 (57)
CMat(s)=1101s3+24661s2+24661s+1101
1001s3+16360s2+8301s+100 (58)
CAbd (s)=
s8+502710.5s7+29376520704.24s6+237079034998476.75s5
+272818468162711600s4+45699052676123180000s3
+1.133e21s2+3.925e21s+3.859e05
0.9976s8+499219.78s7+28916087705.21s6+229048475943137.66s5
+253314527978607520s4+39078673785758040000s3
+822012098401221200000s2+2.039e21s
(59)
and
Ck(s)=s2+2.908s+1
0.7362s2+1.454s+0.2638 (60)
goyal.praveen2011@gmail.com
140 R. El-Khazali et al.
10−4 10−3 10−2 10−1 100101102103104
0
20
40
60
80
Magnitude (dB)
Bode Plot: Magnitude Response
10−2 10−1 100101102
−100
−80
−60
−40
−20
0
Frequency (rad/s)
Angle (deg)
Bode Plot: Phase Response
PI0.4−Khazali
PI0.4−Oustaloup
PI0.4−AbdelAty
PI0.4−Matsuda
Ideal−controller
Fig. 9 Bode diagram of an approximated PI 0.4-controller using Oustaloup’s, Matsuda’s, Abde-
lAty’s, and 2nd-order El-Khazali’s biquadratic form with Kp=Ti=1
Graphically, we make the following comparison in the frequency domain, shown
in Fig. 9, between a classical PI-controller and PI0.4-controller.
Figure 10 shows the same approximation shown in Fig. 9, except for El-Khazali’s
approximation, where the following 4th-order biquadratic form is used to approxi-
mate s0.4instead of a 2nd-order one:
PI0.4=70137s4+1374500s3+3422000s2+1374500s+70137
7977s4+362500s3+1711000s2+1012000s+62160 (61)
Clearly, the frequency response of all approximations match that of the ideal con-
troller except for AbdelAty’s one, which has large deviations in both the magnitude
and phase responses for low frequency. If one considers the simplicity of the real-
ization and construction, one would choose the biquadratic approximation given by
(61) since it depends only on a single parameter, which could be the order of the
integrators or the differentiator.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 141
Fig. 10 Bode diagram of an approximated PI 0.4-controller using (7), (16), (18), and (61) with
Kp=Ti=1
6.5 Fractional-Order PD Controller (PDδ-Controller)
The transfer function of the fractional-order controller (lead controller) of order δis
given by
C(s)=Kp+Td.sδ(62)
As δincreases, the PD
δor the fractional-order lead controller enjoys the following
properties [19]:
•It reduces the overshoot.
•It improves transient response.
•It reduces the settling time.
•It improves the bandwidth of the system.
•It may make noises at high frequencies.
•It does not affect steady-state errors.
Now, to summarize the effects of extending the derivative control actions to the
fractional case; let us explore the case when δ=0.4, Kp=1, and Td=1. Then,
the PD
0.4-controller becomes C(s)=1+s0.4.Nowusing(26), (27), (28), and (29),
goyal.praveen2011@gmail.com
142 R. El-Khazali et al.
Fig. 11 Bode diagram of PD
δ-controller and compared with the ideal one for δ=0.4,
Kp=Td=1
the frequency response of the PD
0.4isshowninFig.11. All approximations yield
a good match with the ideal lead integrator, however, the AbdelAty’s one gives a
perfect match with the ideal lead controller at high frequency only.
7 FOPID Controllers
It is well known that the time response of the PID controllers output is given by the
following expression [7]:
u(t)=Kpe(t)+Tit
0
e(τ).dτ+Td
d
dt e(t)(63)
and its transfer function is given by
C(s)=U(s)
E(s)=Kp+Ti
s+Td.s(64)
where E(s)=L{e(t)}, and U(s)=L{u(t)}are the Laplace transforms of the error
and control signals, respectively.
The Zeigler–Nichols design method is a popular one that can be used to design
integer-order PID controllers for most systems [15,20,21]. The integer-order PID
controller is applicable for many control problems and it often yields satisfactory
performance and, in some cases, it requires parameter tuning [21].
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 143
The concept of the fractional-order PID (FOPID or PIλDδ) controller was intro-
duced by Podlubny in [23]. This controller has an integrator of an order λand a dif-
ferentiator of order δ. It was shown that the fractional-order controller outperforms
its integer counterpart [19,25]. When controlling industrial plants, they require a
complete satisfaction of a wide range of specification, where wide ranges of tech-
niques are needed. Recently, FOPID controllers are used for industrial applications
to improve system performance. They provide extra degrees of freedom by adding
two more parameters to tune (namely, λand δ) to the original three parameters,
(Kp,Ti,Td), thus increasing the complexity of tuning its parameters [20].
7.1 FOPID Controllers
The fractional-order integro-differential equation that describes the FOPID con-
trollers is given by [23]
u(t)=Kpe(t)+Ti.Iλe(t)+Td.Dδe(t)(65)
The Laplace transform of (65) is given by
C(s)=Kp+Ti.s−λ+Td.sδ(66)
Obviously, one can get the classical integer-order PID controller by taking
λ=δ=1. With more freedom in tuning the controller, the four-point PID diagram
can be seen as a PID controller plane, which is depicted in Fig. 12 [7].
Notice that the integer-order controllers are classified as particular cases of the
more general FOPID controller, which provides more flexibility and robustness and
gives the capability for better adjustment of the dynamical properties of fractional-
order control system [21,26]. For example, by assuming δ=0 and Td=0, then a
PIλ- controller is derived, and so on.
The FOPID controller can be considered as an infinite-dimensional linear filter
due to the fractional orders of the differentiators and the integrators. Since PID con-
trollers are ubiquitous in industry process control, then fractional-order PID control
will also be ubiquitous when tuning and implementation techniques are well devel-
Fig. 12 Generalization of
the FOPID controllers
goyal.praveen2011@gmail.com
144 R. El-Khazali et al.
oped [27–35]. As compared to PID controller, a FOPID is supposed to offer the
following advantages [30]:
•If the parameter of a controlled system changes, a FOPID controller is less sensitive
than the classical PID controller.
•Fractional-order controller has two extra variables to tune. This provides extra
degrees of freedom to the dynamic properties of fractional-order system.
Now, we will study the effects of extending the integral and derivative control
actions on the fractional-order case. Let us first explore sλand sδwith λ=δ=0.4
and Kp=Ti=Td=1. Then, the PI0.4D0.4-controller is given by
C(s)=1+1
s0.4+s0.4(67)
Moreover, the PI0.4D0.4-controller expressed in (67) can be approximated by
Oustaloup’s, Matsuda’s, AbdelAty’s, and El-Khazali’s approximations using Eqs.
(26), (27), (28) and (29), respectively, as follows:
COus(s)=
471261s6+13975062s5+121221630s4+206381577s3
+121221630s2+13975062s+471261
63100s6+3405194s5+37483170s4+77336233s3
+37483170s2+3405194s+63100
(68)
CMat(s)=
1112101s6+44358221s5+509457623s4+881186042s3
+509457623s2+44358221s+1112101
100100s6+9945301s5+153010820s4+337568202s3
+153010820s2+9945301s+100100
(69)
CAbd (s)=
s16 +1003151.62s15 +309946795646.82s14 +29742326890849164s13 +1.086e21s12
+1.389e25s11 +6.992e28s10 +1.256e32s9+8.960e34s8+2.301e37s7+2.360e39s6
+8.778e40s5+1.303e42s4+6.770e42s3+1.283e43s2+5.003e42s+7.391e41
0.0024s16 +4700.99s15 +2277382680.28s14 +340180953977873.3s13 +3.491e23s11
+18186425387598578000s12 +2.530e27s10 +6.727e30s9+6.797e33s8+2.523e36s7
+3.548e38s6+1.815e40s5+3.437e41s4+2.223e42s3+4.563e42s2+1.755e42s
(70)
and
Ck(s)=s4+5.4134s3+9.5961s2+5.41399s+1
0.241s4+1.8047s3+3.3834s2+1.8047s+0.241 (71)
A direct comparison in the frequency domain is shown in Fig. 14 for the approx-
imated PI0.4D0.4-controller given by (68), (69), (70), and (71). Observe that for
AbdelAty’s method, an 8th-order approximation was used for simulation purposes.
In spite of using a higher order approximation, a larger magnitude and phase error
is demonstrated for low frequencies.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 145
The following example introduces further comparison in both time and fre-
quency domains between the previous three methods; Oustaloup’s, Matsuda’s, and
El-Khazali’s methods by considering a closed-loop controlled system.
Example 3 Let us consider Fig. 13 with FOPID controller and integer-order plant
transfer function below:
C(s)=18 +13
s0.8+6s1.4(72)
and
G(s)=1
s(s+1)(s+5)(73)
Then, the open-loop transfer function of the controlled system is given by
L(s)=C(s)G(s)=6s2.2+18s0.8+13
s3.8+6s2.8+5s1.8(74)
and the closed-loop transfer function for a unity feedback system is equal to
P(s)=L(s)
1+L(s)=6s2.2+18s0.8+13
s3.8+6s2.8+6s2.2+5s1.8+18s0.8+13 (75)
That is,
P(s)=6s2(s0.2)+18s0.8+13
s3(s0.8)+6s2(s0.8)+6s2(s0.2)+5s(s0.8)+18s0.8+13 (76)
Obviously, (76) includes s0.2and s0.8such that the Oustaloup’s 5th-order approx-
imation of s0.8was given previously in equation (32), while the Oustaloup’s approx-
imation of s0.2is given by
Fig. 13 Block diagram of a FOPID controller with unity gain feedback
goyal.praveen2011@gmail.com
146 R. El-Khazali et al.
Fig. 14 Bode diagram of a frequency response of a classical PID controller and PI λDδcontroller
approximated by Oustaloup’s, Matsuda’s, AbdelAty’s, and El-Khazali’s approximations with λ=
δ=0.4, Kp=Ti=Td=1
s0.2=2.512s5+98.83s4+531.7s3+442.3s2+56.87s+1
s5+56.87s4+442.3s3+531.7s2+98.83s+2.512 (77)
One can obtain an approximate closed-loop transfer function P(s)by substituting
(32) and (77)into(76). Similarly, the Matsuda’s approximation of s0.2is given by
s0.2=3.357s4+161s3+453.9s2+95s+1
s4+95s3+453.9s2+161s+3.357 (78)
The approximated closed-loop transfer function can now be obtained by using (35),
(78) and (76). On the other hand, El-Khazali’s approximation for s0.2is given by
s0.2=2.125s2+5.051s+1.325
1.325s2+5.051s+2.125 (79)
The closed-loop transfer functions of the three approximations are, respectively,
given as follows:
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 147
POus(s)=
0.3786s12 +52.31s11 +2076.37s10 +30424.88s9+190374.23s8
+629279.32s7+1343379.50s6+1723054.78s5+1136020.90s4
+340700.72s3+44471.77s2+2181.14s+33.79
s13 +85.89s12 +2334.45s11 +27839.99s10 +171865.80s9
+587969.75s8+1207365.50s7+1776199.71s6+1881971.42s5
+1162206.67s4+342468.13s3+44515.75s2+2181.46s+33.79
(80)
PMat(s)=
0.0643s11 +55.11s10 +3695.99s9+68507.40s8+371769.57s7
+1075594.57s6+2020717.97s5+1873642.24s4+711212.26s3
+102240.19s2+3913.16s+43.839
s12 +138.98s11 +5049.83s10 +59864.84s9+344954.58s8
+1012528.04s7+1745850.16s6+2331027.67s5+1932393.81s4
+714332.34s3+102286.18s2+3913.21s+43.83
(81)
and
Pk(s)=0.6623s6+13.91s5+58.25s4+142.3s3+202s2+118s+22.84
s7+11.76s6+56.03s5+125.4s4+189.1s3+214.3s2+118.6s+22.84
(82)
The exact step and frequency response of the systems are shown in Figs. 15 and 16.
Obviously, the three approximations yield almost identical step response. However,
using the 2nd-order biquadratic approximation of El-Khazali reduces the size of the
controllers needed for similar cases.
Fig. 15 Bode diagram of the closed-loop controlled system using FPID controllers
goyal.praveen2011@gmail.com
148 R. El-Khazali et al.
Fig. 16 Step responses for the closed-loop controlled system FOPID controllers
8 Conclusions
Four different approximation methods were used to approximate fractional-order
Laplacian operators. These are Oustaloup’s, Matsuda’s, AbdelAty’s, and
EL-Khazali’s methods. Oustaloup yields only odd order of rational-transfer func-
tions, while Matsuda’s method yields an unrealizable model if the sum of poles and
zeros is odd. Both methods require high order of approximation with large coefficients
to generate constant phases in the frequency domain. AbdelAty’s method, however,
is a sum of high-pass filter that yields large phase error for small fractional orders
(say below 0.5). It depends on using optimum phase algorithm (OPA) to calculate
the parameters of the approximations. It yields large parameter values than expected,
which is expensive to design. The fourth method is the biquadratic approximation of
El-Khazali. It yields an exact phase response at the center frequency of the approx-
imation with unity gain. In most cases, lower order approximation of El-Khazali
is very competitive to higher order approximations of other methods, which yields,
most of the time, reasonable parameter values. Notice that one may approximate
fractional-order integrators using El-Khazali’s method by simply using the recipro-
cal approximation of the fractional-order differentiator. This feature is not possible
by the other three approximations. It is worth mentioning that when discretizing
these approximations using bilinear transformation, which is left for further study,
it showed that EL-Khazali’s method gave the most accurate and stable discrete-time
version of the given approximation out of all four methods.
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 149
Acknowledgements We thank the sponsors of the International Conference on Fractional Dif-
ferentiation and its Applications (ICFDA 2018), who provided insight and expertise that greatly
assisted the research to be in hand. We would also like to thank the international editors, Praveen
Agarwal, Dumitru Baleanu, YangQuan Chen, and Tenreiro Machado, for their valuable comments
and remarks.
Appendix: Springer-Author Discount
1. Oustaloup’s Approximation
clear all
alf=input (’Enter value of ALFA = ’)
% frequency range
w_L=0.1; w_H=10.
r=alf; NN=2;
%
[v1,v2,D_N_K,sysO_tf]=ora_foc(r,NN,w_L,w_H);
%%
% Oustaloup-Recursive-Approximation for fractional order differentiator
%
% Input variables:
% r: the fractional order as in sˆr, r is a real number
% N: order of the finite TF approximation for both (num/den)
% (Note: 2N+1 recursive z-p pairs)
% w_L: low frequency limit of the range of the frequency of interest
% w_H: upper freq. limit of the range of the frequency of interest
% Output:
% sys_foc: continuous time transfer function system object (TF)
% Sample values: w_L=0.1;w_H=1000; r=0.5; N=4;
% Existing problem: Be careful when doing "c2d", I met some problems.
function [v1,v2,D_N_K,sys_N_tf]=ora_foc(r,N,w_L,w_H)
w_L=w_L*0.1;w_H=w_H*10; % enlarge the freq. range of interest for % goodness
mu=w_H/w_L; %
w_u=sqrt(w_H*w_L);
alpha=(mu)ˆ(r/(2*N+1));
eta=(mu)ˆ((1-r)/(2*N+1));
k=-N:N;
w_kp=(mu).ˆ( (k+N+0.5-0.5*r)/(2*N+1) )*w_L;
w_k=(mu).ˆ( (k+N+0.5+0.5*r)/(2*N+1) )*w_L;
D_N_K=(w_u/w_H)ˆr * prod(w_k) / prod(w_kp);
D_N_P=-w_k;D_N_Z=-w_kp;
[num,den]=zp2tf(D_N_Z’,D_N_P’,D_N_K);
sys_N_tf=tf(num,den);
v1=num; v2=den;
sys_foc=tf(num,den);
end
2. El-Khazali’s Approximation
%
alf=input (’Enter value of ALFA = ’)
n= input (’No. of Biquadratic Modules = ’)
% wc is the initial selection of the center frequency of the first %biquadratic form
wc=1;
% This is a call statement of the function "Biquad_K" that generates
% a modular structure of biquadratic approximations using EL-Khazali % approximation.
[Nk,Dk,sysk]= Biquad_K (alf,n,wc)
goyal.praveen2011@gmail.com
150 R. El-Khazali et al.
%%
function [Numk,Denk,sysk]=Biquad_K(alf,n,w)
% This function generates a normalized modular biquadratic structure
% structures of order 2*n.
%
et=tan(alf*pi/4);
wc(1)=w;
ao=alfˆalf+2*alf+1;
a2=alfˆalf-2*alf+1;
a1=(a2-ao)*tan((2+alf)*pi/4);
if(n>1)
% The solution of the following polynomial is used to generate
% a recursive formula to select the next center frequency for the
% next modular structure.
Y=roots([ao*a2*et a1*(a2-ao) (a1ˆ2-a2ˆ2-aoˆ2) a1*(a2-ao) ao*a2*et ]);
wx=(abs(max(Y)))
for k=2:n
wc(k)=(wxˆ(2*(k-1)))*wc(1)
end
% normalizing by the geometric mean;
wm=(prod(wc))ˆ(1/n);
sysk=1;
for l=1:n;
N=[ao a1*wc(l)/wm a2*(wc(l)/wm)ˆ2];
D=[a2 a1*wc(l)/wm ao*(wc(l)/wm)ˆ2];
sysk=sysk*(tf(N,D));
end
[Numk,Denk]=tfdata(sysk);
else
Numk=[ao a1*w a2*wˆ2];
Denk=[a2 a1*w ao*wˆ2];
sysk=tf(Numk,Denk);
end
end
References
1. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency band complex noninteger
differentiator: characterization and synthesis. IEEE Trans. Circuit. Syst. I, Fund. Theory and
Appl. 47, 25–39 (2000)
2. Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order
operators used in control theory and applications. Fract. Calculus Appl. Anal. 3, 231–248 (2000)
3. Yce, A., Deniz, F.N., Tan, N.: A new integer-order approximation table for fractional order
derivative operators. Int. Federation Autom. Control 50, 9736–9741 (2017)
4. AbdelAty, A.M., AbdelAty, A.S., Radwan, A.G., Psychalinos, C., Maundy, B.J.: Approxima-
tion of the fractional-order Laplacian sαAs a weighted sum of first-order high-pass filters.
IEEE Trans. Circuit. Sys. II: Express Briefs 65 (2018)
5. El-Khazali, R.: Discretization of Fractional-Order Laplacian Operators. 19th IFAC Congress,
Cape Town, 25/8/2014
6. Atherton, D.P., Tan, N., Yce, A.: Methods for computing the time response of fractional-order
systems. IET Control Theory Appl. 9, 817–830 (2015)
7. Podlubny, I.: Fractional-order systems and PIλDμcontrollers. IEEE Trans. Automat. Control
44, 208–214 (1999)
8. Lin, W., Chongquan, Z.: Design of optimal fractional-order PID controllers using particle
swarm optimization algorithm for DC motor system. In: IEEE Advanced Information Tech-
nology, Electronic and Automation Control Conference (IAEAC), pp. 175–179 (2015)
goyal.praveen2011@gmail.com
Approximation of Fractional-Order Operators 151
9. El-Khazali, R.: Fractional-order PIλDμcontroller design. Comp. Math. App. 66, 639–646
(2013)
10. El-Khazali, R.: Discretization of Fractional-Order Differentiators and Integrators. 19th IFAC
Congress (2014)
11. Tepljakov, A.: Fractional-order Modeling and Control of Dynamic Systems. Tallin University
of Technology, Tallinn (2015)
12. Petras, I., Podlubny, I., OLeary, P.: Analogue Realization of Fractional Order Controllers.
Fakulta BERG, TU Kosice (2002)
13. Anis Kharisovich Gilmutdinov: Pyotr Arkhipovich Ushakov, Reyad El-Khazali: Fractal Ele-
ments and their Applications. Springer, Switzerland (2017)
14. Matsuda, K., Fujii, H.: Optimized wave-absorbing control: analytical and experimental results.
J. Guidance Control Dyn. 16, 1153–1164 (1993)
15. Ogata, K.: Modern Control Engineering. Prentice Hall, New Jersey (2010)
16. Yang, X.S.: Flower pollination algorithm for global optimization in Unconventional computa-
tion and natural computation, pp. 240–249. Springer (2012)
17. El-Khazali, R.: On the biquadratic approximation of fractional-order Laplacian operators. Ana-
log Int. Circuits and Sig. Proc. 82, 503–517 (2015)
18. Astrom, K., Hagglund, T.: PID Controllers; Theory. Design and Tuning. Instrument Society of
America, Research Triangle Park (1995)
19. Saleh, K.: Fractional Order PID Controller Tuning by Frequency Loop-Shaping: Analysis and
Applications. Arizona State University, Arizona (2017)
20. Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control
44, 208–214 (1999)
21. Cech, M., Schlegel, M.: The Fractional-order PID Controller Outperforms the Classical One.
Process control 2006: Pardubice Technical University, pp. 1–6 (2006)
22. Podlubny, I., Dorcak, L., Kostial, I.: On fractional derivatives, fractional-order dynamic systems
and PIλDμcontrollers. In: Proceedings of the 36th Conference on Decision and Control, San
Diego, California, USA (1997)
23. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
24. Rinu, P.S.: Design and analysis of optimal fractional-order PID controller. Int. J. Appl. Eng.
Res. 10, 23000–23002 (2015)
25. Xue, D., Chen, Y.Q., Atherton, D.P.: Linear Feedback Control: Analysis and Design with
MATLAB. Society for Industrial and Applied Mathematics, Philadelphia (2007)
26. Luo, Y., Chen, Y.Q.: Fractional-order [proportional derivative] controller for robust motion
control: tuning procedure and validation. Am. Control Conf., 1412–1417 (2009)
27. Kumar, A.: Controller Design for Fractional Order Systems. National Institute of Technology-
Rourkela, Odisha (2013)
28. Maione, G., Lino, P.: New tuning rules for fractional PIαcontrollers. Nonlinear Dyn. 49,
251–257 (2007)
29. Luo, Y., Chen, Y.Q.: Fractional Order Motion Controls. Wiley, Odisha (2012)
30. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and
Controls: Fundamentals and Applications. Springer, London, New York (2010)
31. Dimeas, I.: Design of an Integrated Fractional-Order Controller. University of Patras, Patras
(2017)
32. Oldham, K.B., Spanier, J.: The fractional calculus: Theory and Applications of Differentiation
and Integration to Arbitrary Order. Academic Press, New York (1974)
33. Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional order Systems: Modelling and
Control Applications. World Scientific, Singapore (2010)
34. Chen, Y.Q.: Ubiquitous fractional order controls. In: Proceedings of the Second IFAC Sympo-
sium on Fractional Derivatives and Applications. Porto (2006)
35. Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order
controllers for industry application. Control Eng. Pract. 16, 798–812 (2008)
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear
Fractional Bloch System Using Adomian
Decomposition Techniques
Asad Freihat, Shatha Hasan, Mohammed Al-Smadi, Omar Abu Arqub
and Shaher Momani
Abstract In this chapter, a superb multistep approach, based on the Adomian decom-
position method (ADM), is successfully implemented for solving nonlinear fractional
Bolch system over a vast interval, numerically. This approach is demonstrated by
studying the dynamical behavior of the fractional Bolch equations (FBEs) at differ-
ent values of fractional order αin the sense of Caputo concept over a sequence of
the considerable domain. Further, the numerical comparison between the proposed
approach and implicit Runge–Kutta method is discussed by providing an illustrated
example. The gained results reveal that the MADM is a systematic technique in
obtaining a feasible solution for many nonlinear systems of fractional order arising
in natural sciences.
Keywords Multistep approach ·Fractional system ·Bolch equations ·Adomian
decomposition method
1 Introduction
During the past few decades, a growing interest in the study of complex systems has
been observed including glasses, amorphous systems, microemulsions, polymers,
biopolymers, and so forth. The term “complex” is due to a broad distinction of the
elementary units, intense interactions between the units, or an irregular evolution of
units over time. Nowadays, such complex systems are investigated on all structural
levels from microscopic to macroscopic and in all fields of physics, biophysics, engi-
A. Freihat ·S. Hasan ·M. Al-Smadi
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816,
Jordan
O. A. Arqub ·S. Momani
Department of Mathematics, Faculty of Science, Jordan University, Amman 11942, Jordan
S. Momani (B
)
Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman
University, Ajman, UAE
e-mail: S.Momani@ju.edu.jo
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_9
153
goyal.praveen2011@gmail.com
154 A. Freihat et al.
neering, chemistry, and medicine. For instance, nuclear magnetic resonance (NMR),
magnetic resonance imaging (MRI), or electron spin resonance (ESR) are physical
phenomena that are widely utilized to study complex systems. Indeed, these systems
have nonlocal interaction and a long memory due to the disorder that appears in the
magnetic relaxation in complex environments. In this aspect, Bloch equations are
a set of macroscopic equations that are used to calculate the nuclear magnetization
M=(Mx,My,Mz)as a function of time in the presence of the relaxation times T1
and T2, where the components Mx(t),My(t), and Mz(t)are the system magnetiza-
tion, T1is the spin–lattice relaxation time that characterizes the rate of which the
longitudinal Mz component recovers exponentially toward the thermodynamic equi-
librium, and T2represents the spin–spin relaxation time that characterizes the signal
decay in the NMR and MRI systems. For more details, see [1–3] and the references
therein.
The standard Bloch equations are a set of first-order ordinary differential equa-
tions that describe the magnetization behavior in static, varying magnetic fields, and
relaxation. However, to study the heterogeneity, complex structure, and memory
effects in the relaxation process, the classical Bloch equations were generalized to
fractional order by extending the first-order time derivative to a derivative of non-
integer order [4]. From this point of view, the fractional operator is considered as a
robust framework to account for anomalous diffusion in structurally heterogeneous
tissues, porous and composite materials. This is due to the nonlocal nature of frac-
tional derivatives. On the other hand, the utility of generalizing the FBEs with their
contributions have recently appeared in the literature. For example, numerical and
simulation models of integer and fractional orders of the BEs have been proposed
in [5]. In [6], the FBEs have been used to describe anomalous NMR relaxation phe-
nomena. Also, these equations have been considered with time delays, and different
stability behaviors for T1and T2processes were analyzed [7].
Our motivation for this chapter has been devoted to studying approximate solu-
tions for nonlinear Bloch models of fractional order utilizing a numerical multistep
approach based on the Adomian decomposition method (ADM). The problem under
consideration is subjected to appropriate initial conditions over a vast domain. Indeed,
the ADM is efficiently used to provide approximate solutions for many nonlinear
fractional problems in convergent series formula with accurately computable struc-
tures. Unfortunately, such approximations are found to be not valid for the large
values of tfor some systems. So, the multistep approach is needed that offers an
accurate solution over a longer time frame compared to standard ADM.
Consider the following fractional-modified transformed model of nonlinear BEs
that govern the evolution of the magnetization:
Dα1x(t)=δy(t)+γz(t)(x(t)sin(c)−y(t)cos(c)−x(t)
2),
Dα2y(t)=−δx(t)−z(t)+γz(t)x(t)cos(c)+y(t)sin(c)−y(t)
2,
Dα3z(t)=y(t)+γsin(c)x2(t)+y2(t)−z(t)−1
1,
(1)
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 155
subject to the initial conditions
x(0)=x0,y(0)=y0,z(0)=z0,(2)
where t≥0,x0,y0,z0are real finite constants, δ,γ,c,
1,and 2are physical
parameters, Dαi,i=1,2,3,are the fractional derivatives of order αiin Caputo
sense that will be introduced in the next section, and x(t), y(t), and z(t)are analytical
unknown functions to be determined.
Anyhow, some of the well-known analytic and numeric techniques were modified
for solving the FBEs such as the finite difference method [8], the homotopy pertur-
bation method [9], the predictor-corrector method [10], the Chebyshev polynomials
method [3], the operational matrix methods based on Legendre scaling and Laguerre
polynomials [11], and the multistep generalized differential transform method [12].
This chapter introduces MADM for fractional nonlinear problems and contains
the following sections:
1. Introduction
2. Preliminaries for fractional calculus
3. Principle of Adomian decomposition method
4. Multistep Adomian decomposition method
5. Nonlinear fractional Bloch equations and its modification
6. Multistep approach for modified fractional Bloch equations.
2 Preliminaries for Fractional Calculus
The subject of fractional calculus is not new. It is a generalization of classical calcu-
lus that deals with the ordinary differentiation and integration of arbitrary order. The
basic idea of fractional calculus goes back to Leibniz in a letter to L‘Hospital in 1695
“Can the meaning of derivatives with integer order be generalized to derivatives with
non-integer orders?”. This concept was developed almost in tandem with the evolu-
tion of the classical ones. Anyhow, the fractional operators highlight the intermediate
behaviors that cannot be modeled by traditional theory [13]. Nowadays, fractional
calculus has become an effective instrument in theoretical and applied fields including
physics, bioengineering, finance, signal processing, and so forth [14–21]. Moreover,
fractional models can be used to describe the memory and transmissibility for mul-
tiple types of materials. So, it plays a vital role in modeling many scientific issues,
especially in the anomalous transport process and Hamiltonian chaos.
Unlike the classical calculus, which has unique definitions and clear geometri-
cal and physical interpretations, there are numerous definitions for the operations
of differentiation and integration of fractional order. Riemann–Liouville, Riesz,
Grünwald–Letnikov, and Caputo are some examples of these definitions. In this
chapter, the Caputo concept was preferred due to the facts that the derivative of any
constant is equal to zero, and the initial conditions are treated in similar form to those
goyal.praveen2011@gmail.com
156 A. Freihat et al.
for integer order [22–31]. Next, some main definitions and results concerned with
fractional calculus theory are briefly mentioned.
It is well known that the Cauchy’s formula for n∈N,a,t∈R,t>aholds such
that
Jn
af(t)=
t
a
τ1
a
...
τn−1
a
f(τn)dτn...dτ2dτ1=1
(n−1)!
t
a
(t−τ)n−1f(τ)dτ.
Thus, if nreplaced by a positive real number αand (n−1)!by Gamma function
(n), then a formula of fractional integration can be obtained as in the following
definition:
Definition 1 The fractional operator Jα
aof order αfor a function f(t)
Jα
af(t)=1
(α)t
a
(t−τ)1−αf(τ)dτ,0≤τ<t,α>0,
is called the Riemann–Liouville fractional integral operator.
The following are some of the interesting properties of the operator Jα
a:
1. For α=0, Jα
ais the identity operator.
2. The operator Jα
ais linear, that is, Jα
a(cf(t)±g(t)) =cJα
af(t)±Jα
ag(t), for any
c∈R.
3. If f(t)is continuous for t≥0, then lim
α→0Jα
af(t)=f(t).
4. Jα1
aJα2
af(t)=Jα1+α2
a(f(t))=Jα2
aJα1
af(t),α1,α2>0.
Definition 2 The fractional operator D∗α
aof order αfor a function f(t)
D∗α
af(t)=1
(n−α)
dn
dtnt
a
f(τ)
(t−τ)α−n+1dτ,n−1<α<n,n∈N,
is called the Riemann–Liouville fractional derivative operator.
Here, it should be observed that if α∈N, then the operator D∗α
ais reduced to
the standard integer-order differential operator Dn=dn
dtn.In 1967, an alternative
operator to the above Riemann–Liouville fractional derivative has been presented by
Caputo as follows:
Definition 3 The fractional operator Dα
0of order αfor a function f(t)
Dαf(t)=1
(n−α)t
a
f(n)(τ)
(t−τ)α−n+1dτ,n−1<α<n,n∈N,
is called the Caputo-fractional derivative operator.
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 157
The following are some of the interesting properties of the operator Dα:
1. For α=n,wehaveDαf(t)=dn
dtnf(t).
2. The operator Dαis linear, that is, Dα(cf(t)±g(t)) =cDαf(t)±Dαg(t), for
any c∈R.
3. Dαc=0 for any constant c∈R.
4. For γ>n−1,we have Dαtγ=(γ+1)
(γ+1−α)tγ−αfor n−1<α<n, and is equal
to zero otherwise.
3 Principle of Adomian Decomposition Method
The Adomian decomposition method (ADM) is an alternative systematic technique
for providing a robust algorithm for analytically approximate solutions and numerical
optimization of several fractional applications in physics and engineering. The main
features of ADM lie in that it can be directly applied for solving nonlinear fractional
problems without the need for unphysical restrictive assumptions such as lineariza-
tion, discretization, perturbation, guessing the initial data, etc. [32–35]. Indeed, the
ADM concept evolved to deal with the linear, nonlinear, stochastic, and determinis-
tic operator problems of Taylor’s analytical series with an easily computable, easily
verifiable, and rapidly convergent sequence of analytic approximate functions.
For understanding the ADM concept, consider the following nonlinear problem
in general form:
u=Nu +f,(3)
where fis the system input, uis the system output, and Nis the nonlinear operator
which is assumed to be analytic.
The ADM decomposes the solution into a series
u=
∞
n=0
un,
and decomposes the nonlinear term Nu into a series
Nu =
∞
n=0
An,
where An’s are called the Adomian polynomials, which are depending on the values
of u0,u1, ..., un.
In the first approach given by Adomian, An’s are obtained from the following
equalities:
q=
∞
n=0
λnun,
Nq =N∞
n=0
λnun=
∞
n=0
λnAn,
goyal.praveen2011@gmail.com
158 A. Freihat et al.
where λis a grouping parameter of convenience.
Formally, the Adomian polynomials An’s for the nonlinearity are obtained by the
following formula:
An=1
n!
dn
dλnN∞
k=0
λkukλ=0
,n=0,1,2, ....
Consequently, the above process leads to the equality
∞
n=0
un=
∞
n=0
An+f,
in which the Adomian polynomials Ancan be listed, inclusively, by
A0=f,
A1=A0(u0),
A2=A1(u0,u1),
.
.
.
An=An−1(u0,u1, ..., un−1).
Therefore, the solution ucan be written as a series of functions unsuch that
∞
n=0
|un|<+∞.
4 Multistep Adomian Decomposition Method
Multistep Adomian decomposition method (MADM) is effectively utilized due to
many advantages in the scientific application. Indeed, since it is based on ADM, so
there is no need for unphysical restrictive assumptions or small and auxiliary param-
eters. However, the approximate solution obtained by ADM is usually converged in
a small interval but it is not valid or completely divergent over the broader term. So,
the MADM is needed to partition the domain of interest into small time steps, which
offers a powerful accuracy, especially for the nonlinear problems [36–40].
For perception of Ms-DTM basic idea, consider a general system of fractional
differential equations
Dα1x1(t)=F1(t,x1(t), ..., xn(t)),
Dα2x2(t)=F2(t,x1(t), ..., xn(t)),
.
.
.
Dαnxn(t)=Fn(t,x1(t), ..., xn(t)),
(4)
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 159
subject to the initial conditions
xi(0)=ci,i=1,2, ..., n,(5)
where 0 ≤t≤T,ci∈R(i=1,2, ..., n), Dαi’s are the Caputo-fractional derivative
of order αi,0<αi1,for i=1,2,...,n,and Fi’s,i=1,2, ..., nare linear or
nonlinear functions in terms of x1(t), ..., xn(t).
To illustrate the MADM for solving such fractional system, the main ideas of the
multistep technique are introduced as follows:
Suppose that the interval [0,T]can be divided into m-subintervals of equal length
t,such as [t0,t1],[t1,t2],[t2,t3], ..., [tm−1,tm]with t0=0,tm=T.Let t∗be the
initial value for each subintervals and let xi,j(t), i=1,2, ..., n,j=1,2, ..., mbe
approximate solutions in each subinterval [tj−1,tj],j=1,2, ..., m.
Consequently, system (4) can be converted equivalently into
Dα1x1,j(t)=F1,j(t,x1,j(t), ..., xn,j(t)),
Dα2x2,j(t)=F2,j(t,x1,j(t), ..., xn,j(t)),
.
.
.
Dαnxn,j(t)=Fn,j(t,x1,j(t), ..., xn,j(t)),
(6)
where 0 <αi≤1,i=1,2,...,n,and j=1,2, ..., m.
By applying Jαion both the sides of (6)for j=1,2, ..., m, it follows that
x1,j(t)=x1,j(t∗)+Jα1F1,j(t,x1,j(t), ..., xn,j(t)),
x2,j(t)=x2,j(t∗)+Jα2F2,j(t,x1,j(t), ..., xn,j(t)),
.
.
.
xn,j(t)=xn,j(t∗)+JαnFn,j(t,x1,j(t), ..., xn,j(t)),
(7)
Here, by employing the ADM to system (7), we have the following equalities:
xi,j(t)=xi,j(t∗)+
∞
k=1
xi,j,k(t), i=1,2,...,n,j=1,2, ..., m,(8)
Fi,j(t,x1,j(t),...,xn,j(t)) =
∞
k=0
Ai,j,k,i=1,2,...,n,j=1,2, ..., m,(9)
where Ai,j,k’s are Adomian polynomials, which are depending on the values of
x1,j,0,..., x1,j,k,x2,j,0,..., x2,j,k,..., xn,j,0,..., xn,j,k.
Consequently, the above process leads to the equality
goyal.praveen2011@gmail.com
160 A. Freihat et al.
∞
k=0
xi,j,k(t)=xi,j(t∗)+Jαi∞
k=0
Ai,j,k(x1,j,0, ..., x1,j,k, ..., xn,j,0, ..., xn,j,k),
(10)
i=1,2,...,n,j=1,2, ..., m.
Now, for i=1,2,...,n,j=1,2, ..., m,and k=0,1,2, ..., set
xi,j,0(t)=xi,j(t∗),
xi,j,1(t)=JαiAi,j,1(x1,j,0,x1,j,1, ..., xn,j,0,xn,j,1),
.
.
.
xi,j,k+1(t)=JαiAi,j,k(x1,j,0, ..., x1,j,k, ..., xn,j,0, ..., xn,j,k).
(11)
To determine the Adomian polynomials Ai,j,kintroduce a parameter qinto (9)
such that
Fi,j(t,
∞
k=0
x1,j,kqk,...,
∞
k=0
xn,j,kqk)=
∞
k=0
Ai,j,kqk,(12)
thus by letting xi,j,q(t)=
∞
k=0
xi,j,kqk,one can get that
Ai,j,k=1
k!Dkα
qFi,j,q(t,x1,j,q, ..., xn,j,q)q=0,q=0,1,2, ....
=1
k!Dkα
qFi,j,q(t,
∞
k=0
x1,j,kqk, ...,
∞
k=0
xn,j,kqk)q=0
,(13)
=1
k!Dkα
qFi,j,q(t,
∞
k=0
x1,j,kqk, ...,
∞
k=0
xn,j,kqk)q=0
.
Hence, for q=0,1,2, ..., j=1,2, ..., m,the following recurrence relations are
satisfied:
xi,j,0(t)=xi,j(t∗),
xi,j,1(t)=JαiFi,j,q(t,
∞
k=0
x1,j,1q, ...,
∞
k=0
xn,j,1q)q=0,
,
.
.
.
xi,j,k+1(t)=Jαi1
k!Dkα
qNi,j(t,
∞
k=0
x1,j,kqk, ...,
∞
k=0
xn,j,kqk)q=0,
.
(14)
The N-term approximate solution xN
i,j(t)can be given by
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 161
xN
i,j(t)=
N
k=0
xi,j,k(t),
such as lim
N−→ ∞ xN
i,j=xi,j.
For the convergence of MADM, if the system (4) admits a unique solution, then
the MADM will produce a unique solution, while if the system (4) does not possess
a unique solution, then the MADM will give a solution among many (possible) other
solutions [41].
The solution of system (4) in each subinterval [tj−1,tj],j=1,2, ..., n,has the
following form:
xi,j(t)=
∞
k=0
xi,j,k(t−tj−1), i=1,2,...,n,j=1,2, ..., m,(15)
while the solution of system (4) in the interval [0,T]can be given as
xi(t)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩xi,1(t)t∈[t0,t1],
xi,2(t)t∈[t1,t2],
.
.
..
.
.
xi,m(t)t∈tm−1,tm,
.(16)
subject to the initial guesses
xi,1(t∗)=ci,
xi,j(t∗)=xi,j(tj−1)=xi,j−1(tj−1), i=1,2, ..., n,j=2,3, ..., m.
5 Nonlinear Fractional Bloch Equations
and its Modification
MR experiments are mostly performed with a large number of electron spins’ reso-
nance and nuclear spins that measure the behaviors and quantities of identical spins.
The BEs describe the spin systems of electronic and nuclear resonance in arbitrary
magnetic fields over the time–space from transient processes to steady states. The
classical BEs is derived from a magnetization Mprocessing in the magnetic induc-
tion field with the presence of a constant radio frequency and given in the following
form:
goyal.praveen2011@gmail.com
162 A. Freihat et al.
dM
x(t)
dt =ω0My(t)−Mx(t)
T2
,
dMy(t)
dt =−ω0Mx(t)−My(t)
T2
,(17)
dM
z(t)
dt =M0−Mz(t)
T1
,
subject to the initial conditions
Mx(0)=0,My(0)=100,Mz(0)=0,(18)
where M0is the equilibrium magnetization, ω0is the resonant frequency in terms
of the static magnetic field B0(z-component) given by the Larmor relationship
ω0=γB0,γ/2πis the gyromagnetic ratio, T1is the spin–lattice relaxation time
that characterizes the rate of which the longitudinal Mz-component recovers expo-
nentially toward the thermodynamic equilibrium, T2is the spin–spin relaxation time
that characterizes the signal decay in the NMR and MRI systems, and Mx(t), My(t),
and Mz(t)represent the system magnetization in x,y,and zcomponent, respectively.
Here, the set of analytical solution is given by
Mx(t)=e−t/T2Mx(0)cos ω0t+My(0)sin ω0t,
My(t)=e−t/T2My(0)cos ω0t−Mx(0)sin ω0t,(19)
Mz(t)=Mz(0)e−t/T1+M01−e−t/T1.
Some fractional models have been proposed for the BEs, for instance, the follow-
ing model is investigated in [42] by utilizing the Caputo sense with fractional order
0<α≤1 to study the spin dynamics and magnetization relaxation, in the simple
case of a single spin particle at resonance in a static magnetic field B0:
DαMx(t)=ω
0My(t)−Mx(t)
T
2
,
DαMy(t)=−ω
0Mx(t)−My(t)
T
2
,(20)
DαMz(t)=M0−Mz(t)
T
1
,
where ω
0=ω0/τα−1
2,T
1=τ1−α
1T1,T
2=τ1−α
2T2,and τ1and τ2are fractional time
constants.
The next anomalous model is investigated in [43] by utilizing the Riemann–
Liouville with fractional orders 0 <α,β≤1 to fit the derived spin–spin relaxation
T2decay curves to relaxation data from normal and trypsin-digested bovine nasal
cartilage:
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 163
dM
x(t)
dt =ω0My(t)−D1−αMx(t)
T2
,
dMy(t)
dt =−ω0Mx(t)−D1−αMy(t)
T2
,(21)
dM
z(t)
dt =D1−βM0−Mz(t)
T1
,
where D1−αand D1−βare the time-fractional Riemann–Liouville derivative. For
more details about these models, see [10] and the references therein.
Furthermore, the following modified model of nonlinear BEs governs the evolu-
tion of the magnetization M:
dM
x
dt =ρMy+GM
z(Mxsin ψ−Mycos ψ)−Mx
T2
,
dMy
dt =−ρMx−ω1Mz+GM
z(Mxcos ψ+Mysin ψ)−My
T2
,(22)
dM
z
dt =ω1My−Gsin ψ(Mx)2+My2−Mz−M0
T1
,
where ρ=ωrf −ω0,ωrf is the frequency of a constant radio frequency field with
intensity ω1/γ,γ/2πis the gyromagnetic ratio, Gis the enhancement factor with
respect to the magnitude of the transverse magnetization, ψis the feedback field, T1
and T2are the longitudinal time and transverse relaxation time, respectively. This
model can be transformed by introducing
t→ω1t,G→G
ω1
=γ,δ→ρ
ω1
,
i→ω1Ti,i=1,2,
and
Mx→Mx
M0
=x,My→My
M0
=y,Mz→Mz
M0
=z,
into dimensionless variables model in the following form:
dx
dt =δy+γz(xsin ψ−ycos ψ)−x
2
,
dy
dt =−δx−z+γz(xcos ψ+ysin ψ)−y
2
,(23)
dz
dt =y−γsin ψx2+y2−z−1
1
.
In this chapter, we consider the modified transformed model by utilizing the
Caputo-fractional derivative of order αi(0<αi≤1,i=1,2,3)described in Eqs.
(1) and (2).
goyal.praveen2011@gmail.com
164 A. Freihat et al.
6 Multistep Approach for Modified Fractional Bloch
Equations
The objective of the section is to obtain the approximate solution of the fractional-
modified transformed model (1) and (2) using the MADM. To perform so, set the
values of the magnetization parameters as follows:
γ=35,δ=−1.26,c=0.173,
1=5 and 2=2.5,(24)
and set the initial conditions as
x(0)=0.5,y(0)=−0.5 and z(0)=0.(25)
For j=1,2,3, ..., n,define the nonlinear terms by
N1,j(q)=δyj(q)+γzj(q)(xj(q)sin(c)−yj(q)cos(c)) −xj(q)
2
=
∞
m=0
A1,j,m,
N2,j(q)=−δxj(q)−zj(q)+γzj(q)(xj(q)cos(c)+yj(q)sin(c)) −yj(q)
2
=
∞
m=0
A2j,m,(26)
N3,j(q)=yj(q)−γsin(c)(x2
j(q)+y2
j(q)) −zj(q)−1
1
=
∞
m=0
A3,j,m,
where
xj(q)=
K
m=0
xj,m(t)qm,yj(q)=
K
m=1
yj,m(t)qm,zj(q)=
K
m=1
zj,m(t)qm,
and the Adomian polynomials Ai,j,m,i=1,2,3,is given by
Ai,j,m=1
m!Dmα
qNi,j(q)q=0,j=1,2, ..., n,m=1,2, ..., K.
So in this case, we have to satisfy the initial conditions at each subinterval
[tj−1,tj],j=1,2,3, ..., n,such that
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 165
x1(t∗)=0.5,xj(t∗)=xj(tj−1)=xj−1(tj−1),
y1(t∗)=−0.5,yj(t∗)=yj(tj−1)=yj−1(tj−1),
z1(t∗)=0,zj(t∗)=zj(tj−1)=zj−1(tj−1),
where t∗is the initial value for each subinterval.
For j=1,2,3, ..., n,m=0,1,2, ...K, the Adomain decomposition series (11)
leads to the following scheme:
xj,0=xj(t∗), xj,m+1=Jα1A1,j,m,
yj,0=yj(t∗), yj,m+1=Jα2A2,j,m,
zj,0=zj(t∗), zj,m+1=Jα3A3,j,m.
The solutions of system (1), (2) in each subinterval [tj−1,tj],j=1,2, ..., n,has
the following form:
xj(t)=
K
m=0
xj,m(t−tj−1),
yj(t)=
K
m=0
yj,m(t−tj−1),
zj(t)=
K
m=0
zj,m(t−tj−1),
and the solution in the interval [0,T]is given by
x(t)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩x1(t)t∈[t0,t1],
x2(t)t∈[t1,t2],
.
.
..
.
.
xn(t)t∈tn−1,tn,
y(t)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩yj(t)t∈[t0,t1],
yj(t)t∈[t1,t2],
.
.
..
.
.
yj(t)t∈tn−1,tn,
z(t)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩z1(t)t∈[t0,t1],
z2(t)t∈[t1,t2],
.
.
..
.
.
zn(t)t∈tn−1,tn.
goyal.praveen2011@gmail.com
166 A. Freihat et al.
Table 1 Numerical results of x(t)at fractional order α=1
tMADM IRKM Absolute error Relative error
0.5 0.0900899 0.09009 2.63948 ×10−82.92983 ×10−7
1.0 0.0100309 0.0100309 1.28886 ×10−81.28489 ×10−6
1.5 0.0362608 0.0362609 1.63743 ×10−84.51568 ×10−7
2.0 0.0289491 0.0289491 1.28465 ×10−84.43762 ×10−7
2.5 0.0289752 0.0289751 5.42791 ×10−91.87330 ×10−7
3.0−0.00164103 −0.00164104 1.23807 ×10−87.54445 ×10−6
3.5 0.0399933 0.0399933 1.02814 ×10−82.57077 ×10−7
4.0−0.0277855 −0.0277855 5.37176 ×10−91.93330 ×10−7
4.5 0.0100665 0.0100664 1.23103 ×10−81.22291 ×10−6
5.0 0.0967854 0.0967853 5.44949 ×10−85.63049 ×10−7
Table 2 Numerical results of y(t)at fractional order α=1
tMADM IRKM Absolute error Relative error
0.5−0.139658 −0.139658 1.54080 ×10−81.10327 ×10−7
1.0−0.0178745 −0.0178746 2.15256 ×10−81.20426 ×10−6
1.5−0.0000591291 −0.0000591296 5.18577 ×10−10 8.77018 ×10−6
2.0 0.0142948 0.0142948 3.69011 ×10−11 2.58144 ×10−9
2.5 0.0231906 0.0231906 1.42967 ×10−86.16489 ×10−7
3.0 0.020148 0.020148 3.34537 ×10−91.66039 ×10−7
3.5−0.0321054 −0.0321054 1.83989 ×10−95.73080 ×10−8
4.0 0.0174674 0.0174673 6.89852 ×10−93.94938 ×10−7
4.5 0.100417 0.100417 4.78751 ×10−94.76763 ×10−8
5.0 0.187399 0.187399 5.27830 ×10−92.81661 ×10−8
For numerical simulation, we have compared the MAD results from the implicit
Runge–Kutta method (IRKM) at the fractional order αi=1,i=1,2,3,over the
interval [0,5]with step size 0.5, K=5, n=800, and the numeric results are listed
in Tables 1,2, and 3. From these tables, it is observed that the accuracy by the MADM
is compatible with the IRKM at the value of fractional order α.All the results are
calculated by using the Mathematica software package.
The graphical results and parametric plots of the MADM and IRKM are given
in Figs. 1,2, and 3at fractional order αi=1,i=1,2,3,over the interval [0,5].
While Figs. 4and 5are given for the MAD solutions at fractional order αi=0.9,
i=1,2,3,over the interval [0,5].
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 167
Table 3 Numerical results of z(t)at fractional order α=1
tMADM IRKM Absolute error Relative error
0.5−0.519828 −0.519828 1.75095 ×10−83.36833 ×10−8
1.0−0.398178 −0.398178 1.51377 ×10−83.80175 ×10−8
1.5−0.263365 −0.263365 1.44217 ×10−85.47591 ×10−8
2.0−0.142461 −0.142461 8.71885 ×10−96.12018 ×10−8
2.5−0.0286219 −0.0286219 1.27388 ×10−84.45072 ×10−7
3.0 0.0786463 0.0786463 1.34023 ×10−81.70413 ×10−7
3.5 0.158251 0.158251 7.90765 ×10−94.99689 ×10−8
4.0 0.237731 0.237731 8.78612 ×10−93.69582 ×10−8
4.5 0.288485 0.288485 1.02007 ×10−83.53596 ×10−8
5.0 0.285525 0.285525 2.49369 ×10−88.73371 ×10−8
Fig. 1 The MAD solutions and the corresponding IRKM at α=1andt∈[0,5]
Fig. 2 The parametric plots of the solution xversus yat α=1
goyal.praveen2011@gmail.com
168 A. Freihat et al.
Fig. 3 The 3D parametric plots of the solutions at α=1
Fig. 4 The parametric plots
of the MADM solutions x
versus yat α=0.9
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 169
Fig. 5 The 3D parametric
plots of the MADM
solutions at α=0.9
References
1. Matuszak, Z.: Fractional Bloch’s equations approach to magnetic relaxation. Curr. Topics
Biophys. 37, 9–22 (2014)
2. Petras, I.: Modeling and numerical analysis of fractional order Bloch equations. Comp. Math.
Appl. 61, 341–56 (2011)
3. Singh, H.: A new numerical algorithm for fractional model of Bloch equation in nuclear mag-
netic resonance. Alexandria Eng. J. 55, 2863–2869 (2016)
4. Ascher, U.M., Mattheij, R.M., Russell, R.D.: Numerical solution of boundary value problems
for ordinary differential equations. Classics Appl. Math. 13 (1995)
5. Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.L.: Transient chaos in fractional Bloch
equations. Comput. Math. Appl. 64, 3367–3376 (2012)
6. Magin, R., Li, W., Velasco, M., Trujillo, J., Reiter, D., Morgenstern, A., Spencer, R.: J. Magn.
Reson. 210(2), 184 (2011)
7. Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.L.: Fractional Bloch equation with
delay. Comput. Math. Appl. 61(5), 1355–1365 (2011)
8. Hajipour, M., Jajarmi, A., Baleanu, D.: An efficient nonstandard finite difference scheme for a
class of fractional chaotic systems. J. Comput. Nonlinear Dyn. 13, (2018). In press
9. Kumar, S., Faraz, N., Sayevand, K.: A fractional model of Bloch equation in NMR and its
analytic approximate solution. Walailak J. Sci. Technol. 11(4), 273–285 (2014)
10. Yu, Q., Liu, F., Turner, I., Burrage, K.: Numerical simulation of the fractional Bloch equations.
J. Comput. Appl. Math. 255, 635–651 (2014)
11. Singh, H., Singh, C.S.: A reliable method based on second kind Chebyshev polynomial for the
fractional model of Bloch equation. Alexandria Eng. J. (2017). In press
12. Abuteen, E., Momani, S., Alawneh, A.: Solving the fractional nonlinear Bloch system using
the multi-step generalized differential transform method. Comput. Math. Appl. 68, 2124–2132
(2014)
13. Herrmann, R.: Fractional calculus: an introduction for physicists. World Scientific (2014)
goyal.praveen2011@gmail.com
170 A. Freihat et al.
14. Al-Smadi, M., Abu Arqub, O.: Computational algorithm for solving fredholm time-fractional
partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math.
Comput. 342, 280–294 (2019)
15. Abu Arqub, O., Al-Smadi, M.: Atangana-Baleanu fractional approach to the solutions of
Bagley-Torvik and Painlevé equations in Hilbert space, Chaos Solitons & Fractals (2018).
In press
16. Abu Arqub, O., Odibat, Z., Al-Smadi, M.: Numerical solutions of time-fractional partial inte-
grodifferential equations of Robin functions types in Hilbert space with error bounds and error
estimates. Nonlinear Dyn., 1–16 (2018)
17. Moaddy, K., Freihat, A., Al-Smadi, M., Abuteen, E., Hashim, I.: Numerical investigation
for handling fractional-order Rabinovich-Fabrikant model using the multistep approach. Soft
Comput. 22(3), 773–782 (2018)
18. Agarwal, P., Chand, M., Baleanu, D., O’Regan,D., Jain, S.: On the solutions of certain fractional
kinetic equations involving k-Mittag-Leffler function. Adv. Differ. Equ. 2018(1), 249 (2018)
19. Khalil, H., Khan, R.A., Al-Smadi, M., Freihat, A.: Approximation of solution of time fractional
order three-dimensional heat conduction problems with Jacobi Polynomials. Punjab Univ. J.
Math. 47(1), 35–56 (2015)
20. Agarwal, P., Al-Mdallal, Q., Cho, Y.J., Jain, S.: Fractional differential equations for the gener-
alized Mittag-Leffler function. Adv. Differ. Equ. 2018(1), 58 (2018)
21. Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation meth-
ods for solving fractional diffusion equation. Phys. A: Stat. Mech. Appl. 500, 40–49 (2018)
22. Alaroud, M., Al-Smadi, M., Ahmad, R.R., Salma Din, U.K.: Computational optimization of
residual power series algorithm for certain classes of fuzzy fractional differential equations.
Int. J. Differ. Equ. 2018, 1–11 (2018). Art. ID 8686502
23. Al-Smadi, M.: Simplified iterative reproducing kernel method for handling time-fractional
BVPs with error estimation. Ain Shams Eng. J. (2017). In press. https://doi.org/10.1016/j.asej.
2017.04.006
24. Abu Arqub, O., Al-Smadi, M.: Numerical algorithm for solving time-fractional partial inte-
grodifferential equations subject to initial and Dirichlet boundary conditions. Numer. Methods
Partial Differ. Equ., 1–21 (2017). https://doi.org/10.1002/num.22209
25. Al-Smadi, M., Abu Arqub, O., Shawagfeh, N., Momani, S.: Numerical investigations for sys-
tems of second-order periodic boundary value problems using reproducing kernel method.
Appl. Math. Comput. 291, 137–148 (2016)
26. Agarwal, P., Choi, J., Jain, S., Rashidi, M.M.: Certain integrals associated with generalized
Mittag-Leffler function. Commun. Korean Math. Soc 32(1), 29–38 (2017)
27. Agarwal, P., Berdyshev, A., Karimov, E.: Solvability of a non-local problem with integral
transmitting condition for mixed type equation with Caputo fractional derivative. Results Math.
71(3–4), 1235–1257 (2017)
28. Agarwal, P., Jain, S., Mansour, T.: Further extended Caputo fractional derivative operator and
its applications. Russian J. Math. Phys. 24(4), 415–425 (2017)
29. Abuteen, E., Freihat, A., Al-Smadi, M., Khalil, H., Khan, R.A.: Approximate series solution
of nonlinear, fractional Klein-Gordon equations using fractional reduced differential transform
method. J. Math. Stat. 12(1), 23–33 (2016)
30. Khalil, H., Khan, R.A., Al-Smadi, M., Freihat, A., Shawagfeh, N.: New operational matrix for
Shifted Legendre polynomials and fractional differential equations with variable coefficients.
Punjab Univ. J. Math. 47(1), 81–103 (2015)
31. Abu-Gdairi, R., Al-Smadi, M., Gumah, G.: An expansion iterative technique for handling
fractional differential equations using fractional power series scheme. J. Math. Stat. 11(2),
29–38 (2015)
32. Adomian, G., Rach, R.C., Meyers, R.E.: Numerical integration, analytic continuation, and
decomposition. Appl. Math. Comput. 88, 95–116 (1997)
33. Wazwaz, A.M.: The modified Adomian decomposition method for solving linear and nonlinear
boundary value problems of tenth-order and twelfthorder. Int. J. Nonlinear Sci. Numer. Simul.
1, 17–24 (2000)
goyal.praveen2011@gmail.com
Multistep Approach for Nonlinear Fractional Bloch System … 171
34. Wazwaz, A.M.: A note on using Adomian decomposition method for solving boundary value
problems. Found. Phys. Lett. 13, 493–498 (2000)
35. Ebadi, G., Rashedi, S.: The extended Adomian decomposition method for fourth order boundary
value problems. Acta Univ. Apulensis 22, 65–78 (2010)
36. Al-Smadi, M., Freihat, A., Khalil, H., Momani, S., Khan, R.A.: Numerical multistep approach
for solving fractional partial differential equations. Int. J. Computat. Methods 14(1750029),
1–15 (2017). https://doi.org/10.1142/S0219876217500293
37. Al-Smadi, M., Freihat, A., Abu Hammad, M., Momani, S., Abu Arqub, O.: Analytical approxi-
mations of partial differential equations of fractional order with multistep approach. J. Comput.
Theor. Nanosci. 13(11), 7793–7801
38. Momani, S., Abu Arqub, O., Freihat, A., Al-Smadi, M.: Analytical approximations for Fokker-
Planck equations of fractional order in multistep schemes. Appl. Comput. Math. 15(3), 319–330
(2016)
39. Odibat, Z.M., Bertelle, C., Aziz-Alaoui, M.A., Duchamp, G.H.E.: A multi-step differential
transform method and application to non-chaotic or chaotic systems. Comput. Math. Appl. 59,
1462–1472 (2010)
40. Gokdogan, A., Merdan, M., Yıldırım, A.: Adaptive multi-step differential transformation
method for solving nonlinear equations. Math. Comput. Model. 55(3–4), 761–769 (2012)
41. Jafari, H., Daftardar-Gejji, V.: Solving a system of nonlinear fractional differential equations
using Adomian decomposition. J. Comput. Appl. Math. 196, 644–651 (2006)
42. Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Concepts Magn.
Reson. Part A 34A(1), 16–23 (2009)
43. Velasco, M., Trujillo, J., Reiter, D., Spencer, R., Li, W., Magin, R.: Anomalous fractional order
models of NMR relaxation, In: Proceedings of FDA’10, the 4th IFAC Workshop Fractional
Differentiation and its Applications, Vol. 1, 2010. Paper number FDA10-058
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional
Ultrasound Waves with Attenuation
in Fractal Media
E. A. Abdel-Rehim and A. S. Hashem
Abstract In this paper, we are interested in studying the propagation of the over
diagnostic ultrasound waves through complex biological vascular networks such as
the tumor tissue. Evidence shows that the over diagnostic wave propagates through
complex media with power law of non-integer order t−ν,1<ν<2. Evidence shows
also that the vascular morphology of the tumor is non-smooth and is a complex media
that means it is a fractal media. The wave propagates through this fractal media
which exhibits with extremely long jumps whose length is distributed according
to the Lévy long tail ∼|x|−1−α,0<α<2. Therefore, the space–time-fractional
forced wave equation with attenuation, or the so-called multi-term wave equation,
mathematically models this medicine problem. This equation mathematically models
many other physical, biological, chemical, and environmental problems. We get the
approximate solution of this model to study the time evolution of the propagated wave
by adopting the backward Grünwald–Letnikov scheme joining with the common
finite difference method. We investigate numerically the effect of the time fractional
on the propagation of the wave as well as the effect of the space-fractional order
for the three cases as: 0 <α<1, 1 <α<2, and α=1. The stability condition of
each approximate solution is also discussed separately. Finally, we prove that the
space-fractional order αhas no effect on the stability condition.
Keywords Attenuation ·Over diagnostic wave equation ·Explicit scheme ·
Caputo time derivative ·Stability ·Space–fractional derivative ·Backward
grünwald–Letnikov scheme ·memory process ·vascular morphology of tumor
Mathematics Subject Classification: 26A33 ·35L05 ·35J05 ·45K05 ·47G30 ·
33E20 ·65N06 ·80-99
E. A. Abdel-Rehim (B
)·A. S. Hashem
Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt
e-mail: entsarabdelrehim@gmail.com;entsarabdelrehim@yahoo.com
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_10
173
goyal.praveen2011@gmail.com
174 E. A. Abdel-Rehim and A. S. Hashem
1 Introduction
The propagation of the ultrasound waves with attenuation in a smooth biological
tissue is mathematically modeled by the forced wave equation with attenuation.
Whereas attenuation means loss of wave amplitude due to all mechanisms including
absorption, scattering through the tissues, and mode conversion [1]. This equation is
defined as
∂2u(x,t)
∂t2=k∂u(x,t)
∂t+a∂2u(x,t)
∂x2−∂
∂x(F(x)u(x,t)) , −R≤x≤R,t>o,
(1.1)
here u(x,t)is the pressure amplitude, a>0 is the general positive constant, and
0<k<1 is the attenuation coefficient. F(x)represents the external force, which
supplies energy to the ultrasound wave.
It is known that the vascular morphology of tumor is significantly different from
normal tissues. Vascular networks are developed and ordered with a hierarchical
vessel arrangement. While the tumor vascular networks randomly consisted of a dis-
orderly tangle of vessels. In other words, tumor vasculature has long been known to
be more chaotic in appearance than normal vasculature. Complexity, irregularities,
and poorly regulated growth are some of the known characteristics of cancer. Tumor
vasculature, in particular, defines the optimized growth patterns of healthy vascula-
ture and is known to contain many tortuous vessels, shunts, vascular loops, widely
variable inter-vascular distances, and large vascular areas. This complex structure
represents a fractal media. That means, when the over diagnostic ultrasound wave
propagates in tumor vasculature, it propagates in fractal medium. For more infor-
mation, see [2–8]. In other words, the over diagnostic ultrasound wave propagates
with large deviation from the stochastic process of Brownian motion. The Lévy sta-
ble motion is the natural generalization of the Brownian motion but it is different
because of the occurrence of the extremely long jumps whose length is proportional
to the Lévy tail ∼|x|−1−α,0<α<2. This long jump requires not finite velocity.
Evidence shows that the velocity of propagation in fractal media has a power law
of non-integer frequency of order t−ν,1<ν<2, see [9–11,18]. Then to study the
space–time-fractional forced diagnostic ultrasound wave propagation with attenua-
tion, Eq. (1.1) should be modified to
D
t∗
βu(x,t)=k∂u(x,t)
∂t+aD
x0
αu(x,t)−∂
∂x(F(x)u(x,t)) , (1.2)
where 1 <β≤2 and 0 <α<2. The used time-fractional derivative operator
D
t∗
βu(x,t)is called the Caputo-fractional operator, see [12] for more informa-
tion about the relation between Caputo-fractional derivative and the Riemann–
Liouville-fractional integral operators. The used D
x0
αu(x,t)is the Riesz–Feller
space-fractional differentiation operator, see [13]. Time-fractional wave equations
with attenuation term have been studied by Szabo [1], Caputo [14], and Treeby and
Cox [15]. The recent works [16] proposed transient-wave propagation in porous
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 175
materials using fractional modeling to take into account the frequency variability of
some dynamic coefficients of the medium like tortuosity and compressibility. Other
authors, like Tarasov [11], gave a space-fractional formulation of the hydrodynamic
equations to describe fluid flow in fractal media. Casasanta and Garra [17] studied
the space-fractional wave equation in relation to the propagation of acoustic waves
with space-dependent sound speed. We have given the approximate solution of the
time-fractional wave, forced wave (shear wave), and damped wave equations, see [9,
18].
In this paper, we are interested in finding the approximate solution of the studied
model (1.2). The approximate solution is given by adopting the backward Grünwald–
Letnikov scheme joined with the common finite difference method. We investigate
numerically the effect of the time fractional on the propagation of the wave as well as
the effect of the space- fractional order for the three cases as: 0 <α<1, 1 <α<2,
and α=1. The paper is organized as follows: Section1is devoted to the introduction.
Section 2is to introduce the approximate solution of the space–time-fractional forced
wave equation with attenuation term for all the fractional-order values. In Sect. 3,the
proof of the stability conditions are studied. Finally, Sect.4is devoted to simulate the
propagation of the waves of the previous model for different values of the parameters
β,α,f(x), and t. The interpretation of the numerical results with investigations of
the effect of the memory is also given.
2 Approximate Solutions
In this section, we discuss the approximate solution of Eq. (1.2) for all values of α
and β. To do so, identify first the used external force F(x). There are many forms
of F(x), which can be used depending on the kind of the model. In this paper, we
consider F(x)=−bx,b>0, i.e., F(x)is a linear attractive force. So far, Eq. (1.2)
takes the form
D
t∗
βu(x,t)=k∂u(x,t)
∂t+aD
x0
αu(x,t)+b∂
∂x(xu(x,t)) , (2.1)
we solve this equation with the initial conditions
u(x,0)=f(x), ut(x,0)=0,(2.2)
and the boundary conditions
u(−R,t)=u(R,t)=0.(2.3)
Discretize xand tby the grid {xj,tn:−R≤j≤R,n≥0}with xj=jh,tn=
nτ. Where h>0, and τ>0 are the steps in space and time, respectively. Introduce
the clump y(n)={y(n)
−R,y(n)
−R+1,··· ,y(n)
R−1,y(n)
R}Tto approximate the integral of the
goyal.praveen2011@gmail.com
176 E. A. Abdel-Rehim and A. S. Hashem
pressure function u(x,t)over the small interval h. The initial value y(0)is obtained
by the aid of initial condition u(x,0)=f(x). To discretize D
t∗
βu(x,t), we utilize
the backward Grünwald–Letnikov scheme as
D
τ∗
βyj(tn+1)=
n+1
m=0
(−1)mβ
myj(tn+1−m)−yj(t0)
τβ,1<β≤2,∀n∈N0.(2.4)
This scheme has been successfully utilized at [9,18–20] for modeling and simulation
the time-fractional diffusion processes. The used space-fractional operator D
x0
α,is
the symmetric Riesz–Feller operator, see [21]. We use here the Zaslavski notation
[13] to define the inverse Riesz–Feller, see also Oldham and Spanier [22], Ross and
Miller [23] and Samko [24], as
D
0
α=−1
2cos(απ/2)[I−α
++I−α
−],0<α≤2,α= 1,(2.5)
where I−α
±are the inverse of the operators Iα
±, being called the Wey l integrals. So far,
the discretization of the Riesz potential operator is as follows:
D
h0
αyj(tn)=−1
2cos απ
2
(I
h+
−α+I
h−
−α)yj(tn), 0<α≤2α= 1j∈Z.(2.6)
This scheme has been effectively used for finding the approximate solutions of the
space-fractional diffusion processes, see also [20] and the references therein. D
x0
α
has three different discretization schemes depending on the values of αas follows:
I
h±
−αyj(tn)=1
hα
∞
s=0
(−1)sα
syj±1∓s,1<α<2,(2.7)
while
I
h±
−αyj(tn)=1
hα
∞
s=0
(−1)sα
syj∓s,0<α<1.(2.8)
Finally, for the case α=1, which is related to the Cauchy distribution and because
of Eq. (2.6) one cannot use the Grünwald–Letnikov to discretize it, instead we use
the method introduced by Gorenflo and Mainardi [25].
Now joining the discretization of the time- and space-fractional differential opera-
tors, one can give the discretization of the space–time-fractional forced wave equation
with damping term for each case. Beginning by 1 <α<2,1<β<2, see for more
detail [26], one gets after minor mathematical manipulating
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 177
y(n+1)
j=bn
1−kτβ−1y(0)
j+1
1−kτβ−1
n
m=2
cmy(n+1−m)
j
+β−kτβ−1
1−kτβ−1+aμ
cos απ
2(1−kτβ−1)α
1y(n)
j
−1
1−kτβ−1bτβ(j−1)
2+aμ
2 cos απ
2
(1+α
2)y(n)
j−1
+1
1−kτβ−1bτβ(j+1)
2−aμ
2 cos απ
2
(1+α
2)y(n)
j+1
−1
1−kτβ−1
aμ
2 cos απ
2
s≥3
(−1)sα
sy(n)
j+1−s+y(n)
j−1+s.(2.9)
For ease of writing, we use the same parameters bnand cmdefining in [19]. Define
the scaling relation as
μ=τβ
hα.(2.10)
For easing the computation, we rewrite Eq. (2.9) in the matrix form as
y(n+1)=bn
1−kτβ−1y(0)+1
1−kτβ−1
n
m=2
cmy(n+1−m)+QTy(n).(2.11)
Since (−1)sα
s→0ass→∞, then we can ignore the terms which are correspond-
ing to large values of s. Define Q={qij}to be the diagonally matrix whose elements
qij are defined as
qij =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
q(1)
ij =aμ
2cosαπ
2(1−kτβ−1)(−1)j−i+1α
|j−i+1|j=i+M,i=−R,··· ,R−M
q(2)
ij =1
1−kτβ−1bτβ(j+1)
2−aμ
2cosαπ
2
(1+α
2)j=i+1,i=−R,··· ,R−1
q(3)
ij =β−kτβ−1
1−kτβ−1+aμ
cos απ
2(1−kτβ−1)α
1j=i,i=−R,··· ,R
q(4)
ij =−1
1−kτβ−1bτβ(j−1)
2+aμ
2cosαπ
2
(1+α
2)j=i−1,i=−R+1,··· ,R
q(5)
ij =aμ
2cosαπ
2(1−kτβ−1)(−1)j−i+1α
|i−j+1|j=i−M,i=−R+M,··· ,R,
(2.12)
where 2 ≤M≤R, to cover all the elements of the matrix. Now, we discuss the case
0<α<1 and 1 <β<2. Joining the discretization (2.8) with the discretization
(2.4) with the common finite difference rules for finding the approximate solution of
Eq. (2.1) and solving for y(n+1)
jafter using the scaling relation (2.10), to get
goyal.praveen2011@gmail.com
178 E. A. Abdel-Rehim and A. S. Hashem
y(n+1)
j=bn
1−kτβ−1y(0)
j+1
1−kτβ−1
n
m=2
cmy(n+1−m)
j
+β−kτβ−1
1−kτβ−1−aμ
cos απ
2(1−kτβ−1)y(n)
j
+1
1−kτβ−1aμ
2 cos απ
2α
1−bτβ(j−1)
2y(n)
j−1
+1
1−kτβ−1aμ
2 cos απ
2α
1+bτβ(j+1)
2y(n)
j+1
+1
1−kτβ−1
aμ
2 cos απ
2
s≥2
(−1)s+1α
sy(n)
j−s+y(n)
j+s.(2.13)
Equation (2.13) can be written in the same matrix form (2.11), where the diagonal
elements of the matrix Qare defined in this case as
qij =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
q(1)
ij =aμ
2cos απ
2(1−kτβ−1)(−1)j−i+1α
|j−i|j=i+M,i=−R,··· ,R−M
q(2)
ij =1
1−kτβ−1aμ
2cos απ
2α
1+bτβ(j+1)
2j=i+1,i=−R,··· ,R−1
q(3)
ij =β−kτβ−1
1−kτβ−1−aμ
cos απ
2(1−kτβ−1)j=i,i=−R,··· ,R
q(4)
ij =1
1−kτβ−1aμ
2cos απ
2α
1−bτβ(j−1)
2j=i−1,i=−R+1,··· ,R
q(5)
ij =aμ
2cos απ
2(1−kτβ−1)(−1)j−i+1α
|i−j|j=i−M,i=−R+M,··· ,R,
(2.14)
where 2 ≤M≤R. Finally, we discuss the singular case α=1 and 1 <β<2. In this
case, Gorenflo and Mainardi [25] deduced the discretization of D
0
1from the Cauchy
density p1(x,0)=1
π
1
1+x2. They replaced the factor (−1)sα
s,s∈Z, in equation
(2.7) and in Eq. (2.8)by −2
πfor s=0, and 1
π|s|(|s|+1)for s= 0,s∈Z. Substituting
with these factors with the scaling relation (2.10)forα=1, and finally with common
finite difference rules in Eq. (2.1), then solving for y(n+1)
j, to get
y(n+1)
j=bn
1−kτβ−1y(0)
j+1
1−kτβ−1
n
m=2
cmy(n+1−m)
j+β−kτβ−1
1−kτβ−1−2aμ
π(1−kτβ−1)y(n)
j
+1
1−kτβ−1aμ
2π−bτβ(j−1)
2y(n)
j−1+1
1−kτβ−1aμ
2π+bτβ(j+1)
2y(n)
j+1
+aμ
π(1−kτβ−1)
s≥2
1
s(s+1)y(n)
j+s+y(n)
j−s.(2.15)
This equation can also be written in the same matrix form (2.11). Where Qis a
matrix whose diagonal elements are defined as
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 179
qij =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
q(1)
ij =aμ
π(1−kτβ−1)
1
(j−i)( j−i+1)j=i+M,i=−R,··· ,R−M
q(2)
ij =1
1−kτβ−1aμ
2π+bτβ(j+1)
2j=i+1,i=−R,··· ,R−1
q(3)
ij =β−kτβ−1
1−kτβ−1−2aμ
π(1−kτβ−1)j=i,i=−R,··· ,R
q(4)
ij =1
1−kτβ−1aμ
2π−bτβ(j−1)
2j=i−1,i=−R+1,··· ,R
q(5)
ij =aμ
π(1−kτβ−1)
1
(i−j)(i−j+1)j=i−M,i=−R+M,··· ,R−1,
(2.16)
where 2 ≤M≤R. In what follows, we give the stability of the above studied dif-
ference schemes.
3 The Proof of the Stability
In this section, we give the necessary conditions of stability, for the three previous
cases separately, according to the Gerschorgin´stheorem. First, we use this theorem
to find the eigenvalues λi, where −R≤i≤R, of the discrete scheme (2.9), of case
a, from the inequality
|λi−qii|≤
j=i
qij =ρ,−R≤j≤R.(3.1)
The eigenvalues are contained in circles centered at
β−kτβ−1
1−kτβ−1+aμ
cos απ
2(1−kτβ−1)α
1 ,
with radius
1
1−kτβ−1bτβ(j+1)−bτβ(j−1)
2−aμ
cos απ
2
(1+α
2)−aμ
cos απ
2
∞
s=3
(−1)sα
s,
such that
|λi−β−kτβ−1
1−kτβ−1+aμ
cos απ
2(1−kτβ−1)α
1|≤ 1
1−kτβ−1
bτβ(j+1)−bτβ(j−1)
2−aμ
cos απ
2
(1+α
2)−aμ
cos απ
2
∞
s=3
(−1)sα
s=ρ,
(3.2)
and as a result
goyal.praveen2011@gmail.com
180 E. A. Abdel-Rehim and A. S. Hashem
1
1−kτβ−1(β−bτβ−kτβ−1+2aμα
cos απ
2
)+aμ
cos απ
2
∞
s=0
(−1)s+1α
s≤λi≤
1
1−kτβ−1(bτβ−kτβ−1+β)+aμ
cos απ
2
∞
s=0
(−1)s+1α
s.(3.3)
But the term
∞
s=∞
(−1)s+1α
s=0, then Eq. (3.3) reduces to
1
1−kτβ−1β−bτβ−kτβ−1+2aμα
cos απ
2≤λi≤1
1−kτβ−1bτβ−kτβ−1+β.
(3.4)
So far, as 0 <k<1 and by taking the limit as τ→0, one gets
1
1−kτβ−1bτβ−kτβ−1+β≈β,
and consequently |λ1|≤β.Forλ2to satisfy the condition |λ2|≤β, the following
condition must be satisfied:
|1
1−kτβ−1β−bτβ−kτβ−1+2aμα
cos απ
2|≤ 1
1−kτβ−1bτβ−kτβ−1+β.
Solving this inequality, one gets the condition of the stability of this model as
0≤μ≤−β+kτβ−1
aαcos απ
2.(3.5)
Second, we use also Gerschorgin´stheorem to find the eigenvalues λiof the discrete
scheme (2.13) from the inequality (3.1). In this case, the eigenvalues are contained
in circles centered at
β−kτβ−1
1−kτβ−1−aμ
cos απ
2(1−kτβ−1)
with radius
1
1−kτβ−1bτβ+aμ
cos απ
2α
1+aμ
cos απ
2
∞
s=2
(−1)s+1α
s,
such that
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 181
|λi−β−kτβ−1
1−kτβ−1−aμ
cos απ
2(1−kτβ−1)|≤ 1
1−kτβ−1
bτβ+aμ
cos απ
2α
1+aμ
cos απ
2
∞
s=2
(−1)s+1α
s=ρ,(3.6)
and as a result
1
1−kτβ−1(β−bτβ−kτβ−1−2aμ
cos απ
2
)−aμ
cos απ
2
∞
s=0
(−1)s+1α
s≤λi≤
1
1−kτβ−1(bτβ−kτβ−1+β)+aμ
cos απ
2
∞
s=0
(−1)s+1α
s.(3.7)
But the term
∞
s=0
(−1)s+1α
s=0, then Eq. (3.7) reduces to
1
1−kτβ−1β−bτβ−kτβ−1−2aμ
cos απ
2≤λi≤1
1−kτβ−1bτβ−kτβ−1+β.
(3.8)
So far, as 0 <k<1 and by taking the limit as τ→0, one gets
1
1−kτβ−1bτβ−kτβ−1+β≈β,
and consequently |λ1|≤β.Forλ2to satisfy the condition |λ2|≤β, the following
condition must be satisfied:
|1
1−kτβ−1β−bτβ−kτβ−1−2aμ
cos απ
2|≤ 1
1−kτβ−1bτβ−kτβ−1+β.
By solving this inequality, one gets the condition of the stability of this model as
0≤μ≤β−kτβ−1
acos απ
2.(3.9)
Finally, we use the same theorem to find the eigenvalues of the discrete scheme
(2.15) from the inequality (3.1). In this case the eigenvalues are contained in circles
centered at β−kτβ−1
1−kτβ−1−2aμ
π(1−kτβ−1)
with radius
goyal.praveen2011@gmail.com
182 E. A. Abdel-Rehim and A. S. Hashem
1
1−kτβ−1bτβ+aμ
π+2aμ
π
∞
s=2
1
s(s+1),
such that
|λi−β−kτβ−1
1−kτβ−1−2aμ
π(1−kτβ−1)|≤ 1
1−kτβ−1bτβ+aμ
π+2aμ
π
∞
s=2
1
s(s+1)=ρ,
(3.10)
and as a result
1
1−kτβ−1(β−bτβ−kτβ−1−4aμ
π)+2aμ
π−2aμ
π
∞
s=1
1
s(s+1)≤λi≤
1
1−kτβ−1(bτβ−kτβ−1+β)+−2aμ
π+2aμ
π
∞
s=1
1
s(s+1).(3.11)
But the term
∞
s=1
1
s(s+1)=1, then Eq. (3.11) reduces to
1
1−kτβ−1β−bτβ−kτβ−1−4aμ
π≤λi≤1
1−kτβ−1bτβ−kτβ−1+β.
(3.12)
Again, as k<1 and by taking the limit as τ→0, one gets
1
1−kτβ−1bτβ−kτβ−1+β≈β,
and consequently |λ1|≤β.Forλ2to satisfy the condition |λ2|≤β, the following
condition must be satisfied:
|1
1−kτβ−1β−bτβ−kτβ−1−4aμ
π|≤ 1
1−kτβ−1bτβ−kτβ−1+β.
By solving this inequality, one gets the condition of the stability of this model is
0<μ≤π(β−kτβ−1)
2a.(3.13)
In what follows, we prove the stability of these difference schemes by using the von
Neumann stability condition but we have to put in mind that, the time-fractional
means that the solution depends on all the history of the approximate solutions, i.e.,
y(n+1)depends on y(n),y(n−1),y(n−2),···, and back to y(0). In other words, the wave
propagation has a memory. Von Neumann method has the Fourier image
y(n)
j=ζ(n)eiκxj,(3.14)
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 183
where ζ=ζ(κ)is a complex number and this method does not depend on the bound-
ary conditions. The approximate solution y(n)is stable if the amplification factor
|ζ|2≤1. Now, we substitute Eq. (3.14) into Eq. (2.9). To prove the stability, one
has to do it on steps. First, we ignore the coefficients of y(n−1),y(n−2),···,y(0),
and substitute Eq. (3.14) on the rest of Eq. (2.9), to get after some mathematical
manipulations
ζ=1
1−kτβ−1{β−kτβ−1+bτβcos(κh)+ijbτβsin(κh)+
aμ
cos απ
2
∞
s=0
(−1)s+1α
scos((1−s)κh)}.(3.15)
Taking the limit as h→0 and then putting the term
∞
s=0
(−1)s+1α
s=0, Eq. (3.15)
reduces to
ζ=1
1−kτβ−1β−kτβ−1+bτβ.(3.16)
Calculate |ζ|2as
|ζ|2=1
(1−kτβ−1)2β2−2βkτβ−1+k2τ2β−2−b2τ2β.(3.17)
So far, as 0 <k<1 and by taking the limit as τ→0, one gets |ζ|2=β2>1.
Second, we take into consideration the dependence of y(n+1)on y(n)and y(n−1), only,
to get after some calculations
ζ2−1
1−kτβ−1{β−kτβ−1+bτβcos(κh)+ijbτβsin(κh)+
aμ
cos απ
2
∞
s=0
(−1)s+1α
scos((1−s)κh)}ζ+1
1−kτβ−1β
2=0.(3.18)
After taking the limit as h→0 and putting
∞
s=0
(−1)s+1α
s=0, then the above
equation reduces to
ζ2−1
1−kτβ−1{β−kτβ−1+bτβ}ζ+1
1−kτβ−1β
2=0.(3.19)
The roots of the above equation are
goyal.praveen2011@gmail.com
184 E. A. Abdel-Rehim and A. S. Hashem
ζ1,2=1
4(τ−kτβ){2βτ −2kτβ+2bτ1+β∓
−8(β2τ−βτ)(τ−2kτβ)+(2kτβ−2βτ −2bτ1+β)2}.(3.20)
After some mathematical manipulations and taking the limit as τ→0, one gets
the roots of the above equation |ζ1|2=|ζ2|2≈|
β(β−1)
2|≤1asβ≤2. The space
fractional order αhas no effect on the stability condition. This proves the stability
condition. The third step is by assuming that y(n+1)depends only on y(n),y(n−1),
and y(n−2). At this step, one gets |ζi|2,i=1,2,3 that are approximately less than
one. In the next step, we add the dependence on y(n−3), and so on till reaching y(0).
At each step, one has to solve the resulted equation and use the previous limits to
get |ζi|2<1,i≥1. So far, the scheme (2.9) is stable for the space–time-fractional
order αand β.
Now, we prove also the stability of the scheme (2.13) by using the von Neumann
method. We substitute equation (3.14) into Eq. (2.13). First, we ignore the coefficients
of y(n−1),y(n−2),···,y(0), and substitute Eq. (3.14) on the rest of equation (2.13), to
get after manipulations
ζ=1
1−kτβ−1{β−kτβ−1+bτβcos(κh)+ijbτβsin(κh)+
aμ
cos απ
2
∞
s=0
(−1)s+1α
scos(κsh)}.(3.21)
After taking the limit as h→0 and putting
∞
s=0
(−1)s+1α
s=0, Eq. (3.21) reduces
to the same Eq. (3.16) with the same condition |ζ|2=β2>1. Second, we take
into consideration the dependence of y(n+1)on y(n)and y(n−1), only, to get after
calculations
ζ2−1
1−kτβ−1{β−kτβ−1+bτβcos(κh)+ijbτβsin(κh)+
aμ
cos απ
2
∞
s=0
(−1)s+1α
scos(κsh)}ζ+1
1−kτβ−1β
2=0.(3.22)
After taking the limit as h→0 and putting
∞
s=0
(−1)s+1α
s=0, Equation (3.22)
reduces to the same equation as (3.19). Then after some mathematical manipulations,
one gets the roots of the above equation |ζ1|2=|ζ2|2≈|
β(β−1)
2|≤1asβ≤2. As
before, the space-fractional order αhas no effect on the stability condition The
third step is to assume that y(n+1)depends only on y(n),y(n−1), and y(n−2).Atthis
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 185
step, one gets |ζi|2,i=1,2,3 are approximately less than one. In the next step, we
add the dependence on y(n−3), and so on till reaching y(0). At each step, one has to
solve the resulted equation and use the previous limits, to get |ζi|2<1,i≥1. So
far, the scheme (2.13) is stable for the space–time-fractional order αand β.
Finally, we apply von Neumann method to prove the stability of the scheme (2.15).
First, we ignore the coefficients of y(n−1),y(n−2),···,y(0), and substitute Eq. (3.14)
on the rest of Eq. (2.15), to get after manipulating
ζ=1
1−kτβ−1{β−kτβ−1+bτβcos(κh)−2aμ
π+ijbτβsin(κh)+
2aμ
π
∞
s=1
1
s(s+1)cos(κsh)}.(3.23)
After taking the limit as h→0 and putting the term
∞
s=1
1
s(s+1)=1, the above equation
reduces to the same Eq. (3.16). That means one gets the condition |ζ|2=β2>1.
Second, we take into consideration the dependence of y(n+1)on y(n)and y(n−1), only,
to get after some calculations
ζ2−1
1−kτβ−1{β−kτβ−1+bτβcos(κh)−2aμ
π+ijbτβsin(κh)+
2aμ
π
∞
s=1
1
s(s+1)cos(κsh)}ζ+1
1−kτβ−1β
2=0.(3.24)
After taking the limit as h→0 and putting
∞
s=1
1
s(s+1)=1, Eq. (3.24) reduces to the
same equation as (3.19). Then after some mathematical manipulations, one gets
|ζ1|2=|ζ2|2≈|
β(β−1)
2|≤1asβ≤2. Again, the space-fractional order αhas no
effect on the stability condition. The third step is to assume that y(n+1)depends only
Table 1 The values of parameters which are used in the calculations
Case Figures f(x)α β μ tdtf
1<α<2Figures 1,2,3,4,5and 6sin(πx
2R+1)2 2 0.9 t=6t=50
Figures 1,2,3,4,5and 6sin(πx
2R+1)1.8 1.9 0.97 t=5t=38
Figures 7,8,9and 10 sin(πx
2R+1)1.7 1.8 0.9 t=5t=38
Figures 11,12,13 and 14 sin(πx
2R+1)1.7 2 1 t=29 t=48
0<α<1Figures 15 and 16 sin(πx
2R+1)0.9 20.3 t=30 t=30
Figures 17 and 18 δ(x)0.8 1.7 0.4 t=50 t=50
α=1Figures 19 and 20 δ(x)1 2 0.92 −t=30
Figures 21 and 22 sin(πx
2R+1)11.7 0.6 −t=20
goyal.praveen2011@gmail.com
186 E. A. Abdel-Rehim and A. S. Hashem
on y(n),y(n−1), and y(n−2). At this step, one gets |ζi|2,i=1,2,3 are approximately
less than one. In the next step, we add the dependence on y(n−3), and so on till reaching
y(0). At each step, one has to solve the resulted equation and use the previous limits, to
get |ζi|2<1,i≥1. So far, the scheme (2.15) is stable for the space–time-fractional
order αand β(Table 1).
Fig. 1 t=1
10 5 5 10
2
1
1
2
Fig. 2 t=2
10 5 5 10
15
10
5
5
10
15
Fig. 3 t=5
10 5 5 10
300
200
100
100
200
300
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 187
Fig. 4 t=10
10 5 5 10
30
20
10
10
20
30
Fig. 5 t=35
10 5 5 10
5000
5000
10 000
15 000
20 000
Fig. 6 t=38
10 5 5 10
100 000
200 000
300 000
400 000
500 000
600 000
goyal.praveen2011@gmail.com
188 E. A. Abdel-Rehim and A. S. Hashem
Fig. 7 t=5
10 5 5 10
300
200
100
100
200
300
Fig. 8 t=8
10 5 5 10
15
10
5
5
10
15
Fig. 9 t=20
10 5 5 10
150
100
50
50
100
150
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 189
Fig. 10 t=38
10 5 5 10
100 000
200 000
300 000
400 000
Fig. 11 t=5
10 5 5 10
600
400
200
200
400
600
Fig. 12 t=29
10 5 5 10
200 000
100 000
100 000
200 000
goyal.praveen2011@gmail.com
190 E. A. Abdel-Rehim and A. S. Hashem
Fig. 13 t=44
10 5 5 10
5.0 106
1.0 107
1.5 107
2.0 107
2.5 107
3.0 107
3.5 107
Fig. 14 t=48
10 5 5 10
1 109
2 109
3 109
Fig. 15 t=10
10 5 5 10
1000
500
500
1000
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 191
Fig. 16 t=30
10 5 5 10
200 000
100 000
100 000
200 000
Fig. 17 t=5
10 5 5 10
1500
1000
500
500
1000
1500
2000
Fig. 18 t=50
10 5 5 10
2 1030
1 1030
1 1030
2 1030
goyal.praveen2011@gmail.com
192 E. A. Abdel-Rehim and A. S. Hashem
Fig. 19 t=20
10 5 5 10
2 1022
1 1022
1 1022
2 1022
Fig. 20 t=30
10 5 5 10
41034
21034
2 1034
4 1034
6 1034
Fig. 21 t=10
10 5 5 10
6 1020
4 1020
2 1020
2 1020
4 1020
6 1020
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 193
Fig. 22 t=20
10 5 5 10
1 1043
5 1042
5 1042
1 1043
4 Convergence to the Stationary Approximate Solutions
We seek to find the stationary approximate solution which does not depend on the
time, i.e., the solution of the space–time-fractional differential equation as t→∞,
for the above-discussed cases. To do so, we omit the terms depending on the time
in the matrix Eq. (2.11), where Qis defined for each cases in Eqs. (2.12), (2.14),
and (2.16). For these cases, we get the same matrix equation of the form z.H=0,
i.e., HT.y=0. The matrix His obtained from the matrices defined in (2.12), (2.14),
and (2.16) after omitting all the terms depending on tand β. The sum of the rows of
the resulted matrix His zero. The matrix HThas an eigenvector y∗of eigenvalue
zero. Our stationary approximation solution ¯y=vy∗with v=1/
R
j=−R
y∗
jis a vector,
whose elements sum to 1. We simulate the stationary approximate solutions at Figs. 23
and 24 for the classical case and for 0 <α<1.
To study the convergence of the approximate solution, we constitute the sequence
d={d(t1), d(t2), ···}, where t1<t2<··· → ∞. The numbers d(ti), i:1→16
is defined as
d(ti)=
R
j=−R
|yj(ti)−¯yj|,i=1,2,··· .
The convergent approximate solutions are simulated at Figs.25,26 and 27. In these
figures, we plot Log10dagainst the number of points. We compare Figs.25,26 and
27 with Fig.28 to prove that the convergent approximate solutions of the discussed
cases are related to e−tby the relation d(t)=c1e−c2t, where c1and c2are positive
constants related to αand βfor each case.
goyal.praveen2011@gmail.com
194 E. A. Abdel-Rehim and A. S. Hashem
Fig. 23 α=β=2
20 10 10 20 x
0.1
0.2
0.3
0.4
y
h
Fig. 24 0<α<1,β=1.7
5 5 x
0.03
0.04
0.05
0.06
0.07
y
h
Fig. 25 Convergence
5 10 15 t
0.2
0.4
0.6
log10d1.9, 1.8
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 195
Fig. 26 Convergence
510 15 t
0.5
1.0
1.5
2.0
2.5
log10d0.8, 1.7
Fig. 27 Convergence
5 10 15 t
0.05
0.10
0.15
0.20
0.25
log10d1, 1.7
Fig. 28 Convergence
510 15 t
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Exp t
goyal.praveen2011@gmail.com
196 E. A. Abdel-Rehim and A. S. Hashem
5 Numerical Results and Discussions
In this section, we give the numerical approximate solution for Eq. (2.1). We give the
evolution of y(n)=y(tn)with different values of tnand with different values of the
space-fractional order αand different values of the time-fractional order β. In these
simulations, we have enlarged the x−axis as −10 ≤x≤10 with h=0.2. We fix the
value of the attenuation coefficient k=0.5. The value of μmust satisfy the required
conditions depending on the values of αand β. Since {t0,t1,t2,···} = {0,1,2,···},
then the iteration index n=tn
τwhile τis calculated from the scaling parameter of
the specified model. We calculate all the numerical results for a=b=1. In the
following table, we summarize the values of f(x),μ,β,α,td,tfbeing used in the
numerical results. Here tdis the time when the wave is starting to damp and tfis the
time when the wave reaches to the get its stationary solution. For all the values of
the fractional orders αand β, the studied approximate solutions have the same start
for t=1 till t=3, i.e., have the same start . For 1 <α<2, the results show that
the effect of the damping force is much bigger than the external force F(x).Also,
we observe that for the fractional values αand β, the propagating waves reach its
stationary solutions faster than as in the classical case, α=2 and β=2, see [9,18].
The convergence of the approximate solutions (the first norm) are simulated at
Figs. 25,26, and 27. Figure 28 represents the plot of the rapid convergent function
e−t. Then by comparing the last four figures, one can deduce that the approximate
solution of the studied model is convergent for all the values of the fractional orders
αand β.
References
1. Szabo, T.L.: Causal theories and data for acoustic attenuation obeying a frequency power-law.
J. Acoust. Soc. Am. 97, 14–24 (1995)
2. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, NewYork (1982)
3. Family, F., Masters, B.R., Platt, D.E.: Fractal pattern formation in the human retinal vessels.
Phys. D: Nonlinear Phenom. J. 38, 98–103 (1989)
4. Lemehaute, A.: Fractal Geometries Theory and Applications. CRC Press, Boca Raton (1991)
5. Daxer, A.A.: Fractals and Retinal vessels. Lancet J., 339–618 (1992)
6. Cross, S.S.: Fractals in pathology. Pathol. J. 182, 1–8 (1997)
7. Gazit, Y., Baish, J.W., Safabakhsh, N., Leunig, M., Baxter, L.T., Jain, R.K.: Fractal Charac-
teristics of tumor vascular architecture during tumor growth and regression. Microcircul. J.,
395–402 (1997)
8. Baish, J.W., Jain, R.K.: Fractals and cancer. Cancer Res., 3683–3688 (2000)
9. Abdel-Rehim, E.A., El-Sayed, A.M.A., Hashem, A.S.: Simulation of the approximate solution
of the time fractional multi-term wave equations. J. Comput.Math. Appl. 73, 1134–1154 (2017)
10. Liebler, M., Ginter, S., Dreyer, T., Riedlinge, R.E.: Full wave modeling of therapeutic ultra-
sound: efficient time-domaint, implemetation of the frequency power-law attenuation. J.
Acoust. Soc. Am. 116, 2742–2750 (2004)
11. Tarasov,V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. J. 318, 286–307
(2005)
goyal.praveen2011@gmail.com
Simulation of the Space–Time-Fractional Ultrasound Waves … 197
12. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional
order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum
Mechanics. Springer, Wien and New York (1997)
13. Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations, solutions and applications. Chaos
7, 743–764 (1997)
14. Caputo, M.: Linear models of dissipation whose Q is almost independent II. Geophys. J. 13,
529–539 (1967)
15. Treeby, B.E., Cox, B.T.: Modeling power law absorption and dispersion for acoustic propaga-
tion using the fractional Laplacian. J. Acoust. Soc. Am., 2741–2748 (2010)
16. Fellah, M.M., Fellah, Z.E.A., Depollier, C.: Transient wave propagation in inhomogeneous
porous materials: application of fractional derivatives. Signal Process. J. 86, 2658–2667 (2006)
17. Casasanta, G., Garra, R.: Fractional calculus approach to the acoustic wave propagation with
space-dependent sound speed. Signal Image Video Process. J. 6, 389–392 (2016)
18. El-Sayed, A.M.A., Abdel-Rehim, E.A., Hashem, A.S.: Time evolution of the approximate and
stationary solutions of the time-fractional forced-damped wave equation. Tbillisi Math. J. 10,
127–144 (2017)
19. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time-Fractional diffusion: a discrete ran-
dom walk approach. J. Nonlinear Dyn. 29, 129–143 (2002)
20. Abdel-Rehim, E.A.: Modelling and simulating of classical and non-classical diffusion pro-
cesses by random walks. Mensch& Buch Verlag (2004)
21. Feller, W.: On a generalization of marcel Riesz potentials and the semi-groups generated by
them, In: Meddelanden Lunds Universitetes Matematiska Seminar-ium (Comm. Sm. Mathm.
Universit de Lund), Tome Suppl. ddi a M. Riesz, Lund, pp. 73–81 (1952)
22. Oldham, K.B.: Spanier Journal of the Fractional Calculus. Academic Press, New York (1974)
23. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential
Equations. Wily (1993)
24. Samko, S.G., Kilbas, A.A., Marichevm, O.I.: Fractional Integrals and Derivatives. Theory and
Applications), OPA, Amsterdam (1993)
25. Gorenflo, R., Mainardi, F.: Approximation of Lévy-Feller diffusion by random walk. J. Anal.
Appl. (ZAA) 18, 231–246 (1999)
26. Abdel-Rehim, E.A.: Explicit approximation solutions and proof of convergence of the space-
time fractional advection dispersion equations. J. Appl. Math. 4, 1427–1440 (2013)
goyal.praveen2011@gmail.com
Certain Properties of Konhauser
Polynomial via Generalized
Mittag-Leffler Function
J. C. Prajapati, N. K. Ajudia, Shilpi Jain, Anjali Goswami
and Praveen Agarwal
Abstract The principal aim of this paper is to establish several new properties
of generalized Mittag-Leffler function via Konhauser polynomials. Properties like
mixed recurrence relations, Differential equations, pure recurrence relations, finite
summation formulae, and Laplace transform have been obtained.
Keywords Konhauser polynomials ·Generalized Mittag-Leffler function ·
Laguerre polynomials ·Laplace transform
AMS(2010) Subject Classification: 33E12, 33C45, 33C47.
J. C. Prajapati
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, Gujarat, India
e-mail: jyotindra18@rediffmail.com
N. K. Ajudia
H & H B Kotak Institute of Science, Saurashtra University, Rajkot, Gujarat, India
e-mail: nka121@gmail.com
S. Jain
Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India
e-mail: shilpijain1310@gmail.com
A. Goswami
College of Science and Theoretical Studies, Main Branch Riyadh, Female, Saudi Electronic
University, Abu Bakr Street, PO. Box: 93499, Riyadh KSA, Saudi Arabia
e-mail: dranjaligoswami09@gmail.com
P. Ag a r w a l ( B
)
Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
e-mail: goyal.praveen2011@gmail.com
Department of Mathematics, Harish Chandra Research Institute, Chhatnag Road, Jhunsi 211019,
Allahabad, India
International Center for Basic and Applied Sciences, Jaipur 302029, India
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_11
199
goyal.praveen2011@gmail.com
200 J. C. Prajapati et al.
1 Introduction
Konhauser polynomial has drawn the attention of several researchers. Recently,
Prajapati et al. [7] introduced a class of polynomials
L(α,β)
m
q(z)=(αm+β+1)
m!
m
q
n=0
(−m)qn
(αn+β+1)
zn
n!,(1.1)
where α,β∈C;m,q∈N,m
qdenotes integral part of m
q,Re(β)>−1.
This is generalized form of Konhauser polynomials (Konhauser [5]),
Zμ
m(x;k)=(km +μ+1)
m!
n
j=0
(−1)jm
jxkj
(kj +μ+1),(1.2)
where μ>−1.
Note that
L(k,μ)
m(zk)=Zμ
m(z;k). (1.3)
The Laguerre polynomials (Rainville [8]) defined as
Lμ
m(x)=(m+μ+1)
m!
n
j=0
(−1)jm
jxj
( j+μ+1),(1.4)
where μ>−1.
This is special case of (1.2)as
Zμ
m(x;1)=Lμ
m(x). (1.5)
In 1970, Prabhakar [6] defined the generalized Mittag-Leffler function as
Eγ
α,β(z)=
∞
n=0
(γ)nzn
(αn+β)n!,Re(α)>0.(1.6)
This is an entire function of order (Re(α))−1.
Kilbas et al. [4] studied the relation between (1.2) and (1.6)as
E−m
k,μ+1(zk)=(m+1)
(km +μ+1)Zμ
m(z;k), (1.7)
where m,k∈N;μ∈Cwith Re(μ)>−1. If k=1 then (1.7) reduces to
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 201
E−m
1,μ+1(z)=(m+1)
(m+μ+1)Lμ
m(z), m∈N;μ∈C.(1.8)
In 2007, Shukla and Prajapati [9] introduced generalized Mittag-Leffler function
as
Eγ,q
α,β(z)=
∞
n=0
(γ)qn
(αn+β)
zn
n!,(1.9)
where α,β,γ∈C;Re(α)>0,Re(β)>0,Re(γ)>0 and q∈(0,1)∪N
In 2014, Prajapati et al. [7] established relation between (1.9) and (1.1)as
E−m,q
k,μ+1(zk)=(m+1)
(km +μ+1)L(k,μ)
m
q(zk). (1.10)
Furthermore, some useful results are obtained in (1.11)–(1.15) (Prajapati et al.
[7]).
L(α,β)
m
q(t)=(αm+β)
1
0
zβ−1L(α,β−1)
m
q(tzα)dz (1.11)
(z−t)βL(α,β)
m
q(z−t)α=(αm+β+1)
(αm+β−γ+1)(γ)×
z
t
(z−u)γ−1(u−t)β−γL(α,β−γ)
m
q(u−t)αdu,(1.12)
where α,β,γ∈Cwith Re(β)>Re(γ)>−1,
∞
m=0
(γ)mL(α,β)
m
q(z)tm
(αm+β+1)=(1−t)−γEγ,q
α,β+1z(−t)q
(1−t)q,|t|<1 (1.13)
where α,β,γ∈Cwith Re(β)>−1 and q∈N,
∞
m=0
L(α,β)
m
q(zα)tm
(αm+β+1)=etW(α;β+1;zα(−t)q), (1.14)
where α,β∈Cwith Re(β)>−1 and q∈N,
and
L(k,β)
m
q(zk)=z
ykm m
r=0
(β+1)km
(β+1)km−kr y
zk−1r
r!L(k,β)
m−r
q(yk), (1.15)
where k∈Nand β∈Cwith Re(β)>−1.
goyal.praveen2011@gmail.com
202 J. C. Prajapati et al.
In 2010, Maged Gumaan Bin-Saad [1] investigated Hermite–Konhauser polyno-
mial as
kHμ
m(x,y;z)=m!
m
n=0
[m−n
2]
r=0
(−1)n+rxrykn+μzm−n−2r
n!r!(m−n−2r)!(kn +μ+1),(1.16)
and the relation between Konhauser polynomial and Hermite–Konhauser polynomial
as
Zμ
m(x;k)=x−μ(km +μ+1)
m!kHμ
m(0,x;1). (1.17)
Konhauser [5] obtained mixed recurrence relation, differential equation, and pure
recurrence relation of (1.2)as(1.18)–(1.20)
xDZμ
m(x;k)=mkZ μ
m(x;k)−k(km −k+μ+1)kZμ
m−1(x;k), (1.18)
Dk[xμ+1DZμ
m(x;k)]=xμ+1DZμ
m(x;k)−mkxμZμ
m(x;k), (1.19)
k
i=0k
i[Dk−ixμ+1][Di+1Zμ
m(x;k)]=−kxμ(km −k+c+1)kZμ
m−1(x;k). (1.20)
Srivastava [10] gives another form of Zμ
m(y;k)as
Zμ
m(x;k)=x
ykm m
r=0μ+km
kr (kr)!
r!y
xk
−1r
Zμ
m−r(y;k). (1.21)
The recurrence relations, differential equations, pure recurrence relations, finite sum-
mation formulae, and Laplace transforms of (1.2) studied by Srivastava [12]as(1.22)
to (1.30)
DZμ
m(x;k)=−kxk−1Zμ+k
m−1(x;k), (1.22)
(x1−kD)nZμ
m(x;k)=(−k)nZμ+kn
m−n(x;k), m≥n≥0,(1.23)
xDZμ
m(x;k)=(mk +μ)Zμ−1
m(x;k)−μZμ
m(x;k), (1.24)
Zμ
m(x;k)−Zμ−1
m(x;k)=k(km +μ)
(k(m−1)+μ+1)Zμ
m−1(x;k), (1.25)
xkZμ+k
m(x;k)=(km +μ+1)kZμ
m(x;k)−(m+1)Zμ
m+1(x;k), (1.26)
kxkZμ+k
m(x;k)=μZμ
m+1(x;k)−(km +μ+k)Zμ−1
m+1(x;k), (1.27)
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 203
Zμ
m(δx;k)=
m
j=0μ+km
kj δk(m−j)(1−δk)jZμ
m−j(x;k), (1.28)
L{tνZμ
m(xt;k);s}= (μ+1)km(ν+1)
sν+1m!k+1Fk−m,ν+1
k,ν+2
k, ..., ν+k
k;
μ+1
k,μ+2
k, ..., μ+k
k;x
sk,(1.29)
Re(s)>0,Re(ν)>−1;
if ν=μthen (1.29) reduces to
L{tμZμ
m(xt;k);s}= (km +μ+1)
skm+μ+1(sk−xk)m,(1.30)
where Re(s)>0,Re(μ)>−1.
In 1970, Prabhakar [6] gives relations as follows:
Dnxμ+nZμ+n
m(x;k)=(km +μ+n+1)
(km +μ+1)xμZμ
m(x;k), μ>−1,(1.31)
Iν[zμZμ
m(z;k)]= (km +μ+1)
(km +μ+ν+1)zμ+νZμ+ν
m(z;k), (1.32)
Re(μ)>−1,Re(ν)>−Re(μ+1), where for suitable fand complex ν,Iνf(x)
denotes the νth-order fractional integral (or fractional derivative) of f(x).Healso
obtained
xk(γ−1)Zμ
m(x;k)=(km +μ+1)
(γ)m2πi
C
tm+γ−1Eγ
k,μ+1(xk−t)
(t−xk)m+1dt,(1.33)
where Cis a circle: |t−xk|=, for small radius .
In 1976, Karande and Thakare [3], studied the relation
σZμ
m(z;k)
(1+μ)km =− Zμ
m−1(z;k)
kk(1+μ)k(m−1)
,(1.34)
where σ=z−k+1
kD
k
i=1θ
k+μ+i
k−1,θ=zD =zd
dz.
Orthogonal property of Konhauser polynomials is given by Konhauser [5]as
∞
0
e−zzμZμ
m(z;k)Yμ
n(z;k)dz =(km +μ+1)
(m+1)δmn,∀m,n∈{0,1,2, ...}(1.35)
where δmn is Kronecker’s delta.
goyal.praveen2011@gmail.com
204 J. C. Prajapati et al.
Srivastava [10] derived summation formulae as
∞
m=0
Zμ
m(x;k)y
kkm tm
(1+μ)km
=exp y
kk
t0Fk−; μ+1
k,μ+2
k, ..., μ+k
k;−xy
k2k
t,(1.36)
and ∞
m=0
Zμ
m(x;k)y
kkm tm
(1+μ)km
=exp y
kk
−x
kkt∞
m=0
Zμ
m(y;k)x
kkm tm
(1+μ)km
(1.37)
Srivastava [11] proved bilateral generating function as
∞
m=0
m!Zμ
m(x;k)Yα−lm
m(y;l)tm
(μ+km +1)=(1+t)−1+α+1
le(x[1−(1+t)
1
l])Hx(1+t)
1
l,−ykt
(1+t),(1.38)
where H[x,t]=
∞
m=0
Yα−lm
m(x;l)tm
(μ+km +1).
Srivastava [12] studied generating function for Konhauser polynomials as
∞
m=0
Zμ
m(z;k)(γ)mtm
(1+μ)km
=(1−t)−γ1Fkγ;μ+1
k,μ+2
k, ..., μ+k
k;x
kkt
t−1(1.39)
and a summation formula as
∞
m=0m+n
mZμ
m+n(z;k)tm
(1+μ)k(m+n)
=
∞
m=nm
ntm−n
m!
(−zk)m
(1+μ)km 1F1[m+1;m−n+1;t](1.40)
In 1981, Karande and Patil [2] obtained double integral in the form of orthogonal
property
∞
0
∞
0
e−(x+y)xμyνZμ+ν+1
m(x+y;k)Yμ+ν+1
n(x+y;k)dxdy,
=(1+μ)(1+ν)(μ+ν+2)km
m!if n=m
0ifm= n.
(1.41)
2 Mixed Recurrence Relations
Consider (1.18)
xDZμ
m(z;k)=mkZ μ
m(z;k)−k(km −k+μ+1)kZμ
m−1(z;k),
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 205
this can be written as
xD(km +μ+1)
(m+1)E−m
k,μ+1(zk)=mk (km +μ+1)
(m+1)E−m
k,μ+1(zk)
−k(km −k+μ+1+k)
(km −k+μ+1)
(km −k+μ+1)
(m)E−m+1
k,μ+1(zk),
i.e., xDE−m
k,μ+1(zk)=mk[E−m
k,μ+1(zk)−E−m+1
k,μ+1(zk)].(2.1)
From (1.20), we have
k
i=0k
i[Dk−izμ+1][Di+1Zμ
m(z;k)]=−kzμ(km −k+μ+1)kZμ
m−1(z;k),
this gives
k
i=0k
i[Dk−izμ+1][Di+1(km +μ+1)
(m+1)E−m
k,μ+1(zk)]
=−kzμ(km −k+μ+1+k)
(km −k+μ+1)
(km −k+μ+1)
(m)E−m+1
k,μ+1(zk),
i.e.,
k
i=0k
i[Dk−izμ+1][Di+1E−m
k,μ+1(zk)]=−kmzμE−m+1
k,μ+1(zk). (2.2)
Now, consider (1.22),
DZμ
m(z;k)=−kzk−1Zμ+k
m−1(z;k),
this reduces to
D(km +μ+1)
(m+1)E−m
k,μ+1(zk)=−kzk−1(km −k+μ+k+1)
(m)E−m+1
k,μ+k+1(zk),
the simplification gives
DE−m
k,μ+1(zk)=−kmzk−1E−m+1
k,μ+k+1(zk). (2.3)
Consider (1.23),
(z1−kD)pZμ
m(z;k)=(−k)pZμ+kp
m−p(z;k),
this leads to
goyal.praveen2011@gmail.com
206 J. C. Prajapati et al.
(z1−kD)pE−m
k,μ+1(zk)=(−k)p(m+1)
(m−p+1)E−m+p
k,μ+kp+1(zk). (2.4)
Consider (1.24)
zDZμ
m(z;k)=(mk +μ)Zμ−1
m(z;k)−μZμ
m(z;k),
this gives
zDE−m
k,μ+1(zk)=E−m
k,μ(zk)−μE−m
k,μ+1(zk). (2.5)
Consider (1.31)
Dnxμ+nZμ+n
m(x;k)=(km +μ+n+1)
(km +μ+1)xμZμ
m(x;k)
with Re(μ)>−1.
Using (1.7), this reduces to
Dnxμ+nE−m
k,μ+n+1(zk)=xμE−m
k,μ+1(zk). (2.6)
Consider (1.32)
Iν[zμZμ
m(z;k)]= (km +μ+1)
(km +μ+ν+1)zμ+νZμ+ν
m(z;k).
Using (1.7), this can be written as
Iν[zμE−m
k,μ+1(zk)]=zμ+νE−m
k,μ+ν(zk). (2.7)
From (1.34), we have
σZμ
m(z;k)
(1+μ)km =− Zμ
m−1(z;k)
kk(1+μ)k(m−1)
.
this gives
σE−m
k,μ+1(zk)(μ+1)
(m+1)=−
(km −k+μ+1)E−m+1
k,μ+1(zk)
(m)kk(1+μ)k(m−1)
,
this reduces to
σE−m
k,μ+1(zk)=−m
kkE−m+1
k,μ+1(zk). (2.8)
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 207
3 Pure Recurrence Relations
In this section, the authors obtained some recurrence relations.
Consider (1.25)
Zμ
m(z;k)−Zμ−1
m(z;k)=k(km +μ)
(k(m−1)+μ+1)Zμ
m−1(z;k),
this follows
(km +μ)E−m
k,μ+1(zk)−E−m
k,μ(zk)=kmE−m+1
k,μ+1(zk). (3.1)
Consider (1.26)
zkZμ+k
m(z;k)=(km +μ+1)kZμ
m(z;k)−(m+1)Zμ
m+1(z;k),
this leads to
zkE−m
k,μ+k+1(zk)=E−m
k,μ+1(zk)−E−m−1
k,μ+1(zk). (3.2)
Consider (1.27)
kzkZμ+k
m(z;k)=μZμ
m+1(z;k)−(km +μ+k)Zμ−1
m+1(z;k),
this gives
(m+1)kzkE−m
k,μ+k+1(zk)=μE−m−1
k,μ+1(zk)−E−m−1
k,μ(zk). (3.3)
4 Differential Equation
Consider (1.19)
Dk[zμ+1DZμ
m(z;k)]=zμ+1DZμ
m(z;k)−kmzμZμ
m(z;k).
Using (1.7), this follows
Dk[zμ+1DE−m
k,μ+1(zk)]=zμ+1DE−m
k,μ+1(zk)−mkzμE−m
k,μ+1(zk). (4.1)
goyal.praveen2011@gmail.com
208 J. C. Prajapati et al.
5 Finite Summation Formulae
From (1.10) and (1.15), we have
E−m,q
k,β+1(zk)=z
ykm m
r=0m
ry
zk
−1r
E−m+r,q
k,β+1(yk), (5.1)
where k∈Nand β∈Cwith Re(β)>−1.
Now, consider (1.21)
Zμ
m(z;k)=z
ykm m
r=0μ+km
kr (kr)!
r!y
zk
−1r
Zμ
m−r(y;k).
Using (1.7), this can be written as
E−m
k,μ+1(zk)=z
ykm m!
(km +μ)!
m
r=0
(μ+km)!
(kr)!(μ+km −kr )!
(kr)!
r!
y
zk
−1r(km −kr +μ+1)
(m−r+1)E−m+r
k,μ+1(yk).
Finally, we arrived at
E−m
k,μ+1(zk)=z
ykm m
r=0m
ry
zk
−1r
E−m+r
k,μ+1(yk). (5.2)
Consider (1.28)
Zμ
m(δz;k)=
m
j=0μ+km
kj (kj)!
j!δk(m−j)(1−δk)jZμ
m−j(z;k).
By using (1.7), this leads to
E−m
k,μ+1((δz)k)=
m
j=0
(m+1)
j!(m−j+1)δk(m−j)(1−δk)jE−m+j
k,μ+1(zk),
this follows:
E−m
k,μ+1((δz)k)=
m
j=0m
jδk(m−j)(1−δk)jE−m+j
k,μ+1(zk). (5.3)
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 209
6 Integral Representation and Orthogonal Property
Keeping α=k∈Nand replacing tby tkin (1.11), we have
L(k,β)
m
q(tk)=(km +β)
1
0
zβ−1L(k,β−1)
m
q(tz)k)dz.
By using (1.10), this reduces to
E−m,q
k,β+1(tk)=
1
0
tβ−1E−m,q
k,β((tz)k)dz.(6.1)
Consider (1.12)
(z−t)βL(α,β)
m
q(z−t)α=(αm+β+1)
(αm+β−γ+1)(γ)×
z
t
(z−u)γ−1(u−t)β−γL(α,β−γ)
m
q(u−t)αdu,(6.2)
where β,γ∈Cwith Re(β)>Re(γ)>−1 and α∈N.
Using (1.10), this gives
(z−t)βE−m,q
k,β+1(z−t)α=1
(γ)
z
t
(z−u)γ−1(u−t)β−γE−m,q
k,β−γ+1(u−t)αdu.(6.3)
Let Cis a circle |t−xk|=for small radius , consider (1.33)
xk(γ−1)Zμ
m(x;k)=(km +μ+1)
(γ)m2πi
C
tm+γ−1Eγ
k,μ+1(xk−t)
(t−xk)m+1dt.
Using (1.7), the above equation can be written in the form
xk(γ−1)E−m
k,μ+1(zk)=(m+1)
(γ)m2πi
C
tm+γ−1Eγ
k,μ+1(xk−t)
(t−xk)m+1dt.(6.4)
From (1.35) and (1.7), we have
goyal.praveen2011@gmail.com
210 J. C. Prajapati et al.
∞
0
e−zzμE−m
k,μ+1(zk)Yμ
n(z;k)dz =δmn,∀m,n∈{0,1,2, ...}.(6.5)
Consider (1.41)
∞
0
∞
0
e−(x+y)xμyνZμ+ν+1
m(x+y;k)Yμ+ν+1
n(x+y;k)dxdy
=(1+μ)(1+ν)(μ+ν+2)km
m!if n=m
0ifm= n
By using (1.7), this immediately follows:
∞
0
∞
0
e−(x+y)xμyνE−m
k,μ+ν+2((x+y)k)Yμ+ν+1
n(x+y;k)dxdy
=B(1+μ,1+ν)if n=m
0ifm= n
where B(m,n)is the usual beta function.
7 Laplace Transform
From equation (1.29), we have
L{tνZμ
m(zt;k);s}= (μ+1)km (ν+1)
sν+1m!k+1Fk−m,ν+1
k,ν+2
k, ..., ν+k
k;
μ+1
k,μ+2
k, ..., μ+k
k;z
sk.
This immediately leads to
LtνE−m
k,μ+1((zt)k);s=(ν+1)
(μ+1)sν+1k+1Fk−m,ν+1
k,ν+2
k, ..., ν+k
k;
μ+1
k,μ+2
k, ..., μ+k
k;z
sk.
If μ=ν, then the above equation follows:
LtνE−m
k,ν+1((zt)k);s=(sk−zk)n
skm+ν+1.
If k=1, then the above results reduces in the form of Laguerre polynomials.
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 211
8 Generating Functions
Keeping α=k∈Nand replacing zby zkin (1.13)gives
∞
m=0
(γ)mL(k,β)
m
q(zk)tm
(km +β+1)=(1−t)−γEγ,q
k,β+1zk(−t)q
(1−t)q,|t|<1.(8.1)
By using (1.10), this gives
∞
m=0
(γ)mE−m,q
k,β+1(zk)tm
m!=(1−t)−γEγ,q
k,β+1zk(−t)q
(1−t)q,|t|<1(8.2)
where β,γ∈Cwith Re(β)>−1 and k,q∈N.
Keeping α=k∈Nthen (1.14) yields
∞
m=0
L(k,β)
m
q(zk)tm
(km +β+1)=etW(k;β+1;zk(−t)q), (8.3)
where β∈Cwith Re(β)>−1 and k,q∈N,
Using (1.10), (8.3) can be written in the form
∞
m=0
E−m,q
k,β+1(zk)tm
m!=etW(k;β+1;zk(−t)q). (8.4)
Again from (1.38) and (1.7), we have
∞
m=0
E−m
k,μ+1(zk)Yα−lm
m(y;l)tm=(1+t)−1+α+1
le(x[1−(1+t)1
l])Hx(1+t)1
l,−ykt
(1+t),
where H[x,t]=
∞
m=0
Yα−lm
m(x;l)tm
(μ+km +1).
From (1.39) and (1.7), we get
∞
m=0
E−m
k,μ+1(zk)(γ)mtm=(1−t)−γ
(1+μ)1Fkγ;μ+1
k,μ+2
k, ..., μ+k
k;x
kkt
t−1.(8.5)
goyal.praveen2011@gmail.com
212 J. C. Prajapati et al.
9 Miscellaneous
Equations (1.17) and (1.7)gives
E−m
k,μ+1(zk)=x−μkHμ
m(0,x;1).
Consider (1.36)
∞
m=0
Zμ
m(x;k)y
kkm tm
(1+μ)km
=exp y
kk
t0Fk−; μ+1
k,μ+2
k, ..., μ+k
k;−xy
k2k
t.
By using (1.7), the above equation can be written as
∞
m=0
E−m
k,μ+1(zk)y
kkm (μ+1)tm
m!=exp y
kk
t0Fk−; μ+1
k,μ+2
k, ..., μ+k
k;−xy
k2k
t.
From (1.37) and (1.7), we have
∞
m=0
E−m
k,μ+1(xk)y
kkm tm
m!=exp y
kk−x
kkt∞
m=0
E−m
k,μ+1(yk)x
kkm tm
m!.(9.1)
Consider (1.40)
∞
m=0m+n
mZμ
m+n(z;k)tm
(1+μ)k(m+n)
=
∞
m=nm
ntm−n
m!
(−zk)m
(1+μ)km 1F1[m+1;m−n+1;t].
Using (1.7), the above equation yields
∞
m=0
E−(m+n)
k,μ+1(zk)tm
m!=
∞
m=n
tm−n
(m−n)!
(−zk)m
(km +μ+1)1F1[m+1;m−n+1;t].
The simplification gives
∞
m=0
E−(m+n)
k,μ+1(zk)tm
m!=
∞
j=0
1F1[j+n+1;j+1;t](−zk)j+n
(k(j+n)+μ+1)
tj
j!.(9.2)
Acknowledgements This work was supported to third author [S Jain] by the Science & Engi-
neering Research Board (SERB), India (No: MTR/2017/000194) and fifth author’s [P Agar-
wal] research grant is supported by the Department of Science & Technology(DST), India
(No: INT/RUS/RFBR/P-308) and Science & Engineering Research Board (SERB), India (No:
TAR/2018/000001).
goyal.praveen2011@gmail.com
Certain Properties of Konhauser Polynomial via Generalized Mittag-Leffler Function 213
References
1. Bin-Saad, M.G., New, A.: Class of Hermite-Konhauser polynomials together with differential
equations. Kyungpook Math. J. 50, 237–253 (2010)
2. Karande, B.K., Patil, K.R.: Note on Konhauser Biorthogonal polynomials. Indian J. pure appl.
Math. 12(2), 222–225 (1981)
3. Karande, B.K., Thakare, N.K.: Some Results for Konhauser-Biorthogonal polynomials and
dual series equations. Indian J. Pure Appl. Math. 7(6), 635–646 (1976)
4. Kilbas, A.A., Saigo, M., Saxsena, R.K.: Generalized Mittag-Leffler function and generalized
fractional calculus operators. Integral Transforms Spec. Funct. 15(1), 31–49 (2004)
5. Konhauser, J.D.E.: Biorthogomal polynomials suggested by the laguerre polynomials. Pacific
J. Math. 21(2), 303–314 (1967)
6. Prabhakar, T.R.: On a set of polynomials suggested by laguerre polynomials. Pacific J. Math.
35(1), 213–219 (1970)
7. Prajapati, J.C., Ajudia, N.K., Agarwal, P.: Some results due to Konhauser polynomials of first
kind and laguerre polynomials. Appl. Math. Comput. 247, 639–650 (2014)
8. Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)
9. Shukla, A.K., Prajapati, J.C.: On a generalization of Mittag-Leffler function and its properties.
J. Math. Anal. Appl. 336(2), 779–811 (2007)
10. Srivastava, H.M.: On the Konhauser sets of Biorthogonal polynomials suggested by the laguerre
polynomials. Pacific J. Math. 49(2), 489–492 (1973)
11. Srivastava, H.M.: A Note on the Konhauser Sets of Biorthogonal polynomials suggested by
the laguerre polynomials. Pacific J. Math. 90(1), 197–200 (1980)
12. Srivastava, H.M.: Some Biorthogonal polynomials suggested by the Laguerre polynomials.
Pacific J. Math. 98(1), 235–250 (1982)
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based
on the Tau Method for the Eigenvalue
Problems
Maryam Attary and Praveen Agarwal
Abstract We consider the (presumably new) effective numerical scheme based on
the Legendre polynomials for an approximate solution of eigenvalue problems. First,
a new operational matrix, which can be represented by a sparse matrix defined by
using the Tau method and orthogonal functions. Sparse data is by nature more com-
pressed and thus requires significantly less storage. A comparison of the results for
some examples reveals that the presented method is convenient and effective, also
we consider the problem of column buckling to show the validity of the proposed
method.
Keywords Eigenvalue problems ·Legendre polynomials ·Numerical treatment
Mathematics Subject Classifications 65L15, 65L05, 65L10, 65N35.
1 Introduction
A special class of boundary-value problems are eigenvalue problems. They are used
in a wide variety of engineering contexts beyond boundary-value problems and play
a very important role in many scientific fields such as vibrations, elasticity, and other
oscillating systems. A simple context to illustrate how eigenvalues occur in physical
problems is the mass–spring system. Detailed description and application of these
problems may be found in [3] and references therein.
M. Attary
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
P. Ag a r w a l ( B
)
Department of Mathematics, ANAND International College of Engineering, Jaipur 303012, India
e-mail: goyal.praveen2011@gmail.com
Department of Mathematics, Harish Chandra Research Institute, Chhatnag Road, Jhunsi 211019,
Allahabad, India
International Center for Basic and Applied Sciences, Jaipur 302029, India
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_12
215
goyal.praveen2011@gmail.com
216 M. Attary and P. Agarwal
The numerical solvability of eigenvalue problems and other related equations
has been considered by several authors. In [5], a software package has been intro-
duced and discussed, which deals with the computation of the eigenvalue of Strum–
Liouville problems. Gamel et al. [4] were concerned with the Chebyshev method for
solving eigenvalue problems of fourth order of ODEs.
Due to the good approximation properties of spectral methods, these methods have
been discussed intensively in recent years. A special case of them is the Tau method,
which has been applied for the numerical solution of many operator equations.
In this paper, we intend to introduce a new Tau approach by using Legendre
polynomials to solve the following eigenvalue problems:
u(k)(x)+λ2u(x)=0,k=2or 4,(1.1)
k−1
r=0αj,ru(r)(cr)=0,j=1, ..., k.(1.2)
s.t.
cr=−1,r=0,
1,o.w.
where αj,r∈Rare constants, (r=0,···,k−1)and λis a solution to be deter-
mined.
The rest of the article is organized as follows: In Sect.2, we describe some pre-
liminaries of the Legendre polynomials and their properties. Section 3explains the
new scheme and introduces matrix representation of the method for problem (1.1).
To clarify the efficiency of the method, the proposed algorithm is applied to some
numerical experiments and also the obtained results are compared with some existing
methods in the literature.
2 Basic Definitions of the Legendre Polynomials
We recall some definitions, which are required for the present study (see, for example,
[1]).
Definition 2.1 The starting point of Legendre polynomials is Rodrigues formula,
which is introduced as
Pn(x)=1
2nn!
dn
dxn(x2−1)n,
the orthogonality of Legendre polynomials in [−1,1]with r(x)=1 can be shown.
Definition 2.2 The Legendre polynomials Pn(x)satisfy the recurrence relation:
Pn+1(x)=1
n+1[(2n+1)xP
n(x)−nP
n−1(x)],n=1,2,···.(2.1)
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based on the Tau Method … 217
Also, Legendre polynomials Pn(x)can be represented in the following recurrence
form:
P0(x)=P
1(x),
Pn(x)=1
2n+1[P
n+1(x)−P
n−1(x)],n=1,2,···.(2.2)
Using Rodrigues’ formula, the orthogonality of Legendre polynomials can be
obtained as follow
1
−1
Pm(x)Pn(x)dx =0,m= n,
2
2n+1m=n.
Since the Legendre polynomials are defined on the interval [−1,1], for using
these polynomials on the interval [a,b], we convert it to [−1,1]by introducing
x=b−a
2t+b+a
2.
3 Numerical Treatment of the Problem
In this section, we replace the differential part of Eq. (1.1) by an operational matrix.
Our main result is asserted by Theorem 1.
Theorem 1 Let Pi(x)be a Legendre polynomials in [−1,1]. Suppose that functions
u(x)and u(x)can be expressed as
u(x)=∞
i=0
aiPi(x)=aPx,(3.1)
u(x)=∞
i=0
b1,iPi(x)=b1Px,(3.2)
where a=[a0,a1,a2,...]T,b1=[b1,0,b1,1,b1,2,...]Tand Px=[P0,P1,P2,···].
Then we have:
u(x)=Mapx,(3.3)
where
M=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0101010···
0030303···
0005050···
0000707···
.
.
..
.
..
.
..
.
..
.
..
.
..
.
....
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
goyal.praveen2011@gmail.com
218 M. Attary and P. Agarwal
Proof Taking the derivative of (3.1) and due to (3.2), we can write
∞
i=1
aiP
i(x)=∞
i=0
b1,iPi(x)=b1,0P0(x)+∞
i=1
b1,iPi(x). (3.4)
Using (2.2), we rewrite the above relation as
∞
i=1
aiP
i(x)=b1,0P
1(x)+∞
i=1
b1,i
2i+1[P
i+1(x)−P
i−1(x)].(3.5)
Therefore
a1=b1,0−b1,2
5,a2=b1,1
3−b1,3
7,..., an=b1,n−1
2n−1−b1,n+1
2n+3,..., (3.6)
and so
b1,n=∞
i=n+1
(2n+1)λn,iai,s.t.λn,i=1,i+nodd,
0,o.w. (3.7)
(3.7) can be transformed to the following matrix form:
b1=Ma,(3.8)
where
M=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0101010···
0030303···
0005050···
0000707···
.
.
..
.
..
.
..
.
..
.
..
.
..
.
....
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Due to the last equation, (3.2) can be written as
u(x)=b1Px=MaPx.(3.9)
Lemma 1 Let u(n)(x)=∞
i=0bn,iPi(x)=bnPx, be a Legendre polynomiyal with
bn=[bn,0,bn,1,bn,2,...]T, and Mis a matrix, which is defined in Theorem1. Then
we have
u(n)(x)=MnaPx.(3.10)
Proof According to Theorem 1, the validity of Lemma 1for n =1 is obvious. From
(2.2), we can write
pi(x)=1
2i+1[P
i+1(x)−P
i−1(x)].
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based on the Tau Method … 219
Given the assumption, it follows that
u(2)(x)=∞
i=0
b2,iPi(x)=b2Px.(3.11)
Using the given scheme in Theorem 1, we conclude
b2=Mb1,(3.12)
Dueto(3.8) and the last equation, we get b2=M2a. Therefore, by repeating this
scheme, it follows that bn=Mna.
Finally,
u(n)(x)=MnaPx.(3.13)
We are now ready to obtain the algebraic form of the eigenvalue problems (1.1)
based on the operational matrix of the Legendre polynomials. We define um(x)as
an approximation function of the exact solution u(x)as follows:
um(x)=
m
i=0
aiPi(x)=amPx,m.(3.14)
First, we consider the following form of (1.1):
u(2)(x)+λ2u(x)=0.(3.15)
We define u(2)
m(x)as an approximation function of the exact solution u(2)(x)as
follows:
u(2)
m(x)=M2
mamPx,m,(3.16)
where Mmand M2
mare finite forms of Mand M2, respectively.
Also boundary conditions of (3.15) can be written as
α1,0m
i=0aiPi(−1)+α1,1m
i=0aiP
i(1)=0,
α2,0m
i=0aiPi(−1)+α2,1m
i=0aiP
i(1)=0,(3.17)
or equivalently ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
m
i=0ai[α1,0Pi(−1)+α1,1P
i(1)]
di,1
=0,
m
i=0ai[α2,0Pi(−1)+α2,1P
i(1)]
di,2
=0.(3.18)
Due to (3.14) and (3.16), Eq. (3.15) rewritten as
goyal.praveen2011@gmail.com
220 M. Attary and P. Agarwal
M2
mamPx,m+λ2ImamPx,m=0.(3.19)
Since (3.15) has two boundary conditions, we need to remove the last two equations
from (3.19) and replace boundary conditions (3.18) instead of them.
Due to orthogonality of {Pi(x)}∞
i=0and using simple computations, we derive
⎧
⎨
⎩
M2
mam=−λ2Imam,
di,1am=0,
di,2am=0,
(3.20)
where M2
mand Imare obtained by removing the last two rows of M2
mand Im, respec-
tively. Also, Imis a m+1-dimensionl identity matrix.
(3.20) can be symbolically expressed as
Ham=λ2Gam,(3.21)
where Hand Gare defined as
H=⎡
⎣
M2
m
di,1
di,2
⎤
⎦,G=⎡
⎣−Im
0
0
⎤
⎦.(3.22)
In the following, we consider another form of (1.1)
u(4)(x)+λ2u(x)=0.(3.23)
In a similar manner, let u(4)
m(x)be an approximation function of the exact solution
u(4)(x)as follows:
u(4)
m(x)=M4
mamPx,m,(3.24)
where M4
mbe a finite form of M4. by substituting (3.14) and (3.24)in(3.23), we have
M4
mamPx,m+λ2ImamPx,m=0.(3.25)
Also boundary conditions of (3.23) can be written as
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based on the Tau Method … 221
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
m
i=0ai[α1,0Pi(−1)+α1,1P
i(1)+α1,2P
i(1)+α1,3P
i(1)]
ei,1
=0,
m
i=0ai[α2,0Pi(−1)+α2,1P
i(1)+α2,2P
i(1)+α2,3P
i(1)]
ei,2
=0,
m
i=0ai[α3,0Pi(−1)+α3,1P
i(1)+α3,2P
i(1)+α3,3P
i(1)]
ei,3
=0,
m
i=0ai[α4,0Pi(−1)+α4,1P
i(1)+α4,2P
i(1)+α4,3P
i(1)]
ei,4
=0.
(3.26)
These boundary conditions should be applied in Eq. (3.25), so we conclude
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
M4
mam=−λ2Imam,
ei,1am=0,
ei,2am=0,
ei,3am=0,
ei,4am=0,
(3.27)
or equivalently
am=λ2am,(3.28)
where and are defined as
=
⎡
⎢
⎢
⎢
⎢
⎣
M4
m
ei,1
ei,2
ei,3
ei,4
⎤
⎥
⎥
⎥
⎥
⎦
,=
⎡
⎢
⎢
⎢
⎢
⎣
−Im
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎦
.(3.29)
The same as before, M4
mand Imare obtained by removing the last four rows of
M4
mand Im, respectively. Finally, due to (3.21) and (3.28), the values of λcan be
computed.
4 Numerical Results
In this section, we present some examples to show the accuracy of the proposed
method. These examples are solved by Legendre polynomials. Numerical results are
compared with some existing numerical methods. Obtained results are reported in
Tables 1,2,3,4, and 5.
Example 1 Consider the following second-order eigenvalue problem:
u(x)+λ2u(x)=0,(4.1)
goyal.praveen2011@gmail.com
222 M. Attary and P. Agarwal
with the conditions:
u(0)=0,
u(1)+u(1)=0,
and the exact solution for λis λ=−tan λ.
Here, we consider computational details of the presented method for Example 1.
As we pointed out, the Legendre polynomials are defined in the interval [−1,1].So
(4.1), which is stated on the interval [0,1], will be converted to the interval [−1,1]
by choosing x=1
2(t+1)or t=2x−1.
Therefore, above example can be written as
⎧
⎪
⎨
⎪
⎩
(dt
dx )2d2u
dt2+λ2u=0,
u(−1)=0,
(( dt
dx )du
dt )(1)+u(1)=0,
or ⎧
⎪
⎨
⎪
⎩
4d2u
dt2+λ2u=0,
u(−1)=0,
2du
dt (1)+u(1)=0
by choosing m=5, for numerical implimentation of the proposed method, we will
obtain the following matrices :
M5=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
010101
003030
000505
000070
000009
000000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,M2
5=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
003 0 10 0
00015 0 42
000 0 35 0
000 0 0 63
000 0 0 0
000 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
H=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0030 100
00015 042
0000 350
0000 063
1−11−1−11
1 3 713 2131
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,G=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−100000
0−10 000
00−1000
000−100
000000
000000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
According to (3.21), we obtain the following values of λ:
λ=[−,−,16.3648,8.54625,4.92652,2.02877].
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based on the Tau Method … 223
Table 1 Numerical results of Example 1, using proposed method
λim=5 m =6 m =7 m =8 m =9Exact sol.
λ12.02877 2.02876 2.02876 2.02876 2.02876 2.02876
λ24.92652 4.9145 4.9132 4.91318 4.91318 4.91318
λ38.54625 8.06465 7.9931 7.97981 7.97877 7.97867
λ416.3648 12.391 11.3681 11.1517 11.0946 11.0855
Table 2 Numerical results of Example 2, using the proposed method
λim=14 m=15 m=17 Exact sol.
λ1237.72106753 237.72106753 237.72106753 237.72106753
λ22496.48743849 2496.48743786 2496.48743786 2496.48743786
λ310867.583360842 10867.5824827024 10867.582217387 10867.58221698
λ431782.7593574787 31780.1535527804 31780.096714447 31780.09645408
The obtained numerical results for different values of mhave been reported in Table 1.
Our proposed method has produced highly numerical results and the reported results,
show that we can obtain good numerical results for m≥9.
Example 2 Consider the following fourth-order eigenvalue problem:
u(4)(x)−λu(x)=0(4.2)
with the conditions:
u(0)=u(0)=0,
u(1)=u(1)=0,
and the exact solution for λis tan √λ−tanh √λ=0.
We have reported the obtained results for m=14,15,17, in Table 2. Also, as we
expected the reported results show that high accuracy is obtained in comparison to
the numerical results in [2] and [6]. Table3represents that we can achieve better
results for a lower values of m.
Example 3 To show the validity of the proposed method, consider the problem of
column buckling. A slender column which is subjected to a concentric axial com-
pressive load, P, as it shown in Fig. 1, and is simply supported at its both ends. The
equation representing the bending of the column is
u(2)(x)=M
EI,(4.3)
goyal.praveen2011@gmail.com
224 M. Attary and P. Agarwal
Table 3 Numerical results of Example 2, using the proposed method
λiPresented method m=22
m=18 m=21 Method in [2]Method in [6]
λ1237.72106753 237.72106753 237.72106753 237.72106753
λ22496.48743786 2496.48743786 2496.48743784 2496.48743843
λ310867.58221698 10867.58221698 10867.59367145 10867.58221699
λ431780.09651687 31780.09645408 31475.48355038 31780.09650785
Table 4 Numerical results of Example 3, using proposed method
λiPresented method Method in [3]Analytical sol.
m=6 m =8 h =3/4 h =3/5
λ11.0472 1.0472 1.0205 1.0301 1.0472
λ22.10198 2.09443 1.8856 1.9593 2.0944
λ33.18373 3.14225 2.4637 2.6967 3.1416
λ46.49987 4.31943 −3.1702 4.1888
λ59.06094 5.53544 − − 5.2360
where u(2)(x)specifies the curvature, Mis the bending moment, Eis the modulus
of elasticity, and Iis the moment of inertia of the cross section about its neutral
axis. Corresponding to Fig. 1b, it is clear that the total moment at the free side of the
column is equal to M=−Pu, by substituting the moment into the equation (4.3),
the following second-order differential equation for Euler buckling will be obtained:
u(2)(x)+k2u(x)=0,(4.4)
where
k2=P
EI,
with the conditions:
u(0)=0,u(L)=0,
and k=nπ
Lare the eigenvalues for the column.
For this example, we take E=10 ×109pa, I=1.25 ×10−5m4, and L=3. The
numerical results can be seen from Table 5. Table 4shows our results in comparison
with the results of [3]. By increasing m, additional eigenvalues are determined and
the previously determined values become progressively more accurate.
goyal.praveen2011@gmail.com
An Effective Numerical Technique Based on the Tau Method … 225
Fig. 1 a A slender rod. bA free- body diagram of a rod
Table 5 Numerical results of Example 3
λim=10 m=12 m=14
λ11.0472 1.0472 1.0472
λ22.0944 2.0944 2.0944
λ33.1416 3.1416 3.1416
λ44.1933 4.1888 4.1888
λ55.2540 5.2364 5.2360
5 Conclusion
In this research, a numerical technique based on the Tau method was presented for
solving the eigenvalue problems. This method converts eigenvalue problems into
a system of algebraic equations. The comparison of the obtained results with the
other numerical methods and the exact solution indicates that the desired accuracy
is obtained. By using some modifications, the proposed method can be applied to
solve other ordinary differential equations.
goyal.praveen2011@gmail.com
226 M. Attary and P. Agarwal
Acknowledgements This work was supported to the second author [P Agarwal] by the research
grant supported by the Department of Science & Technology(DST), India (No:INT/RUS/RFBR/P-
308) and Science & Engineering Research Board (SERB), India (No:TAR/2018/000001).
References
1. Agarwal, R.P., Regan, D.O.: Ordinary and Partial Differential Equations. Springer (2009)
2. Attili, B., Lesnic, D.: An efficient method for computing eigenelements of Sturm-Liouville
fourth-order boundary value problems. Appl. Math. Comput. 182(2), 1247–1254 (2006)
3. Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill (2010)
4. EI-Gamel, M., Sameeh, M.: An efficient technique for Finding the Eigenvalues of fourth-order
Sturm-Liouville problems. Appl. Math. 3, 920–925 (2012)
5. Greenberg, L., Marletta, M.: Algorithm 775: The code SLEUTH for solving fourth-order Sturm-
Liouville problems. ACM Trans. Math. Softw. 23(4), 453–493 (1997)
6. Syam, M., Siyyam, H.: An efficient technique for finding the Eigenvalues of fourth-order Sturm-
Liouville problems. Chaos Solitons Fractals 39, 659–665 (2009)
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type
Inequalities for Coordinated Convex
Mappings Utilizing Generalized
Fractional Integrals
Hüseyin Budak and Praveen Agarwal
Abstract In this chapter, we obtain the Hermite–Hadamard-type inequalities for
coordinated convex function via generalized fractional integrals, which generalize
some important fractional integrals such as the Riemann–Liouville fractional inte-
grals, the Hadamard fractional integrals, and Katugampola fractional integrals. The
results given in this chapter provide a generalization of several inequalities obtained
in earlier studies.
Keywords Hermite–Hadamard’s inequalities ·Generalized fractional integral ·
Coordinated convex ·Integral inequalities
2010 Mathematics Subject Classification. 26D07, 26D10, 26D15, 26B15,
26B25.
1 Introduction
The Hermite–Hadamard inequality discovered by Hermite and Hadamard see, e.g.,
[13,28], p. 137) is one of the most well-established inequalities in the theory of
convex functions with a geometrical interpretation and many applications. These
inequalities state that if f:I→Ris a convex function on the interval Iof real
numbers and a,b∈Iwith a<b, then
H. Budak (B
)
Department of Mathematics, Anand International College of Engineering,
Near Kanota, Agra Road, Jaipur 303012, Rajasthan, India
e-mail: hsyn.budak@gmail.com
P. Ag a r w a l
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
e-mail: goyal.praveen2011@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
P. Ag a r w a l e t a l. ( ed s. ), Fractional Calculus, Springer Proceedings
in Mathematics & Statistics 303, https://doi.org/10.1007/978-981- 15-0430-3_13
227
goyal.praveen2011@gmail.com
228 H. Budak and P. Agarwal
fa+b
2≤1
b−a
b
a
f(x)dx ≤f(a)+f(b)
2.(1.1)
Both inequalities hold in the reverse direction if fis concave. We note that Hermite–
Hadamard inequality may be regarded as a refinement of the concept of convexity and
it follows easily from Jensen’s inequality. Hermite–Hadamard inequality for convex
functions has received renewed attention in recent years and a remarkable variety of
refinements and generalizations have been studied (see, for example, [3,14–16,18,
20,27,34,35,40,47,48]).
A formal definition for coordinated convex function may be stated as follows:
Definition 1 A function f:→Ris called coordinated convex on , for all
(x,u), (y,v)∈and t,s∈[0,1], if it satisfies the following inequality:
f(tx +(1−t)y,su +(1−s)v) (1.2)
≤ts f (x,u)+t(1−s)f(x,v)+s(1−t)f(y,u)+(1−t)(1−s)f(y,v).
The mapping fis a coordinated concave on if the inequality (1.2) holds in
reverse direction for all t,s∈[0,1]and (x,u), ( y,v) ∈.
In [12], Dragomir proved the following inequalities which is Hermite–Hadamard-
type inequalities for coordinated convex functions on the rectangle from the plane
R2.
Theorem 1 Suppose that f :→Ris a coordinated convex, then we have the
following inequalities:
fa+b
2,c+d
2≤1
2⎡
⎢
⎣1
b−a
b
a
fx,c+d
2dx +1
d−c
d
c
fa+b
2,ydy⎤
⎥
⎦
≤1
(b−a)(d−c)
b
a
d
c
f(x,y)dydx (1.3)
≤1
4⎡
⎢
⎣1
b−a
b
a
f(x,c)dx +1
b−a
b
a
f(x,d)dx
+1
d−c
d
c
f(a,y)dy +1
d−c
d
c
f(b,y)dy⎤
⎥
⎦
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4.
The above inequalities are sharp. The inequalities in (1.3) hold in reverse direction
if the mapping f is a coordinated concave mapping.
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 229
For the other Hermite–Hadamard-type inequalities for coordinated convex func-
tions, please refer to [2,4,24,26,41,44].
In the following, we give the definition of Riemann–Liouville fractional integrals:
Definition 2 Let f∈L1[a,b].The Riemann–Liouville fractional integrals Jα
a+f
and Jα
b−fof order α>0 with a≥0 are defined by
Jα
a+f(x)=1
(α)
x
a
(x−t)α−1f(t)dt,x>a
and
Jα
b−f(x)=1
(α)
b
x
(t−x)α−1f(t)dt,x<b
respectively. Here, (α)is the Gamma function and J0
a+f(x)=J0
b−f(x)=f(x).
More details on Riemann–Liouville fractional integrals, one can see
[19,23,25,30].
It is remarkable that Sarikaya et al. [32] first gave the following interesting inte-
gral inequalities of Hermite–Hadamard-type involving Riemann–Liouville fractional
integrals.
Theorem 2 Let f :[a,b]→Rbe a positive function with 0≤a<b and f ∈
L1[a,b].If f is a convex function on [a,b], then the following inequalities for
fractional integrals hold:
fa+b
2≤(α+1)
2(b−a)αJα
a+f(b)+Jα
b−f(a)≤f(a)+f(b)
2(1.4)
with α>0.
Moreover, Hermite–Hadamard-type inequality for coordinated convex functions
utilizing Riemann–Liouville fractional integrals is obtained by Sarıkaya in [36]. One
can find some recent Hermite–Hadamard inequalities for the function of one and two
variables via Riemann–Liouville fractional integrals in [1,5–11,17,22,31,33,37–
39,45,49,50].
Hadamard fractional integrals are as follows:
Definition 3 Let f∈L1([a,b]).The Hadamard fractional integrals Hα
a+f,and
Hα
b−fof order α>0 with a≥0 are defined by
Hα
a+f(x):= 1
(α)
x
aln x
tα−1f(t)
tdt,x>a,
goyal.praveen2011@gmail.com
230 H. Budak and P. Agarwal
and
Hα
b−f(x):= 1
(α)
b
x
ln t
xα−1f(t)
tdt,x<b,
respectively.
Recently, some papers are devoted to Hermite–Hadamard inequalities via
Hadamard fractional integrals, see [29,42,43,51,52].
Now we give following generalized fractional integrals:
Definition 4 Let g:[a,b]→Rbe an increasing and positive monotone function
on (a,b]having a continuous derivative g(x)on (a,b). The left-side (Iα
a+;gf(x))
and right-side (Iα
b−;gf(x)) fractional integral of fwith respect to the function gon
[a,b]of order α<0 are defined by
Iα
a+;gf(x)=1
(α)
x
a
g(t)f(t)
[g(x)−g(t)]1−αdt,x>a
and
Iα
b−;gf(x)=1
(α)
b
x
g(t)f(t)
[g(t)−g(x)]1−αdt,x<b
respectively.
Jleli and Samet establish following Hermite–Hadamard inequalities:
Theorem 3 ([21]) Let g:[a,b]→Rbe an increasing and positive monotone func-
tion on (a,b], having a continuous derivative g(x)on (a,b)and let α>0.If f is
a convex function on [a,b],then
ϕa+b
2≤(α+1)
4[g(b)−g(a)]αIα
a+;g(b)+α
b−;gf(a)≤ϕ(a)+f(b)
2
(1.5)
where (x)=ϕ(x)+ϕ(x)and ϕ(x)=ϕ(a+b−x)for x ∈[a,b].
Hadamard fractional integrals of a function with two variables can be given as
follows:
Definition 5 Let f∈L1([a,b]×[c,d]).The Hadamard fractional integrals
Jα,β
a+,c+f,Jα,β
a+,d−f,Jα,β
b−,c+fand Jα,β
b−,d−fof order α,β>0 with a,c≥0are
defined by
Jα,β
a+,c+f(x,y):= 1
(α)(β)
x
a
y
cln x
tα−1ln y
sβ−1f(t,s)
ts dsdt,x>a,y>c,
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 231
Jα,β
a+,d−f(x,y):= 1
(α)(β)
x
a
d
yln x
tα−1ln s
yβ−1f(t,s)
ts dsdt,x>a,y<d,
Jα,β
b−,c+f(x,y):= 1
(α)(β)
b
x
y
cln t
xα−1ln y
sβ−1f(t,s)
ts dsdt,x<b,y>c,
and
Jα,β
b−,d−f(x,y):= 1
(α)(β)
b
x
d
yln t
xα−1ln y
sβ−1f(t,s)
ts dsdt,x<b,y<d,
respectively.
Now, we give following generalized fractional integral operators:
Definition 6 Let g:[a,b]→Rbe an increasing and positive monotone function
on (a,b], having a continuous derivative g(x)on (a,b)and let w:[c,d]→R
be an increasing and positive monotone function on (c,d], having a continuous
derivative w(y)on (c,d). Let f∈L1([a,b]×[c,d]). The generalized fractional
integral operators for functions of two variables are defined by
Jα,β
a+,c+;g,w f(x,y):= 1
(α)(β)
x
a
y
c
g(t)
[g(x)−g(t)]1−α
w(s)
[w(y)−w(s)]1−βf(t,s)dsdt,x>a,y>c,
Jα,β
a+,d−;g,w f(x,y):= 1
(α)(β)
x
a
d
y
g(t)
[g(x)−g(t)]1−α
w(s)
[w(s)−w(y)]1−βf(t,s)dsdt,x>a,y<d,
Jα,β
b−,c+;g,w f(x,y):= 1
(α)(β)
b
x
y
c
g(t)
[g(t)−g(x)]1−α
w(s)
[w(y)−w(s)]1−βf(t,s)dsdt,x<b,y>c,
and
Jα,β
b−,d−;g,w f(x,y):= 1
(α)(β)
b
x
d
y
g(t)
[g(t)−g(x)]1−α
w(s)
[w(s)−w(y)]1−βf(t,s)dsdt,x<b,y<d.
Similar to the above definitions, we can give the following integrals:
Jα
a+;gfx,c+d
2:= 1
(α)
x
a
g(t)
[g(x)−g(t)]1−αft,c+d
2dt,x>a,
goyal.praveen2011@gmail.com
232 H. Budak and P. Agarwal
Jα
b−;gfx,c+d
2:= 1
(α)
b
x
g(t)
[g(t)−g(x)]1−αft,c+d
2dt,x<b,
Jβ
c+;wfa+b
2,y:= 1
(β)
y
c
w(t)
[w(y)−w(s)]1−βfa+b
2,yds,y>c,
and
Jβ
d−;wfa+b
2,y:= 1
(β)
y
c
w(t)
[w(s)−w(y)]1−βfa+b
2,yds,y<d,
If we choose g(t)=tρ
ρand w(s)=sσ
σ
ρin Definition 6, then we have the following
Katugampola fractional integrals for function with two variables similar to definitions
given by Yaldiz in [46]:
Definition 7 Let f∈L1([a,b]×[c,d]). The Katugampola fractional integrals for
function with two variables are defined by
ρ,σIα,β
a+,c+f(x,y):= ρ1−ασ1−β
(α)(β)
x
a
y
c
tρ−1
[xρ−tρ]1−α
sσ−1
[yσ−sσ]1−βf(t,s)dsdt,x>a,y>c,
ρ,σIα,β
a+,d−f(x,y):= ρ1−ασ1−β
(α)(β)
x
a
d
y
tρ−1
[xρ−tρ]1−α
sσ−1
[sσ−yσ]1−βf(t,s)dsdt,x>a,y<d,
ρ,σIα,β
b−,c+f(x,y):= ρ1−ασ1−β
(α)(β)
b
x
y
c
tρ−1
[tρ−xρ]1−α
sσ−1)
[yσ−sσ]1−βf(t,s)dsdt,x<b,y>c,
and
ρ,σIα,β
b−,d−f(x,y):= ρ1−ασ1−β
(α)(β)
b
x
d
y
tρ−1
[tρ−xρ]1−α
sσ−1
[sσ−yσ]1−βf(t,s)dsdt,x<b,y<d.
The aim of this study is to establish Hermite–Hadamard-type integral inequalities
for a coordinated convex function involving generalized fractional integrals. The
results presented in this paper provide extensions of those given in earlier works.
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 233
2 Main Results
Let f:=[a,b]×[c,d]→R.First, we define the following functions which will
be used frequently:
f1(x,y)=f(a+b−x,y),
f2(x,y)=f(x,c+d−y),
f3(x,y)=f(a+b−x,c+d−y),
G(x,y)=f(x,y)+
f2(x,y)(2.1)
H(x,y)=f(x,y)+
f1(x,y)
K(x,y)=
f1(x,y)+
f3(x,y)
L(x,y)=
f2(x,y)+
f3(x,y)
F(x,y)=
f1(x,y)+
f2(x,y)+
f3(x,y)+f(x,y)
=G(x,y)+H(x,y)+K(x,y)+L(x,y)
2
for (x,y)∈[a,b]×[c,d].
Theorem 4 Let g:[a,b]→Rbe an increasing and positive monotone function on
(a,b], having a continuous derivative g(x)on (a,b)and let w:[c,d]→Rbe an
increasing and positive monotone function on (c,d], having a continuous derivative
w(y)on (c,d).If f :→Ris a coordinated convex on , then for α,β>0the
following Hermite–Hadamard-type inequality hold:
fa+b
2,c+d
2(2.2)
≤(α+1)(β+1)
16 [g(b)−g(a)]α[w(d)−w(c)]β
×Jα,β
a+,c+;g,w F(b,d)+Jα,β
a+,d−;g,w F(b,c)+Jα,β
b−,c+;g,w F(a,d)+Jα,β
b−,d−;g,w F(a,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4,
where the function F is defined as in (2.1).
Proof Since fis a coordinated convex mapping on , we have
fu+v
2,p+q
2≤f(u,p)+f(u,q)+f(v, p)+f(v, q)
4(2.3)
for (u,p), (v, q)∈. Now, for t,s∈[0,1],let u=ta +(1−t)b,v =(1−t)a+
tb,p=cs +(1−s)dand q=(1−s)c+sd.Then we have
goyal.praveen2011@gmail.com
234 H. Budak and P. Agarwal
fa+b
2,c+d
2(2.4)
≤1
4f(ta +(1−t)b,cs +(1−s)d)+1
4f(ta +(1−t)b,(1−s)c+sd)
+1
4f((1−t)a+tb,cs +(1−s)d)+1
4f((1−t)a+tb,(1−s)c+sd).
Multiplying both sides of (2.4)by
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−β,
and integrating the resulting inequality with respect to t,sover (0,1)×(0,1),we
get
(b−a)(
d−c)
(α)(β)fa+b
2,c+d
21
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βdsdt
≤(b−a)(
d−c)
4(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf(ta +(1−t)b,cs +(1−s)d)dsdt
+(b−a)(
d−c)
4(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf(ta +(1−t)b,(1−s)c+sd)dsdt
+(b−a)(
d−c)
4(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf((1−t)a+tb,cs +(1−s)d)dsdt
+(b−a)(
d−c)
4(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf((1−t)a+tb,(1−s)c+sd)dsdt.
By a simple calculation, we have
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βdsdt =[g(b)−g(a)]α[w(d)−w(c)]β
αβ(b−a)(d−c).
Using the change of variables τ=(1−t)a+tb and η=(1−s)c+sd,we obtain
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 235
[g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1)fa+b
2,c+d
2
≤1
4(α)(β)
1
0
1
0
g(τ)
[g(b)−g(τ)]1−α
w(η)
[w(d)−w(η)]1−βf(a+b−τ,c+d−η)dηdτ
+1
4(α)(β)
1
0
1
0
g(τ)
[g(b)−g(τ)]1−α
w(η)
[w(d)−w(η)]1−βf(a+b−τ,η)dηdτ
+1
4(α)(β)
1
0
1
0
g(τ)
[g(b)−g(τ)]1−α
w(η)
[w(d)−w(η)]1−βf(τ,c+d−η)dηdτ
+1
4(α)(β)
1
0
1
0
g(τ)
[g(b)−g(τ)]1−α
w(η)
[w(d)−w(η)]1−βf(τ,η)dηdτ
=1
4Jα,β
a+,c+;g,w
f3(b,d)+Jα,β
a+,c+;g,w
f1(b,d)+Jα,β
a+,c+;g,w
f2(b,d)+Jα,β
a+,c+;g,w f(b,d)
=1
4Jα,β
a+,c+;g,w F(b,d).
That is, we have
[g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1)fa+b
2,c+d
2≤1
4Jα,β
a+,c+;g,w F(b,d). (2.5)
Similarly, multiplying both sides of (2.4)by
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w((1−s)c+sd)−w(c)]1−β
and integrating the obtained inequality with respect to t,sover (0,1)×(0,1),we
obtain
[g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1)fa+b
2,c+d
2≤1
4Jα,β
a+,d−;g,w F(b,c). (2.6)
Moreover, multiplying both sides of (2.4)by
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g((1−t)a+tb)−g(s)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−β
and
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g((1−t)a+tb)−g(s)]1−α
w((1−s)c+sd)
[w((1−s)c+sd)−w(c)]1−β
goyal.praveen2011@gmail.com
236 H. Budak and P. Agarwal
then integrating the established inequalities with respect to t,sover (0,1)×(0,1),
we have the following inequalities:
[g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1)fa+b
2,c+d
2≤1
4Jα,β
b−,c+;g,w F(a,d)(2.7)
and
[g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1)fa+b
2,c+d
2≤1
4Jα,β
b−,d−;g,w F(a,c), (2.8)
respectively.
Summing the inequalities (2.5)–(2.8), we get
fa+b
2,c+d
2
≤(α+1)(β+1)
16 [g(b)−g(a)]α[w(d)−w(c)]β
×Jα,β
a+,c+;g,w F(b,d)+Jα,β
a+,d−;g,w F(b,c)+Jα,β
b−,c+;g,w F(a,d)+Jα,β
b−,d−;g,w F(a,c).
This completes the proof of first inequality in (2.2).
For the proof of the second inequality in (2.2), since fis a coordinated convex,
we have
f(ta +(1−t)b,cs +(1−s)d)+f(ta +(1−t)b,(1−s)c+sd)(2.9)
+f((1−t)a+tb,cs +(1−s)d)+f((1−t)a+tb,(1−s)c+sd)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d).
Multiplying both sides of (2.9)by
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−β,
and integrating the resulting inequality with respect to t,sover (0,1)×(0,1),we
get
(b−a)(
d−c)
(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf(ta +(1−t)b,cs +(1−s)d)dsdt
+(b−a)(
d−c)
(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf(ta +(1−t)b,(1−s)c+sd)dsdt
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 237
+(b−a)(
d−c)
(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf((1−t)a+tb,cs +(1−s)d)dsdt
+(b−a)(
d−c)
(α)(β)
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βf((1−t)a+tb,(1−s)c+sd)dsdt
≤(b−a)(
d−c)
(α)(β)[f(a,c)+f(a,d)+f(b,c)+f(b,d)]
×
1
0
1
0
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−βdsdt.
Then, we get
Jα,β
a+,c+;g,w
f3(b,d)+Jα,β
a+,c+;g,w
f1(b,d)+Jα,β
a+,c+;g,w
f2(b,d)+Jα,β
a+,c+;g,w f(b,d)
≤[f(a,c)+f(a,d)+f(b,c)+f(b,d)][g(b)−g(a)]α[w(d)−w(c)]β
(α+1)(β+1),
that is,
(α+1)(β+1)
[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,c+;g,w F(b,d)≤f(a,c)+f(a,d)+f(b,c)+f(b,d).
(2.10)
Similarly, multiplying both sides of (2.9)by
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g(b)−g((1−t)a+tb)]1−α
w((1−s)c+sd)
[w((1−s)c+sd)−w(c)]1−β,
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g((1−t)a+tb)−g(a)]1−α
w((1−s)c+sd)
[w(d)−w((1−s)c+sd)]1−β
and
(b−a)(
d−c)
(α)(β)
g((1−t)a+tb)
[g((1−t)a+tb)−g(a)]1−α
w((1−s)c+sd)
[w((1−s)c+sd)−w(c)]1−β
integrating the resulting inequalities with respect to t,sover (0,1)×(0,1),we
establish the following inequalities:
(α+1)(β+1)
[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,d−;g,w F(b,c)≤f(a,c)+f(a,d)+f(b,c)+f(b,d),
(2.11)
goyal.praveen2011@gmail.com
238 H. Budak and P. Agarwal
(α+1)(β+1)
[g(b)−g(a)]α[w(d)−w(c)]βJα,β
b−,c+;g,w F(a,d)≤f(a,c)+f(a,d)+f(b,c)+f(b,d),
(2.12)
and
(α+1)(β+1)
[g(b)−g(a)]α[w(d)−w(c)]βJα,β
b−,d−;g,w F(a,c)≤f(a,c)+f(a,d)+f(b,c)+f(b,d),
(2.13)
respectively.
By adding the inequalities (2.10)–(2.13), we have the inequality:
(α+1)(β+1)
[g(b)−g(a)]α[w(d)−w(c)]β(2.14)
×Jα,β
a+,c+;g,w F(b,d)+Jα,β
a+,d−;g,w F(b,c)+Jα,β
b−,c+;g,w F(a,d)+Jα,β
b−,d−;g,w F(a,c)
≤4[f(a,c)+f(a,d)+f(b,c)+f(b,d)]
If we divide the both sides of inequality (2.14)by16,then we have the second
inequality in (2.2).
This completes the proof.
Remark 1 If we choose g(t)=tand w(s)=sin Theorem 4, then we have the
following inequalities for Riemann–Liouville fractional integrals
fa+b
2,c+d
2
≤(α+1)(β+1)
4(b−a)α(d−c)βJα,β
a+,c+f(b,d)+Jα,β
a+,d−f(b,c)+Jα,β
b−,c+f(a,d)+Jα,β
b−,d−f(a,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4
which are proved by Sarikaya in [36].
Corollary 1 Under assumption of Theorem 4with g(t)=ln t and w(s)=ln s,then
we have the following inequalities for Hadamard fractional integrals:
fa+b
2,c+d
2
≤(α+1)(β+1)
16 ln b
aαln d
cβJα,β
a+,c+F(b,d)+Jα,β
a+,d−F(b,c)+Jα,β
b−,c+F(a,d)+Jα,β
b−,d−F(a,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4.
Corollary 2 Under assumption of Theorem 4with g(t)=tρ
ρand w(s)=sσ
σ,then
we have the following inequalities for Katugampola fractional integrals:
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 239
fa+b
2,c+d
2
≤(α+1)(β+1)ρασβ
16 [bρ−aρ]α[dσ−cσ]βρ,σIα,β
a+,c+F(b,d)+ρ,σIα,β
a+,d−F(b,c)+ρ,σIα,β
b−,c+F(a,d)+ρ,σIα,β
b−,d−F(a,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4.
Theorem 5 Let g:[a,b]→Rbe an increasing and positive monotone function on
(a,b], having a continuous derivative g(x)on (a,b)and let w:[c,d]→Rbe an
increasing and positive monotone function on (c,d], having a continuous derivative
w(y)on (c,d).If f →Ris a coordinated convex on , then for α,β>0the
following Hermite–Hadamard-type inequality holds:
fa+b
2,c+d
2(2.15)
≤(α+1)
8[g(b)−g(a)]αJα
a+;gHb,c+d
2+Jα
b−;gHa,c+d
2
+(β+1)
8[w(d)−w(c)]βJβ
c+;wGa+b
2,d+Jβ
d−;wGa+b
2,c
≤(α+1)(β+1)
16 [g(b)−g(a)]α[w(d)−w(c)]β
×Jα,β
a+,c+;g,w F(b,d)+Jα,β
a+,d−;g,w F(b,c)+Jα,β
b−,c+;g,w F(a,d)+Jα,β
b−,d−;g,w F(a,c)
≤(α+1)
16 [g(b)−g(a)]αJα
a+;gH(b,c)+Jα
a+;gH(b,d)+Jα
b−;gH(a,c)+Jα
b−;gH(a,d)
+(β+1)
16 [w(d)−w(c)]βJβ
c+;wG(a,d)+Jβ
c+;wG(b,d)+Jβ
d−;wG(a,c)+Jβ
d−;wG(b,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4
where the function H,F, and G are defined as in Eq. 2.1.
Proof Since fis a coordinated convex on , if we define the mapping h1
x:[c,d]→
R,h1
x(y)=f(x,y), then h1
x(y)is convex for all x∈[a,b]and H1
x(y)=h1
x(y)+
h1
x(y)=f(x,y)+
f2(x,y)=G(x,y). If we apply the inequalities (1.5)forthe
convex function h1
x(y), then we have
h1
xc+d
2≤(β+1)
4[w(d)−w(c)]βJβ
c+;wH1
x(d)+Jβ
d−;wH1
x(c)≤h1
x(c)+h1
x(d)
2,
that is,
goyal.praveen2011@gmail.com
240 H. Budak and P. Agarwal
fx,c+d
2(2.16)
≤β
4[w(d)−w(c)]β⎡
⎣
d
c
w(y)
[w(d)−w(y)]1−βG(x,y)dy +
d
c
w(y)
[w(y)−w(c)]1−βG(x,y)dy⎤
⎦
≤f(x,c)+f(x,d)
2.
Multiplying the inequalities (2.16)by
α
[g(b)−g(a)]α
g(x)
[g(b)−g(x)]1−α,
and α
[g(b)−g(a)]α
g(x)
[g(x)−g(a)]1−α,
then by integrating the obtained results with respect to xfrom ato b,we get
(α+1)
[g(b)−g(a)]αJα
a+;gfb,c+d
2(2.17)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,c+;g,w G(b,d)+Jα,β
a+,d−;g,w G(b,c)
≤(α+1)
2[g(b)−g(a)]αJα
a+;gf(b,c)+Jα
a+;gf(b,d),
and
(α+1)
[g(b)−g(a)]αJα
b−;gfa,c+d
2(2.18)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
b−,c+;g,w G(a,d)+Jα,β
b−,d−;g,w G(a,c)
≤(α+1)
2[g(b)−g(a)]αJα
b−;gf(a,c)+Jα
b−;gf(a,d),
respectively.
On the other hand, since fis a coordinated convex on , if we define the map-
ping h2
x:[c,d]→R,h2
x(y)=
f1(x,y), then h2
x(y)is convex for all x∈[a,b]and
H2
x(y)=h2
x(y)+
h2
x(y)=
f1(x,y)+
f3(x,y)=K(x,y). If we apply the inequal-
ities (1.5) for the convex function h2
x(y), then we have
h2
xc+d
2≤(β+1)
4[w(d)−w(c)]βJβ
c+;wH2
x(d)+Jβ
d−;wH2
x(c)≤h2
x(c)+h2
x(d)
2,
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 241
i.e.,
f1x,c+d
2(2.19)
≤β
4[w(d)−w(c)]β⎡
⎣
d
c
w(y)
[w(d)−w(y)]1−βK(x,y)dy +
d
c
w(y)
[w(y)−w(c)]1−βK(x,y)dy⎤
⎦
≤
f1(x,c)+
f1(x,d)
2.
Similarly, multiplying the inequalities (2.19)by
α
[g(b)−g(a)]α
g(x)
[g(b)−g(x)]1−α,
and α
[g(b)−g(a)]α
g(x)
[g(x)−g(a)]1−α,
then by integrating the obtained results with respect to xfrom ato b,we get
(α+1)
[g(b)−g(a)]αJα
a+;g
f1b,c+d
2(2.20)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,c+;g,w K(b,d)+Jα,β
a+,d−;g,w K(b,c)
≤(α+1)
2[g(b)−g(a)]αJα
a+;g
f1(b,c)+Jα
a+;g
f1(b,d),
and
(α+1)
[g(b)−g(a)]αJα
b−;g
f1a,c+d
2(2.21)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
b−,c+;g,w K(a,d)+Jα,β
b−,d−;g,w K(a,c)
≤(α+1)
2[g(b)−g(a)]αJα
b−;g
f1(a,c)+Jα
b−;g
f1(a,d),
respectively.
Moreover,if we define the mapping h1
y:[a,b]→R,h1
y(x)=f(x,y), then h1
y(x)
is convex for all y∈[c,d]and H1
y(x)=h1
y(x)+
h1
y(x)=f(x,y)+
f1(x,y)=
G(x,y). Applying the inequalities (1.5) for the convex function h1
y(x), then we
have
goyal.praveen2011@gmail.com
242 H. Budak and P. Agarwal
h1
ya+b
2≤(α+1)
4[g(b)−g(a)]αJα
a+;gH1
y(b)+Jα
b−;wH1
y(a)≤h1
y(a)+h1
y(b)
2,
that is,
fa+b
2,y(2.22)
≤α
4[g(b)−g(a)]α⎡
⎣
b
a
g(x)
[g(b)−g(x)]1−αH(x,y)dx +
b
a
g(x)
[g(x)−g(a)]1−αH(x,y)dx⎤
⎦
≤f(a,y)+f(b,y)
2.
Multiplying the inequalities (2.22)by
β
[w(d)−w(c)]β
w(y)
[w(d)−w(y)]1−β
and β
[w(d)−w(c)]β
w(y)
[w(y)−w(c)]1−β
then integrating the established results with respect to yfrom cto d, we obtain the
following inequalities:
(β+1)
[w(d)−w(c)]βJβ
c+;wfa+b
2,d(2.23)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,c+;g,w H(b,d)+Jα,β
b−,c+;g,w H(a,d)
≤(β+1)
2[w(d)−w(c)]βJβ
c+;wf(a,d)+Jβ
c+;wf(b,d),
and
(β+1)
[w(d)−w(c)]βJβ
d−;wfa+b
2,c(2.24)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,d−;g,w H(b,c)+Jα,β
b−,d−;g,w H(a,c)
≤(β+1)
2[w(d)−w(c)]βJβ
d−;wf(a,c)+Jβ
d−;wf(b,c),
respectively.
Furthermore, if we define the mapping h2
y:[a,b]→R,h2
y(x)=
f2(x,y), then
h2
y(x)is convex for all y∈[c,d]and H2
y(x)=h2
y(x)+
h2
y(x)=
f2(x,y)+
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 243
f3(x,y)=L(x,y). Applying the inequalities (1.5) for the convex function h2
y(x),
then we have
h2
ya+b
2≤(α+1)
4[g(b)−g(a)]αJα
a+;gH2
y(b)+Jα
b−;wH2
y(a)≤h2
y(a)+h2
y(b)
2,
i.e.,
f2a+b
2,y(2.25)
≤α
4[g(b)−g(a)]α⎡
⎣
b
a
g(x)
[g(b)−g(x)]1−αL(x,y)dx +
b
a
g(x)
[g(x)−g(a)]1−αL(x,y)dx⎤
⎦
≤
f2(a,y)+
f2(b,y)
2.
Similarly, multiplying the inequalities (2.25)by
β
[w(d)−w(c)]β
w(y)
[w(d)−w(y)]1−β
and β
[w(d)−w(c)]β
w(y)
[w(y)−w(c)]1−β,
then integrating the obtained results with respect to yfrom cto d, we obtain the
following inequalities:
(β+1)
[w(d)−w(c)]βJβ
c+;w
f2a+b
2,d(2.26)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,c+;g,w L(b,d)+Jα,β
b−,c+;g,w L(a,d)
≤(β+1)
2[w(d)−w(c)]βJβ
c+;w
f2(a,d)+Jβ
c+;w
f2(b,d)
and
(β+1)
[w(d)−w(c)]βJβ
d−;w
f2a+b
2,c(2.27)
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]βJα,β
a+,d−;g,w L(b,c)+Jα,β
b−,d−;g,w L(a,c)
≤(β+1)
2[w(d)−w(c)]βJβ
d−;w
f2(a,c)+Jβ
d−;w
f2(b,c),
respectively.
goyal.praveen2011@gmail.com
244 H. Budak and P. Agarwal
Summing the inequalities (2.17), (2.18), (2.20), (2.21), (2.23), (2.24), (2.26) and
(2.27), we have the following inequalities:
(α+1)
[g(b)−g(a)]αJα
a+;gfb,c+d
2+Jα
b−;gfa,c+d
2
+Jα
a+;g
f1b,c+d
2+Jα
b−;g
f1a,c+d
2
+(β+1)
[w(d)−w(c)]βJβ
c+;wfa+b
2,d+Jβ
d−;wfa+b
2,c
+Jβ
c+;w
f2a+b
2,d+Jβ
d−;w
f2a+b
2,c
≤(α+1)(β+1)
4[g(b)−g(a)]α[w(d)−w(c)]β
×Jα,β
a+,c+;g,w G(b,d)+Jα,β
a+,d−;g,w G(b,c)+Jα,β
b−,c+;g,w G(a,d)+Jα,β
b−,d−;g,w G(a,c)
+Jα,β
a+,c+;g,w K(b,d)+Jα,β
a+,d−;g,w K(b,c)+Jα,β
b−,c+;g,w K(a,d)+Jα,β
b−,d−;g,w K(a,c)
+Jα,β
a+,c+;g,w H(b,d)+Jα,β
b−,c+;g,w H(a,d)+Jα,β
a+,d−;g,w H(b,c)+Jα,β
b−,d−;g,w H(a,c)
+Jα,β
a+,c+;g,w L(b,d)+Jα,β
b−,c+;g,w L(a,d)+Jα,β
a+,d−;g,w L(b,c)+Jα,β
b−,d−;g,w L(a,c)
≤(α+1)
2[g(b)−g(a)]αJα
a+;gf(b,c)+Jα
a+;gf(b,d)+Jα
b−;gf(a,c)+Jα
b−;gf(a,d)
+Jα
a+;g
f1(b,c)+Jα
a+;g
f1(b,d)+Jα
b−;g
f1(a,c)+Jα
b−;g
f1(a,d)
+(β+1)
[w(d)−w(c)]βJβ
c+;wf(a,d)+Jβ
c+;wf(b,d)+Jβ
d−;wf(a,c)+Jβ
d−;wf(b,c)
+Jβ
c+;w
f2(a,d)+Jβ
c+;w
f2(b,d)+Jβ
d−;w
f2(a,c)+Jβ
d−;w
f2(b,c).
That is, we have
(α+1)
[g(b)−g(a)]αJα
a+;gHb,c+d
2+Jα
b−;gHa,c+d
2
+(β+1)
[w(d)−w(c)]βJβ
c+;wGa+b
2,d+Jβ
d−;wGa+b
2,c
≤(α+1)(β+1)
2[g(b)−g(a)]α[w(d)−w(c)]β
×Jα,β
a+,c+;g,w F(b,d)+Jα,β
a+,d−;g,w F(b,c)+Jα,β
b−,c+;g,w F(a,d)+Jα,β
b−,d−;g,w F(a,c)
≤(α+1)
2[g(b)−g(a)]αJα
a+;gH(b,c)+Jα
a+;gH(b,d)+Jα
b−;gH(a,c)+Jα
b−;gH(a,d)
+(β+1)
2[w(d)−w(c)]βJβ
c+;wG(a,d)+Jβ
c+;wG(b,d)+Jβ
d−;wG(a,c)+Jβ
d−;wG(b,c)
which completes the proof of the second and third inequalities in (2.15).
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 245
On the other hand, from the first inequality in (1.5), we have
ϕa+b
2(2.28)
≤α
4[g(b)−g(a)]α⎡
⎣
b
a
g(x)
[g(b)−g(x)]α[ϕ(x)+ϕ(a+b−x)]dx
+
b
a
g(x)
[g(x)−g(a)]α[ϕ(x)+ϕ(a+b−x)]dx⎤
⎦.
Since fis a coordinated convex on , by using the inequality (2.28), we obtain
fa+b
2,c+d
2(2.29)
≤α
4[g(b)−g(a)]α⎡
⎢
⎣
b
a
g(x)
[g(b)−g(x)]αfx,c+d
2+fa+b−x,c+d
2dx
+
b
a
g(x)
[g(x)−g(a)]αfx,c+d
2+fa+b−x,c+d
2dx⎤
⎥
⎦
=(α+1)
[g(b)−g(a)]αJα
a+;gHb,c+d
2+Jα
b−;gHa,c+d
2,
and similarly we have
fa+b
2,c+d
2(2.30)
≤β
4[w(d)−w(c)]β⎡
⎢
⎣
d
c
w(y)
[w(d)−w(y)]αfa+b
2,y+fa+b
2,c+d−ydy
+
d
c
w(y)
[w(y)−w(c)]αfa+b
2,y+fa+b
2,c+d−ydy⎤
⎥
⎦
=(β+1)
[w(d)−w(c)]βJβ
c+;wGa+b
2,d+Jβ
d−;wGa+b
2,c.
Combining the inequalities (2.29) and (2.30), we obtain the first inequality in (2.15).
From the second inequality in (1.5), we have
goyal.praveen2011@gmail.com
246 H. Budak and P. Agarwal
α
4[g(b)−g(a)]α⎡
⎣
b
a
g(x)
[g(b)−g(x)]α[ϕ(x)+ϕ(a+b−x)]dx (2.31)
+
b
a
g(x)
[g(x)−g(a)]α[ϕ(x)+ϕ(a+b−x)]dx⎤
⎦
≤ϕ(a)+ϕ(b)
2.
By using the inequality (2.31), we obtain the following inequalities:
(α+1)
4[g(b)−g(a)]αJα
a+;gH(b,c)+Jα
b−;gH(a,c)≤f(a,c)+f(b,c)
2,(2.32)
(α+1)
4[g(b)−g(a)]αJα
a+;gH(b,d)+Jα
b−;gH(a,d)≤f(a,d)+f(b,d)
2,(2.33)
(β+1)
4[w(d)−w(c)]βJβ
c+;wG(a,d)+Jβ
d−;wG(a,c)≤f(a,c)+f(a,d)
2
(2.34)
and
(β+1)
4[w(d)−w(c)]βJβ
c+;wG(b,d)+Jβ
d−;wG(b,c)≤f(b,c)+f(b,d)
2.
(2.35)
Combining the inequalities (2.32)–(2.35), we obtain the last inequality in (2.15).
This completes the proof completely.
Remark 2 If we choose g(t)=tand w(s)=sin Theorem 5, then we have the
following inequalities for Riemann–Liouville fractional integrals:
fa+b
2,c+d
2
≤(α+1)
4(b−a)αJα
a+fb,c+d
2+Jα
b−fa,c+d
2
+(β+1)
4(d−c)βJβ
c+fa+b
2,d+Jβ
d−fa+b
2,c
≤(α+1)(β+1)
4(b−a)α(d−c)βJα,β
a+,c+f(b,d)+Jα,β
a+,d−f(b,c)+Jα,β
b−,c+f(a,d)+Jα,β
b−,d−f(a,c)
≤(α+1)
8(b−a)αJα
a+f(b,c)+Jα
a+f(b,d)+Jα
b−f(a,c)+Jα
b−f(a,d)
+(β+1)
8(d−c)βJβ
c+f(a,d)+Jβ
c+f(b,d)+Jβ
d−f(a,c)+Jβ
d−f(b,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4,
which are proved by Sarikaya in [36].
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 247
Corollary 3 Under assumption of Theorem 4with g(t)=ln t and w(s)=ln s, then
we have the following inequalities for Hadamard fractional integrals
fa+b
2,c+d
2
≤(α+1)
8ln b
aαJα
a+Hb,c+d
2+Jα
b−Ha,c+d
2
+(β+1)
8ln d
cβJβ
c+Ga+b
2,d+Jβ
d−Ga+b
2,c
≤(α+1)(β+1)
16 ln b
aαln d
cβJα,β
a+,c+F(b,d)+Jα,β
a+,d−F(b,c)+Jα,β
b−,c+F(a,d)+Jα,β
b−,d−F(a,c)
≤(α+1)
16 ln b
aαJα
a+H(b,c)+Jα
a+H(b,d)+Jα
b−H(a,c)+Jα
b−H(a,d)
+(β+1)
16 ln d
cβJβ
c+G(a,d)+Jβ
c+G(b,d)+Jβ
d−G(a,c)+Jβ
d−G(b,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4.
Corollary 4 Under assumption of Theorem 4with g(t)=tρ
ρand w(s)=sσ
σ,then
we have the following inequalities for Katugampola fractional integrals:
fa+b
2,c+d
2
≤(α+1)ρα
8[bρ−aρ]αρIα
a+Hb,c+d
2+ρIα
b−Ha,c+d
2
+(β+1)σβ
8[dσ−cσ]βσIβ
c+Ga+b
2,d+σIβ
d−Ga+b
2,c
≤(α+1)(β+1)ρασβ
16 [bρ−aρ]α[dσ−cσ]βρ,σIα,β
a+,c+F(b,d)+ρ,σIα,β
a+,d−F(b,c)+ρ,σIα,β
b−,c+F(a,d)+ρ,σIα,β
b−,d−F(a,c)
≤(α+1)ρασβ
16 [bρ−aρ]αρIα
a+H(b,c)+ρIα
a+H(b,d)+ρIα
b−H(a,c)+ρIα
b−H(a,d)
+(β+1)σβ
16 [dσ−cσ]βσIβ
c+G(a,d)+σIβ
c+G(b,d)+σIβ
d−G(a,c)+σIβ
d−G(b,c)
≤f(a,c)+f(a,d)+f(b,c)+f(b,d)
4.
References
1. Akkurt, A., Sarikaya, M.Z., Budak, H., Yildirim, H.: On the Hadamard’s type inequalities for
co-ordinated convex functions via fractional integrals. J. King Saud Univ. Sci. 29, 380–387
(2017)
2. Alomari, M., Darus, M.: The Hadamards inequality for s-convex function of 2-variables on
the coordinates. Int. J. Math. Anal. 2(13), 629–638 (2008)
goyal.praveen2011@gmail.com
248 H. Budak and P. Agarwal
3. Azpeitia, A.G.: Convex functions and the Hadamard inequality. Rev. Colombiana Math. 28,
7–12 (1994)
4. Bakula, M.K.: An improvement of the Hermite-Hadamard inequality for functions convex on
the coordinates. Aust. J. Math. Anal. Appl. 11(1), 1–7 (2014)
5. Belarbi, S., Dahmani, Z.: On some new fractional integral inequalities. J. Ineq. Pure and Appl.
Math. 10(3) (2009), Art. 86
6. Budak, H., Sarikaya, M.Z.: Hermite-Hadamard type inequalities for s-convex mappings via
fractional integrals of a function with respect to another function. Fasciculi Mathematici 27,
25–36 (2016)
7. Chen, F.X.: On Hermite-Hadamard type inequalities for s-convex functions on the coordinates
via Riemann-Liouville fractional integrals. J. Appl. Math. 2014, Article ID 248710, 8 p., 2014
8. Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497
(2010)
9. Dahmani, Z.: On Minkowski and Hermite-Hadamard integral inequalities via fractional inte-
gration. Ann. Funct. Anal. 1(1), 51–58 (2010)
10. Dahmani, Z., Tabharit, L., Taf, S.: Some fractional integral inequalities. Nonl. Sci. Lett. A 1(2),
155–160 (2010)
11. Deng, J., Wang, J.: Fractional Hermite-Hadamard inequalities for (α,m)-logarithmically con-
vex functions. J. Inequal. Appl. 2013, Article ID 364 (2013)
12. Dragomir, S.S.: On Hadamards inequality for convex functions on the co-ordinates in a rect-
angle from the plane. Taiwan. J. Math. 4, 775–788 (2001)
13. Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite-Hadamard Inequalities and Appli-
cations. Victoria University, RGMIA Monographs (2000)
14. Dragomir, S.S.: Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces.
Proyecciones J. Math. 37(4), 343–341 (2015)
15. Dragomir, S.S.: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl.
167, 49–56 (1992)
16. Dragomir, S.S., Pecaric, J., Persson, L.E.: Some inequalities of Hadamard type. Soochow J.
Math. 21, 335–341 (1995)
17. Farid, G., Rehman, A.U., Zahra, M.: On Hadamard inequalities for k-fractional integrals.
Nonlinear Funct. Anal. Appl. 21(3), 463–478 (2016)
18. Farissi, A.E.: Simple proof and re nement of Hermite-Hadamard inequality. J. Math. Inequal.
4, 365–369 (2010)
19. Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Frac-
tional Order, pp. 223–276. Springer, Wien (1997)
20. Iqbal, M., Qaisar, S., Muddassar, M.: A short note on integral inequality of type Hermite-
Hadamard through convexity. J. Comput. Anal. Appl. 21(5), 946–953 (2016)
21. Jleli, M., Samet, B.: On Hermite-Hadamard type inequalities via fractional integrals of a func-
tion with respect to another function. J. Nonlinear Sci. Appl. 9, 1252–1260 (2016)
22. Katugampola, U.N.: New approach to a generalized fractional integrals. Appl. Math. Comput.
218(4), 860–865 (2011)
23. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential
Equations, North-Holland Mathematics Studies, 204. Elsevier Sci. B.V, Amsterdam (2006)
24. Latif, M.A.: Hermite-Hadamard type inequalities for GA-convex functions on the co-ordinates
with applications. Proc. Pak. Acad. Sci. 52(4), 367–379 (2015)
25. Miller, S., Ross, B.: An introduction to the Fractional Calculus and Fractional Differential
Equations, p. 2. Wiley, USA (1993)
26. Ozdemir, M.E., Yildiz, C., Akdemir, A.O.: On the co-ordinated convex functions Appl. Math.
Inf. Sci. 8(3), 1085–1091 (2014)
27. Pavic, Z.: Improvements of the Hermite-Hadamard inequality. J. Inequal. Appl. 2015, 222
(2015)
28. Peˇcari´c, J.E., Proschan, F., Tong, Y.L.: Convex Functions. Academic Press, Boston, Partial
Orderings and Statistical Applications (1992)
goyal.praveen2011@gmail.com
On Hermite–Hadamard-Type Inequalities for Coordinated Convex Mappings … 249
29. Peng, S., Wei, W., Wang, J.-R.: On the Hermite-Hadamard inequalities for convex functions
via Hadamard fractional integrals. Facta Universitatis Series Math. Inf. 29(1), 55–75 (2014)
30. Podlubni, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
31. Sarikaya, M.Z., Budak, H.: Generalized Hermite-Hadamard type integral inequalities for frac-
tional integrals. Filomat 30(5), 1315–1326 (2016)
32. Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite -Hadamard’s inequalities for fractional
integrals and related fractional inequalities. Math. Comput. Modell. 57, 2403–2407 (2013).
https://doi.org/10.1016/j.mcm.2011.12.048
33. Sarikaya, M.Z., Filiz, H., Kiris, M.E.: On some generalized integral inequalities for Riemann-
Liouville fractional integrals. Filomat 29(6), 1307–1314 (2015). https://doi.org/10.2298/
FIL1506307S
34. Sarikaya, M.Z., Budak, H.: Generalized Hermite-Hadamard type integral inequalities for func-
tions whose 3rd derivatives are s-convex. Tbilisi Math. J. 7(2), 41–49 (2014)
35. Sarikaya, M.Z., Kiris, M.E.: Some new inequalities of Hermite-Hadamard type for s−convex
functions. Miskolc Math. Notes 16(1), 491–501 (2015)
36. Sarikaya, M.Z.: On the Hermite-Hadamard-type inequalities for co-ordinated convex function
via fractional integrals. Integral Trans. Spec. Funct. 25(2), 134–147 (2014)
37. Sarikaya, M.Z., Ogunmez, H.: On new inequalities via Riemann-Liouville fractional integra-
tion. Abstr. Appl. Anal. 2012 (2012), Article ID 428983, 10 p
38. Sarikaya, M.Z., Budak, H.: Some Hermite-Hadamard type integral inequalities for twice dif-
ferentiable mappings via fractional integrals. Facta Universitatis Ser: Math. Inform. 29(4),
371–384 (2014)
39. Set, E., Sarikaya, M.Z., Ozdemir, M.E., Yildirim, H.: The Hermite-Hadamard’s inequality for
some convex functions via fractional integrals and related results. JAMSI 10(2), 69–83 (2014)
40. Tseng, K.L., Hwang, S.R.: New Hermite-Hadamard inequalities and their applications. Filomat
30(14), 3667–3680 (2016)
41. Wang, D.Y., Tseng, K.L., Yang, G.S.: Some Hadamard’s inequalities for co-ordinated convex
functions in a rectangle from the plane. Taiwan. J. Math. 11, 63–73 (2007)
42. Wang, J.R., Li, X., Zhu, C.: Refinements of Hermite-Hadamard type inequalities involving
fractional integrals. Bull. Belg. Math. Soc. Simon Stevin 20, 655–666 (2013)
43. Wang, J.R., Zhu, C., Zhou, Y.: New generalized Hermite-Hadamard type inequalities and
applications to special means. J. Inequal. Appl. 2013, 325 (2013)
44. Xi, B.Y., Hua, J., Qi, F.: Hermite-Hadamard type inequalities for extended s-convex functions
on the co-ordinates in a rectangle. J. Appl. Anal. 20(1), 1–17 (2014)
45. Yaldiz, H., Sarıkaya, M.Z., Dahmani, Z.: On the Hermite-Hadamard-Fejer-type inequalities
for co-ordinated convex functions via fractional integrals. Int. J. Optim. Control: Theor. Appl.
(IJOCTA) 7(2), 205–215 (2017)
46. Yaldiz, H.: On Hermite-Hadamard type inequalities via Katugampola fractional integrals.
ResearchGate paper, Available online at: https://www.researchgate.net/publication/321947179
47. Yang, G.S., Tseng, K.L.: On certain integral inequalities related to Hermite-Hadamard inequal-
ities. J. Math. Anal. Appl. 239, 180–187 (1999)
48. Yang, G.S., Hong, M.C.: A note on Hadamard’s inequality. Tamkang J. Math. 28, 33–37 (1997)
49. Yıldırım, M.E., Akkurt, A., Yıldırım, H.: Hermite-Hadamard type inequalities for co-ordinated
(α1,m1)−(α2,m2)-convex functions via fractional integrals. Contemp. Anal. Appl. Math.
4(1), 48–63 (2016)
50. Zhang, Y., Wang, Y.: On some new Hermite-Hadamard inequalities involving Riemann-
Liouville fractional integrals. J. Inequal. Appl. 2013, Article ID 220 (2013)
51. Zhang, Z., Wei, W., Wang, J.: Generalization of Hermite-Hadamard inequalities involving
Hadamard fractional integrals. Filomat 29(7), 1515–1524 (2015)
52. Zhang, Z., Wang, J.R., Deng, J.H.: Applying GG-convex function to Hermite-Hadamard
inequalities involving Hadamard fractional integrals. Int. J. Math. Math. Sci. 2014, Article
ID 136035, 20 p
goyal.praveen2011@gmail.com
