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International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 6, December 2020, pp. 6521~6530
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i6.pp6521-6530 6521
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
On solving fuzzy delay differential equation
using bezier curves
Ali F. Jameel1, Sardar G. Amen2, Azizan Saaban3, Noraziah H. Man4
1,2,3,4 School of Quantitative Sciences, College of Art and Sciences, Universiti Utara Malaysia (UUM), Malaysia
2Department of Financial and Banking, Collage of Business Administration and Financial Science,
Al-Kitab University, Iraq
Article Info
ABSTRACT
Article history:
Received Apr 14, 2020
Revised May 27, 2020
Accepted Jun 12, 2020
In this article, we plan to use Bezier curves method to solve linear fuzzy
delay differential equations. A Bezier curves method is presented and
modified to solve fuzzy delay problems taking the advantages of the fuzzy
set theory properties. The approximate solution with different degrees is
compared to the exact solution to confirm that the linear fuzzy delay
differential equations process is accurate and efficient. Numerical example is
explained and analyzed involved first order linear fuzzy delay differential
equations to demonstrate these proper features of this proposed problem.
Keywords:
Bezier control points
Delay differential equations
Fuzzy differential equations
Fuzzy numbers
Fuzzy set theory
Residual function
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Ali F. Jameel,
School of Quantitative Sciences, College of Art and Sciences,
Universiti Utara Malaysia (UUM),
Sintok, 06010 Kedah, Malaysia.
Email: alifareed@uum.edu.my
1. INTRODUCTION
Fuzzy set theory is a powerful instrument for modeling uncertainty in a wide range of real issues and
for processing vague or subjective information in mathematical patterns. DDEs are a type of differential
equation in which the derivative of the unknown function at a certain time is given in terms of the values of
the function at a previous time. Often called DDEs time–delay systems with or with dead impact–time,
inherited process equations with deviating argument [1, 2].
The fundamental theory of steady works and key theory variables such as unique solutions are found
in [1-3]. Next, a large number of the Delay Differential Equation have been extensively investigated in
the novel, and monographs were published, including considerable on in [4], and so forth. The research
advantage of the differential delays is because many systems have been the prototype of better differential
delays in engineering, economics, science, etc. The difference equations of delays are delayed. Nevertheless,
they are not realistic to regulate problems. Most of these equations obviously cannot be precisely solved.
Efficient numerical methods must therefore be designed to approach their solutions. Ishiwata et al. used
the rational approximation method and the collocation method [5-7] to compute numerical solutions of DDEs
with proportional delays. Hu et al. [8] applied linear multi-step methods to compute numerical solutions for
neutral DDEs. Other method obtained approximate solutions for variety of DDEs such as Runge Kutta
methods, block methods and one- leg θ-methods in [8-12]. Moreover, the DDEs solved approximately via
some approximation methods in many fields of mathematics using approximation methods: for example,
the homotopy analysis method [13, 14], Adomian decomposition method [15] and homotopy perturbation
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Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530
6522
method [16]. Fuzzy DDE problem will model when the crisp model is not complete and its premasters or
conditions under fuzzy properties. FDDEs were solved by multiple researchers in recent years with an
approximate solution in [17, 18]. We will present in this article new plans for the approximate solution of
FDDEs by means of the curves of Bezier method in fuzzy domain and analyzed the fuzzy solutions in
different degree of approximations.
The outline of this paper will be as follows: FDDEs will be introduced in section 2. In section 3,
Problem of Fuzzy Delay System will be declared. Introduced proportional delay with FDDEs in section 4.
In sections 5 and 6 respectively degree elevation and Bezier curves and will be declared. Using Bezier
control points for Solving FDDE aforementioned method and suggested will be implemented on it in
section 7. In section 8 solved numerical problems, appeared the accuracy and adequacy of the method.
Lastly, the conclusion briefly will be given in section 9.
2. DESCRIPTION OF DELAY FUZZY DIFFERENTIAL EQUATIONS
Many DDEs are increasing, fundamentally optimistic in the models of epidemiology and population
dynamics. It is therefore worth noting that positive initial data lead to positive solutions [15]. Consider
the following FDDE:
(1)
where for all fuzzy level sets we have the following defuzzifications:
- The fuzzy functions [19] is denoted as ,
- The fuzzy delay functions is denoted as
- The fuzzy first order H-derivative, see [19]
,
Next, assume that the fuzzy function in (1) can be written as:
such that
By using Zadeh extension principles [20], we have the following membership function
{,
,
where
(2)
with a single delay . For each, suppose that and, are continuous on .
Let be continuous where be given. Require the solution of (1) satisfying
, (3)
and satisfying (1) on for some Note: should be explain as the right-hand
derivative at z. Now demonstrate a material system design problem that shows phenomenon of time delay.
The question picked in this section is exactly the right one in the test (1).
Int J Elec & Comp Eng ISSN: 2088-8708
On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel)
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3. PROBLEM OF TIME FUZZY DELAY SYSTEM
The existence of lags in economic systems is completely normal since a decision for the results of
article should be given a fixed period after. In one sample [21] of total economy and suppose
be
the proceeds which can divide into autonomous expenditure, consumption, and investment.
From section 2, we have:
(4)
where is a consumption fuzzy coefficient following the properties of triangular fuzzy number [17].
From (4),
(5)
Assuming that, following a decision to run
there is limited time between the production and
ordering of capital instruments. In expression of the paper of capital savings, we have
(6)
=
(7)
For each fuzzy level set in crisp domain the economic rationale suggests that
is
given by rate of saving proportionate to
and by the capital paper such that
(8)
where and is direction factor. Combining (6) and (7) to obtain the following
(9)
From (6) and (9), we get
(10)
By combining (8)-(10), we can yield
(11)
Express acceptance rate new appointment information. It is a delayed-type template operational FDDE.
4. FDDES WITH PROPORTIONAL DELAY
In this research, the Bezier control point method can finish off approximate analytical solutions with
a high level of reliability. Consider the following neutral functional FDEE with proportional delays [21, 22],
(12)
with the fuzzy initial conditions
, (13)
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Here, and k are given analytical fuzzy functions, and , , ,
and denote given fuzzy constants with , The presence and singularity of
the multi pantograph equation analytical solution is demonstrated [21], the solution Dirichlet sequence is
constructed and the asymptotic stability of the analytical solution is sufficiently defined.. It is proved that
the θ-methods with a variable step size are asymptotically stable if
. There are several examples
that show the properties of the θ-methods. In order to apply the Bezier control point method,
we rewrite (12) as
, .
A particular class of crisp DDE represents neutral functional DEEs with proportional delays. The mathematical
modeling of real-world phenomena takes such functioning DEEs on a significant role [12].
5. BEZIER CURVES IN FUZZY DOMAIN
From the definition of Bezier curve polynomial of degree [23] and according to sections 2-4,
we have the following fuzzy analysis
(14)
is control points of Bezier coefficient and are the polynomial of Bernstein on interval [a1, a2]
per each fuzzy level set , see Figure 1. In particular
(15)
(16)
where is a fuzzy parametric Bezier curve when it’s polynomial of vector valued. Figure 1 shows
the comprise line segments with control polygon of a Bezier curve If
polynomial of a scalar valued, the function is call then from [23, 24] an explicit Bezier curve
denoted by .
Figure 1. Degree 5 bezier curve with control polygon
6. SOLUTION OF FDDE USING BEZIER CONTROL POINTS
Consider the following boundary value problem
,, (17)
, , (18)
P0
P1
P2 P3 P4
P5
Int J Elec & Comp Eng ISSN: 2088-8708
On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel)
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where is differential operator with proportional delay, is also a polynomial in x, and
(k = 0, 1, ···, m) [24, 25]. We propose to represent the approximate solution of eq. (18) in fuzzy
Bezier form. The preference between Bezier and B-Spline is that the Bezier form is easier to carry out
multiplication, contrast and degree elevation operations symbolically than B-Spline. We choose the sum of
squares of the Bezier control points of the residual to be the measure quantity. Minimizing this quantity gives
the approximate solution. Therefore, the obvious spotlight is in the following, if the minimizing of
the quantity is zero, so the residual function is zero, which implies that the solution is the exact solution.
We call this approach the control point based method. By following [25] detailed steps of the method are
as follows:
Step 1. Choose a degree n and symbolically express the solution in the degree m () Bezier form
(19)
where the control points are to be de-termined.
Step 2. Substituting the approximate solution into the (19), we obtain the residual function
This is a polynomial in with degree ≤ h, where
So the residual function can be expressed in fuzzy Bezier form as well,
(20)
where for each fuzzy level set the control points are linear functions in the unknowns
. These functions are derived using the operations of multiplication, degree elevation and differentiation for
Bezier form.
Step 3. Construct the objective function
.
Then is also a fuzzy function of.
Step 4. Solve the constrained optimization problem:
,
,, (21)
by some optimization techniques, such as Lagrange multipliers method, we can be used to solve (21).
Step 5. Substituting the minimum solution back into (19) arrives at the approximate solution to
the differential equation.
7. NUMERICAL EXAMPLE
In this part, we used the mentioned control-point-based method on Bezier control points to solve
DDE’s and system of DDE’s. As a practical example, we consider Evens and Raslan [6] the following
pantograph delay equation in fuzzy form:
(22)
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The exact solution is given by
.)2();(,);( xx errxurerxu
According to [5] (21) can be written in defuzzfication form
(23)
For numerical implementation, we consider the approximate solution using Bezier curves of
degree 3 (m=3) and 8 (m=8) respectively as given in (15). In order to obtain the residual function, we also
approximate ex in Taylor polynomial of order 6. The detail results are as follows.
7.1. Degree-3 Bezier curve
Let,
3
1
3
3
1
3
)();(
)();(
ii
i
ii
i
xBarxu
xBarxu
, (24)
where
10 x
and
3...,,0,and iaa i
i
are the Bezier control points need to be determined. Substitute
into (23) and the residual functions can be obtained, i.e.
))(
))
2
()(;
2
exp(
2
1
))(();(
))(
))
2
()(;
2
exp(
2
1
))(();(
3
1
3
3
1
3
3
1
3
3
1
3
3
1
3
3
1
3
iii
iii
iii
ii
i
ii
i
ii
i
xBa
x
Bar
x
xBa
dx
d
rxR
xBa
x
Bar
x
xBa
dx
d
rxR
(25)
The right-hand side of (25) is a polynomial of degree 8 and therefore the residual function can be
represented in the form of (20) with as follows.
)();(
)();(
8
8
0
8
8
0
xBbrxR
xBbrxR
i
ii
i
ii
(26)
To obtain the Bezier control points in (24), we follow the step 3 to step 5 as stated in section 6.
The approximate solutions are available in Tables 1 and 2 and the comparsion of degree 3 bezier curve
solution with exact solution of equation (22) is illustrated in Figure 2 such that:
)]71821024.2
)1(4528665.5)1(0126515.4)1([);(
3
223
x
xxxxxrrxu
)])71821024.243642047.4(
)1)(4528665.5090573311.1(
)1)(0126516.4025303.8()1)(2();(
3
2
23
xr
xxr
xxrxrrxu
Int J Elec & Comp Eng ISSN: 2088-8708
On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel)
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Table 1. Appoximate and exact values for lower
solution,
);( rxu
(degree 3 bezier curve)
r
approx
exact
abs. error
0
0
0
0
0.2
0.5436420472
0.5436563657
1.4318499128x 10-5
0.4
1.0872840944
1.0873127314
2.8636998257x10-5
0.6
1.6309261416
1.6309690971
4.2955497386x 10-5
0.8
2.1745681888
2.1746254628
5.7273996514x 10-5
1.0
2.718210236
2.7182818285
7.1592495642x 10-5
Table 2. Approximate and exact values for upper
solution,
);( rxu
(degree 3 bezier curve)
r
approx
exact
abs. error
0
5.436420472
5.4365636569
1.4318499128 x 10-4
0.2
4.892778425
4.8929072912
1.2886649216x 10-4
0.4
4.349136378
4.3492509255
1.1454799303 x 10-4
0.6
3.805494330
3.8055945598
1.002294939 x 10-4
0.8
3.261852283
3.2619381942
8.591099477 x 10-5
1.0
2 .71821024
2.7182818285
7.159249564 x 10-5
Figure 2. Approximate and exact solution of at t = 1 (degree 3 bezier curves)
7.2. Degree-8 Bezier curve
Let,
8
0
3
8
0
8
)();(
)();(
ii
i
ii
i
xBarxu
xBarxu
, (27)
where
10 x
and
8...,,0,and iaa ii
are the Bezier control points need to be determined.
Substitute into (23) and the residual functions can be obtained, i.e.
))(
))
2
()(;
2
exp(
2
1
))(();(
))(
))
2
()(;
2
exp(
2
1
))(();(
8
0
8
8
0
8
8
0
8
8
0
8
8
0
8
8
0
3
iii
iii
iii
ii
i
ii
i
ii
i
xBa
x
Bar
x
xBa
dx
d
rxR
xBa
x
Bar
x
xBa
dx
d
rxR
(28)
The right-hand side of (25) is a polynomial of degree 13 and therefore the residual function can be
represented in the form of (20) with as follows.
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(r;1)
r
FDDE using Bezier at t = 1
approximate
exact
ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530
6528
)();(
)();(
13
13
0
13
13
0
xBbrxR
xBbrxR
i
ii
i
ii
(29)
To obtain the Bezier control points in (27), we also use the step 3 to step 5 as stated in section 6.
The approximate solutions are in Tables 3 and 4 and the comparsion of degree 8 bezier curve solution with
exact solution of equation (22) is illustrated n Figure 3 such that:
]7182791.2)1(0279716.19
)1(44305288.58
)1(84166368.102)1(3750016.113
)1(16666616.80)1(5000002.35
)1(9)1[();(
87
26
3544
5362
78
ttt
tt
tttt
tttt
tttrrxu
8
7
26
35
44
53
62
78
)7182791.24365582.5(
)1()0279716.190559432.38(
)1()4430588.588861176.116(
)1()84166368.10268332736.205(
)1()3750016.1137500039.226(
)1()16666616.8033333232.160(
)1()5000002.5.300000012.71(
)1()98.1()1)(2();(
tr
ttr
ttr
ttr
ttr
ttr
ttr
ttrtrrxu
Table 3. Approximate and exact values for lower
solution,
);( rxu
(degree 8 bezier curve)
r
approx
exact
abs. error
0
0
0
0
0.2
0.543655820
0.5436563657
5.45269859x10-7
0.4
1.087311648
1.0873127314
1.090539719x 10-6
0.6
1.630967461
1.6309690971
1.635809578x 10-6
0.8
2.174623282
2.1746254628
2.181079437x 10-6
1.0
2.718279102
2.7182818285
2.726349297x 10-6
Table 4. Approximate and exact values for upper
solution,
);( rxu
(degree 8 bezier curve)
r
approx
exact
abs. error
0
5.4365582042
5.4365636569
5.452698593 x10-6
0.2
4.8929023838
4.8929072912
4.907428734 x10-6
0.4
4.3492465634
4.3492509255
4.362158874 x10-6
0.6
3.8055907430
3.8055945598
3.816889015 x10-6
0.8
3.2619349225
3.2619381941
3.271619156 x10-6
1.0
2.7182791021
2.7182818284
2.726349297 x10-6
Figure 3. Approximate and exact solution of u(t) at t = 1 (degree 8 bezier curves)
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(r;1)
r
FDDE using Bezier at t = 1
approximate
exact
Int J Elec & Comp Eng ISSN: 2088-8708
On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel)
6529
8. CONCLUSION
This work has successfully implemented and applied Bezier control points to overcome linear and
fuzzy DDEs. A general method framework has been successfully developed and evaluated using fuzzy sets
properties to obtain rough solutions for fuzzy DDEs. Details have been provided regarding the BCP
convergence mechanism related to the approximate first-order fuzzy DDEs solution. Studies of first-order
linear fizzy DDEs by BCP have shown that the system is capable and reliable studies are obtained that match
the properties of the solution of the fizzy differential equation in the form of the triangle fuzzy numbers with
varying degrees of precision.
ACKNOWLEDGEMENTS
The authors are very grateful to the Ministry of Higher Education of Malaysia for providing us with
the Fundamental Research Grant Scheme (FRGS) S/O number 14188 to enable us to pursue this research.
REFERENCES
[1] G. Adomian and R. Rach, “Nonlinear Stochastic Differential Delay Equation,” Journal of Mathematical Analysis and
Applications, vol. 91, no. 1, pp. 94-101, 1983.
[2] F. M. Asl and A. G. Ulsoy, “Analysis of a System of Linear Delay Differential Equations,” Journal of Dynamic
Systems, Measurement and Control, vol. 125, no. 2, pp. 215-223, 2003.
[3] K. A. G. Azevedo, “Existence and Uniqueness of Solution for Abstract Differential Equations with State-Dependent
Time Impulses,” Mediterranean Journal of Mathematics, vol. 16, no. 42. pp. 1-10, 2019.
[4] W. G. Aiello, et al., “Analysis of a model representing stage-structured population growth with state-dependent time
delay,” SIAM Journal on Applied Mathematics, vol. 52, no. 3, pp. 855-869, 1992.
[5] E. Ishiwata, “On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with
Proportional Delays,” Computing, vol. 64, pp. 207-222, 2000.
[6] E. Ishiwata and Y. Muroya, “Rational Approximation Method for Delay Differential Equations with Proportional
Delay,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 741-747, 2007.
[7] E. Ishiwata, et al., “A Super-Attainable Order in Collocation Methods for Differential Equations with Proportional
Delay,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 227-236, 2008.
[8] P. Hu, et al., “Asymptotic Stability of Linear Multistep Methods for Nonlinear Neutral Delay Differential Equations,”
Applied Mathematics and Computation, vol. 211, no. 1, pp. 95-101, 2009.
[9] Z. Fan, et al., “Existence and Uniqueness of the Solutions and Convergence of Semi-Implicit Euler Methods for Stochastic
Pantograph Equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1142-1159, 2007.
[10] W. Wang, et al., “Stability of One-Leg θ- Methods for Nonlinear Neutral Differential Equations with Proportional
Delay,” Applied Mathematics and Computation, vol. 213, no. 1, pp. 177-183, 2009.
[11] W. Wang, et al., “Stability of Continuous Runge-Kutta-Type Methods for Nonlinear Neutral Delay- Differential
Equations,” Applied Mathematical Modelling, vol. 33, no. 8, pp. 3319-3329, 2009.
[12] S. Z. Ahmad, et al., “Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods,” Journal of
Mathematics, vol. 2018, pp. 285-301, 2018.
[13] A. K. Alomari, et al., “Solution of Delay Differential Equation by Means of Homotopy Analysis Method,” Acta
Applicandae Mathematicae, vol. 108, no. 2, pp. 395-412, 2009.
[14] S. Liao, “Series Solutions of Unsteady Boundary-Layer Flows Over a Stretching Flat Plate,” Studies in Applied
Mathematics, vol. 117, no. 3, pp. 239-263, 2006.
[15] D. J. Evans and K. R. Raslan, “The Adomian Decomposition Method for Solving Delay Differential Equation,”
International Journal of Computer Mathematics, vol. 82, no. 1, pp. 49-54, 2005.
[16] F. Shakeri and M. Dehghan, “Solution of Delay Diffrential Equation via a Homotopy Perturbation Method,”
Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 486-498, 2008.
[17] A. F. Jameel, et al., “A New Approximate Solution of the Fuzzy Delay Differential Equations,” International Journal
of Mathematical Modelling and Numerical Optimisation, vol. 9, no. 3, pp. 221-240, 2019.
[18] S. Narayanamoorthy and T. L. Yookesh, “Approximate method for solving the linear fuzzy delay differential
equations,” Discrete Dynamics in Nature and Society, vol. 2015, pp. 1-9, 2015.
[19] X. Guo, et al., “Fuzzy approximate solutions of second-order fuzzy linear boundary value problems,” Journal of
Boundary Value Problems, vol. 2013, pp. 1-17, 2013.
[20] L. A. Zadeh, “Toward a generalized theory of uncertainty,” Information Sciences, vol. 172, no. 1-2, pp. 1-40, 2005.
[21] J. Biazar and B. Ghanbari, “The homotopy perturbation method for solving neutral functional-differential equations
with proportional delays,” Journal of King Saud University—Science, vol. 24, no. 1, pp. 33-37, 2012.
[22] X. Chen and L. Wang, “The Variational Iteration Method for Solving a Neutral Functional Differential Equation with
Proportional Delays,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2696-2702, 2010.
[23] T. Kushimoto and M. Hosaka, “Description of Intersection Curves by Differential Equation and their Tracing,”
Journal of the Japan Society for Precision Engineering, vol. 57, no. 8, pp. 1375-1380, 1991.
[24] Wubshet I. and Getu K., “Magnetohydrodynamic flow of a nanofluid due to a nonlinearly curved stretching surface
with high order slip flow,” Heat Transfer-Asian Research, vol. 48, no. 6, pp. 1-25, 2019.
[25] Z. C. Li and H. T. Huang, “Blending curves for landing problems by numerical differential equations II,” Numerical
methods, Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 165-187, 2000.
