Zeyad Published paper 2014- AMC-Elsevier

Full text

The general (vector) solutions of such linear (coupled) matrix
fractional differential equations by using Kronecker structures
Zeyad Abdel Aziz Al-Zhour
Department of Basic Sciences and Humanities, College of Engineering, University of Dammam, P.O. Box 1982, Dammam 34151, Saudi Arabia
article info
Keywords:
Kronecker products
Vector-operator
Mittag-Leffler matrix
Matrix fractional differential systems
abstract
In this paper, we demonstrate how Kronecker products and vector operator can be com-
bined beautifully to solve and fit the non-homogenous linear matrix fractional differential
equations and such coupled linear matrix fractional differential equations. The way exists
which transform the given (coupled) matrix fractional differential equations by using
Kronecker structures into forms for which solutions may be readily computed. Some
important and interesting special cases of the general system of non-homogeneous linear
matrix fractional differential equations of fractional order are also considered which
includes the non-homogenous linear fractional dynamical system with delays in control.
Finally, a brief comparison between the fractional and integer order (vector) solutions
and some examples are given to illustrate our new approaches.
Ó2014 Elsevier Inc. All rights reserved.
1. Introduction
A matrix is simply a rectangular array of numbers or other mathematical object, but despite their simplicity matrices and
their generalizations are important tools in mathematics. One of the principal reasons is that matrices arise naturally in solv-
ing the linear system Ax ¼band linear differential system x
=
ðtÞ¼AxðtÞ. Another is because that any matrix of order mn
can be valued as representing a linear map from an n-dimensional vector space to an m-dimensional vector space, and con-
versely, all such linear maps can be represented as mnmatrix.
Linear and non-linear matrix and matrix differential equations are very important in various fields which including
engineering, mathematics, physics, statistics, optimization, economic, linear system and linear differential system problems
[1–19]. The simplest non-homogeneous linear matrix differential equation is given by:
X
=
ðtÞ¼AXðtÞþUðtÞ;Xð0Þ¼C;ð1-1Þ
where X
=
ðtÞstands to the first derivative of matrix function XðtÞ. In fact the general solution of (1-1) is given by:
XðtÞ¼e
At
CþZ
t
0
e
AðtsÞ
UðsÞds:ð1-2Þ
Most of the existing results, however, are connected with particular systems of such matrix differential equations. In
addition, many interesting problems lead to the following general class of non-homogeneous linear matrix differential
equations:
http://dx.doi.org/10.1016/j.amc.2014.01.079
0096-3003/Ó2014 Elsevier Inc. All rights reserved.
E-mail addresses: zalzhour@ud.edu.sa,zeyad1968@yahoo.com
Applied Mathematics and Computation 232 (2014) 498–510
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

X
=
ðtÞ¼X
r
i¼1
A
i
XðtÞB
i
þUðtÞ;Xðt
0
Þ¼C;ð1-3Þ
where A
i
2M
m
,B
i
2M
n
(i¼1;2;...;r) and C2M
m;n
are given scalar matrices; UðtÞ2M
m;n
is a given matrix function and
XðtÞ2M
m;n
is the unknown matrix to be solved. (where M
m;n
is the set of all mnmatrices over the complex number C
or real number R and when m¼nwe write M
m
instead of M
m;m
).
Coupled matrix and matrix differential equations have also been widely used in stability theory of differential equations,
control theory, communication systems, perturbation analysis of linear and non-linear matrix equations and other fields of
pure and applied mathematics and also recently in the context of the analysis and numerical simulation of descriptor sys-
tems. For instance, the canonical system:
X
=
ðtÞ¼AXðtÞþBY ðtÞ;Y
=
ðtÞ¼CXðtÞA
T
YðtÞ;ð1-4Þ
has been used to the solution of optimal control problem with the performance index [11]. In addition, many interesting
problems lead to the following general system of non-homogeneous linear matrix differential equations:
X
=
1
ðtÞ¼A
11
X
1
ðtÞB
11
þA
12
X
2
ðtÞB
12
þþA
1p
X
p
ðtÞB
1p
þU
1
ðtÞ
X
=
2
ðtÞ¼A
21
X
1
ðtÞB
21
þA
22
X
2
ðtÞB
22
þþA
2p
X
p
ðtÞB
2p
þU
2
ðtÞ



X
=
p
ðtÞ¼A
p1
X
1
ðtÞB
p1
þA
p2
X
2
ðtÞB
p2
þþA
pp
X
p
ðtÞB
pp
þU
p
ðtÞ
;ð1-5Þ
where X
i
ðt
0
Þ¼C
i
such that A
ij
,B
ij
,C
i
2M
m
are given scalar matrices, U
i
ðtÞ2M
m
is a given matrix function and X
i
ðtÞ2M
m
is
the unknown matrix function to be solved (i;j¼1;2;...;p). Examples of such situation are coupled matrix differential equa-
tions [1,2,7–10,12,17,19]. Depending on the problem considered, different coupling terms may appear and difficult to solve.
Due to the introduction of fractional calculus which is a generalization of ordinary (integer order) differentiation and inte-
gration to its fractional (non-integer) order counterpart, almost every problem in calculus can be revisited at a whole new
level, where one does not necessarily restrict oneself to an integer order derivative or integral, which allows much more flex-
ibility in solving real-life problems. The topic of fractional calculus has attracted many scientists because of its several appli-
cations in applied sciences, engineering, economics, and many other fields. For a detail survey with collections of
applications in various fields, see for example [20–29]. Furthermore, the fractional derivatives technique has been employed
for solving linear and non-linear fractional and ordinary differential equations [30–36,37]. The solution of differential equa-
tions of fractional order is much involved and some analytical methods are presented such as the popular Laplace transform
method [21,38]; the Fourier transform method [20]; the variational iteration method [39] and Green function method
[40,41]. Numerical schemes for solving fractional differential equations are studied from many researchers [42–44]. Re-
cently, a great deal of effort has been expended over the last years in attempting to find robust and stable numerical as well
as analytical methods for solving fractional differential equations of physical interest. The Adomian decomposition method
[45]; homotopy perturbation method [46]; homotopy analysis method [47]; differential transform method [48], and varia-
tional iteration method [39] are relatively new approaches to provide an analytical approximate solution to linear and non-
linear fractional differential equations. The existence and uniqueness of solutions of initial value problems for fractional
order differential equations have been studied in the literature [49–53].
Numerous problems in physics, chemistry, and engineering are modeled mathematically by systems of fractional differ-
ential equations. Studying such systems has been carried out by various researchers. For example, the existence and unique-
ness of solutions of linear fractional order systems have been discussed in [54,55]. The stability results for linear fractional
order systems have been investigated in [56–58]. The analytical solutions for fractional order systems have been derived by
Lakshmikantham and Vatsala [53] and Bonilla et al. [59]. The numerical solutions for some classes of fractional order systems
have been constructed by Momani and Odibat [60] and Abdulaziz and Noor [61]. The limit cycle is generated in fractional
order systems have been shown by Wang and Li [62]. Finally, some important fractional order systems have been studied
by Grigorenko and Grigorenko [63] and Lu and Chen [64].
Nowadays, however, it was found that many systems in interdisciplinary fields, such as Lorenz systems [63], dielectric
polarization [65] and electromagnetic wave [66] can be described by fractional differential equations. These research efforts
have shown that fractional derivatives provide an excellent tool for describing the memory and hereditary properties of var-
ious materials and processes. It has been shown that some fractional-order dynamical systems, as generalizations of many
well-known integer-order systems, can also behave chaotically, for example, the fractional-order Duffing system [67]; frac-
tional-order Chua system [68]; fractional-order Lorenz system [63]; fractional-order Rossler system [69]; fractional-order
Chen system [70]; fractional order Lü system [71], and the fractional-order unified system [72].
Recently, due to its potential applications in secure communication and control processing, synchronization of fractional-
order chaotic systems has received a great deal of attention by Lu and Chen [64]; Odibat [73] presented the analytic solution
of linear system of fractional differential equations with constant coefficients; Bonilla et al. [59] introduced a new direct
method for solving the homogenous and non-homogenous system of linear fractional differential equations with constant
Z.A.A. Al-Zhour / Applied Mathematics and Computation 232 (2014) 498–510 499

coefficients; Wang and Zhang [74] studied the synchronized motions in a star network of coupled fractional-order systems in
which the major element is coupled to each of the non-interacting individual elements; Wang et al. [75] investigated the
existence and uniqueness of the solution of the coupled system of nonlinear fractional differential equations with m-point
boundary conditions on an unbounded domain and Kilicman and Al-Zhour [37] studied several operational matrices for frac-
tional integration and differentiation and expanded the Kronecker convolution product to the Riemann–Liouville fractional
integral of matrices.
More recently, Rida and Arafa [76] solved linear system of fractional differential equations with constant coefficients by
using generalized Mittag-Leffler function method in the Caputo sense; Balachandran and Kokila [77] introduced the control-
lability of linear and nonlinear fractional dynamical systems in finite dimensional spaces and Balachandran et al. [78] intro-
duced the relative controllability of fractional dynamical systems with multiple delays in control.
To our best knowledge, the general solutions of the following general system of non-homogeneous linear matrix frac-
tional differential equations of fractional order
a
have been not fully investigated:
X
a
1
ðtÞ¼A
11
X
1
ðtÞB
11
þA
12
X
2
ðtÞB
12
þþA
1p
X
p
ðtÞB
1p
þU
1
ðtÞ
X
a
2
ðtÞ¼A
21
X
1
ðtÞB
21
þA
22
X
2
ðtÞB
22
þþA
2p
X
p
ðtÞB
2p
þU
2
ðtÞ



X
a
p
ðtÞ¼A
p1
X
1
ðtÞB
p1
þA
p2
X
2
ðtÞB
p2
þþA
pp
X
p
ðtÞB
pp
þU
p
ðtÞ
:ð1-6Þ
In this paper, we present the general exact solutions of non-homogenous linear matrix fractional differential equations
and such coupled linear matrix fractional differential equations based on the Kronecker structure method. The way exists
which transform the given (coupled) matrix fractional differential equations into forms for which solutions may be readily
computed. Some important and interesting special cases of the general system in (1-6) of order 0 <
a
<1 are also considered
which includes the non-homogenous linear fractional dynamical system with delays in control. Finally, a brief comparison
between the fractional and integer order (vector) solutions and some examples are given to illustrate our new approaches.
2. Preliminaries and definitions
In this section, we recall the main definitions and some important properties of the Kronecker products, fractional deriv-
ative operators and the Mittag-Leffler matrix. Furthermore, some nice new properties for Mittag-Leffler matrix are also
established that will be useful in our investigation of the solution of matrix fractional differential equations.
2.1. Kronecker prouducts of matrices
Matrices can be multiplied in different ways. The Kronecker products used in many fields are almost as important as the
usual product. One of the principle reasons is that Kronecker products affirming their capability of solving a wide range of
problems and playing important tools in many fields such as control theory, system theory, statistics, physics, communica-
tion systems, optimization, economics, and engineering. These include signal processing, image processing, semi definite
programming, matrix equations, matrix differential equations and many other applications [1–3,7,8,15,16,19,79–81]. The
following three matrix operations are studied by many researchers [7,16,81–84] and defined as follow:
(i) Kronecker product
AB¼ða
ij
BÞ
ij
2M
mp;nq
;ð2-1Þ
where A¼ða
ij
Þ2M
m;n
and B¼ðb
kl
Þ2M
p;q
.
(ii) Kronecker sum
AB¼ðAI
n
ÞþðI
m
BÞ2M
mn
;ð2-2Þ
where A¼ða
ij
Þ2M
m
and B¼ðb
kl
Þ2M
n
.
(iii) Vector operator
Vec A ¼ða
11
a
21
...a
m1
a
12
a
22
...a
m2
...a
1n
a
2n
...a
mn
Þ
T
2M
mn;1
;ð2-3Þ
where A¼ða
ij
Þ2M
m;n
.
The Kronecker product and vector-operator are related by [1,5,7,8].
VecðAXBÞ¼ðB
T
AÞVecX;ð2-4Þ
where A2M
m;n
,B2M
p;q
and X2M
n;p
.
For any compatibly matrices A,B,C,D, and X, we shall make frequent use the following properties of the above three
matrix operations that will be useful in our investigation of the solution of matrix fractional differential equations
[1–3,7,16,19,79–84].
500 Z.A.A. Al-Zhour/ Applied Mathematics and Computation 232 (2014) 498–510

ðiÞðABÞ
T
¼ðA
T
B
T
Þ;ð2-5Þ
ðiiÞðABÞðCDÞ¼AC BD;ð2-6Þ
ðiiiÞI
m
I
n
¼I
n
I
m
¼I
mn
;ð2-7Þ
where I
m
is an identify matrix of order mm.
ðivÞfðAI
n
Þ¼fðAÞI
n
;fðI
n
AÞ¼I
n
fðAÞ;ð2-8Þ
where fis analytic function on the region containing the eigenvalues of A2M
m
such that fðAÞexist.
2.2. Fractional differential operators
There are several definitions of a fractional derivative of order
a
>0[38,45,77,78,85]. The two most commonly used are:
(i) Riemann–Liouville fractional operators
I
a
fðtÞ¼ 1
C
ð
a
ÞZ
t
0
ðt
s
Þ
a
1
fð
s
Þd
s
;ð2-9Þ
D
a
fðtÞ¼D
a
I
n
a
fðtÞ;ð2-10Þ
where
a
>0, n1<
a
6n(n2N), Dis the differential operator, Iis the integral operator and fðtÞis a suitable function for
t>0.
Some properties of the operators I
a
and D
a
can be found [29,38,86–88], we recall only the following two properties for
a
;bP0 and
m
>1:
ðaÞI
a
I
b
fðtÞ¼I
a
þb
fðtÞ;ð2-11Þ
ðbÞI
a
t
m
¼
C
ð
m
þ1Þ
C
ð
a
þ
m
þ1Þt
a
þ
v
:ð2-12Þ
(ii) Caputo fractional operators
D
a
fðtÞ¼I
n
a
D
n
fðtÞ¼ 1
C
ðn
a
ÞZ
t
0
f
ðnÞ
ð
s
Þ
ðt
s
Þ
a
nþ1
d
s
;ð2-13Þ
where
a
>0, t>0 and n1<
a
6n(n2N).
The fractional derivative of fðtÞin the Caputo sense is defined for 0 <
a
<1as
D
a
fðtÞ¼ 1
C
ð1
a
ÞZ
t
0
f
=
ð
s
Þ
ðt
s
Þ
a
d
s
:ð2-14Þ
Note that the Caputo’s definition has the advantage of dealing property with initial value problems in which the initial
conditions are given in terms of the field variables and their integer order which is the case most physical processes. Fortu-
nately, the Laplace transform of the Caputo fractional derivative is
£fD
a
fðtÞg ¼ s
a
FðsÞX
n1
k¼0
f
ðkÞ
ð0
þ
Þs
a
k1
;ð2-15Þ
where n1<
a
6n(n2N), F(s) = £ ffðtÞg and sis the Laplace variable. The Laplace transform of the Caputo fractional deriv-
ative requires the knowledge of the (bounded) initial values of the function on its integer derivatives of order
k¼1;2;...;n1.
2.3. Mittag-Leffler matrix
The exponential function e
t
plays a very important role in the theory of integer-order differential equations. It is one
parameter generalization, the function which is defined by
E
a
ðzÞ¼X
1
k¼0
z
k
C
ðk
a
þ1Þ;
a
>0;ð2-16Þ
and was introduced by Mittag-Leffler and studied also by many researchers [29,73,77,78]. The matrix extension of the men-
tioned Mittag-Liffler function for A2M
m
is defined as in the following representation:
E
a
ðAt
a
Þ¼X
1
k¼0
A
k
t
a
k
C
ðk
a
þ1Þ;
a
>0:ð2-17Þ
Z.A.A. Al-Zhour / Applied Mathematics and Computation 232 (2014) 498–510 501

Note that for
a
¼1, we obtain the exponential matrix function e
At
.
Here the Mittag-Liffler matrix of A2M
m
is defined for
a
>0 as follows:
E
a
ðAÞ¼X
1
k¼0
A
k
C
ðk
a
þ1Þ¼I
m
þA
C
ð
a
þ1ÞþA
2
C
ð2
a
þ1Þþ:ð2-18Þ
Since we make use the spectral decomposition of E
a
ðAÞand E
a
ðAt
a
Þ, then we get the following representations:
E
a
ðAÞ¼X
m
k¼0
x
k
y
T
k
E
a
ðk
k
Þ;E
a
ðAt
a
Þ¼X
m
k¼0
x
k
y
T
k
E
a
ðk
k
t
a
Þ;ð2-19Þ
where fx
1
;x
2
;...;x
m
gand fy
1
;y
2
;...;y
m
gare the eigenvectors corresponding to the eigenvalues fk
1
;k
2
;...;k
m
gof Aand A
T
,
respectively. Now if A,B2M
m
and
a
>0. Then it is easy to prove the following nice properties of Mittag-Leffler matrices
E
a
ðAÞand E
a
ðAt
a
Þ:
ðiÞE
a
ðAÞis always invertible and positive definite matrix:
ðiiÞðE
a
ðAÞÞ
1
¼E
a
ðAÞand ðE
a
ðAÞÞ
T
¼E
a
ðA
T
Þ:ð2-20Þ
ðiiiÞIf A¼diagða
11
;a
22
;...;a
mm
Þis a diagonal matrix;then E
a
ðAÞ¼diagðE
a
ða
11
Þ;E
a
ða
22
Þ;...;E
a
ða
mm
ÞÞ:ð2-21Þ
ðivÞE
a
ðAþBÞ¼E
a
ðAÞE
a
ðBÞif and only if AB ¼BA:ð2-22Þ
ðvÞE
a
ðABÞ¼E
a
ðAÞE
a
ðBÞ:ð2-23Þ
ðviÞD
a
E
a
ðAt
a
Þ¼X
1
k¼1
A
k
t
ðk1Þ
a
C
ððk1Þ
a
þ1Þ;
as in Caputo sense:
ðviiÞE
a
ðAI
n
Þ¼E
a
ðAÞI
n
and E
a
ðI
n
AÞ¼I
n
E
a
ðAÞ:ð2-25Þ
Some important special cases include (2-25) for A2M
m
:
ðiÞe
AI
n
¼e
A
I
n
and e
I
n
A
¼I
n
e
A
:ð2-26Þ
ðiiÞsinhðAI
n
Þ¼sinhðAÞI
n
and sinhðI
n
AÞ¼I
n
sinhðAÞ:ð2-27Þ
ðiiiÞcoshðAI
n
Þ¼coshðAÞI
n
and coshðI
n
AÞ¼I
n
coshðAÞ:ð2-28Þ
3. Linear (coupled) matrix fractional differential equations
In this Section, we present our main results for finding the general exact solutions of some important non-homogenous
linear matrix fractional differential equations and such coupled linear matrix fractional differential equations based on the
Kronecker structure method. The solutions procedure presented here are based on the idea of Kronecker products and vector
operator.
Firstly, we use our knowledge of the solution of the of simplest non-homogenous linear fractional dynamical system with
delays in control of order 0 <
a
<1:
x
a
ðtÞ¼AxðtÞþuðtÞ;xð0Þ¼x
0
;ð3-1Þ
where A2M
n
is a given constant matrix, x
0
and uðtÞ2M
n;1
are given vectors, and xðtÞ2M
n;1
is unknown vector to be solved.
In fact, the unique solution of the system in (3-1) can be expressed in the following form [78]:
xðtÞ¼E
a
ðAt
a
Þx
0
þZ
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞuðsÞds:ð3-2Þ
In particular, if uðtÞis a zero vector, then the unique solution of the homogenous linear fractional
x
a
ðtÞ¼AxðtÞ;xð0Þ¼x
0
;ð3-3Þ
is given by [77,78]
xðtÞ¼E
a
ðAt
a
Þx
0
:ð3-4Þ
502 Z.A.A. Al-Zhour/ Applied Mathematics and Computation 232 (2014) 498–510

Lemma 1. Let A 2M
n
and C 2M
n;m
be given scalar matrices, UðtÞ2M
n;m
be a given matrix function and XðtÞ2M
n;m
be the
unknown matrix function to be solved. Then general solution of the following non-homogenous linear matrix fractional dynamical
differential equation with delays in control of order 0<
a
<1:
X
a
ðtÞ¼AXðtÞþUðtÞ;Xðt
0
Þ¼C;ð3-5Þ
is given by
XðtÞ¼E
a
ðAðtt
0
Þ
a
ÞCþZ
t
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞds:ð3-6Þ
In particular, if UðtÞ¼0, then the general solution of the homogenous case:
X
a
ðtÞ¼AXðtÞ;Xðt
0
Þ¼C;ð3-7Þ
is given by:
XðtÞ¼E
a
ðAðtt
0
Þ
a
ÞC:ð3-8Þ
Proof. If we use the Vecð:Þ– notation of (3-5), we have the following system:
VecX
a
ðtÞ¼ðI
m
AÞVecXðtÞþVecUðtÞ:ð3-9Þ
This equation can be rewrite as follows:
x
a
ðtÞ¼HxðtÞþuðtÞ:xðt
0
Þ¼c;ð3-10Þ
where x
a
ðtÞ¼VecX
a
ðtÞ,H¼I
m
A,xðtÞ¼VecXðtÞ,uðtÞ¼VecUðtÞand c¼VecC.
Due to (3-2) and using properties of Kronecker products, we have:
VecXðtÞ¼xðtÞ¼E
a
ðHðtt
0
Þ
a
ÞcþZ
t
t
0
ðtsÞ
a
1
E
a
ðHðtsÞ
a
ÞuðsÞds
¼E
a
ððI
m
AÞðtt
0
Þ
a
ÞcþZ
t
t
0
ðtsÞ
a
1
E
a
ððI
m
AÞðtsÞ
a
ÞuðsÞds
¼½I
m
E
a
ðAðtt
0
Þ
a
ÞcþZ
t
t
0
ðtsÞ
a
1
½I
m
E
a
ðAðtsÞ
a
ÞuðsÞds
¼½I
m
E
a
ðAðtt
0
Þ
a
ÞVecC þZ
t
t
0
ðtsÞ
a
1
½I
m
E
a
ðAðtsÞ
a
ÞVecUðsÞds
¼Vec½E
a
ðAðtt
0
Þ
a
ÞCI
m
þZ
t
t
0
ðtsÞ
a
1
Vec½E
a
ðAðtsÞ
a
ÞUðsÞI
m
ds
¼Vec½E
a
ðAðtt
0
Þ
a
ÞCþVec½Z
t
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞds
¼Vec½E
a
ðAðtt
0
Þ
a
ÞCþZ
t
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞds:
This completes the proof of Lemma 1.h
If we set
a
¼1inLemma 1, we get the following corollary.
Corollary 1. Let A 2M
n
and C 2M
n;m
be given scalar matrices, UðtÞ2M
n;m
be a given matrix function and XðtÞ2M
n;m
be the
unknown matrix function to be solved. Then general solution of the following non-homogenous linear matrix dynamical ordinary
differential equations with delays in control:
X
=
ðtÞ¼AXðtÞþUðtÞ;Xðt
0
Þ¼C;ð3-11Þ
is given by
XðtÞ¼e
Aðtt
0
Þ
CþZ
t
t
0
e
AðtsÞ
UðsÞds:ð3-12Þ
Theorem 1. Let A 2M
n
,B2M
m
, and C 2M
n;m
be given scalar matrices, UðtÞ2M
n;m
be a given matrix function and XðtÞ2M
n;m
be
the unknown matrix function to be solved. Then general solution of the following non-homogenous linear matrix fractional differ-
ential equation of order 0<
a
<1:
X
a
ðtÞ¼AXðtÞþXðtÞBþUðtÞ;Xðt
0
Þ¼C;ð3-13Þ
Z.A.A. Al-Zhour / Applied Mathematics and Computation 232 (2014) 498–510 503

is given by
XðtÞ¼E
a
ðAðtt
0
Þ
a
ÞCE
a
ðBðtt
0
Þ
a
ÞþZ
t
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞE
a
ðBðtsÞ
a
Þds:ð3-14Þ
In particular, if UðtÞ¼0, then the general solution of the homogenous case:
X
a
ðtÞ¼AXðtÞþXðtÞB;Xðt
0
Þ¼C;ð3-15Þ
is given by:
XðtÞ¼E
a
ðAðtt
0
Þ
a
ÞCE
a
ðBðtt
0
Þ
a
Þ:ð3-16Þ
Proof. If we use the Vecð:Þ- notation of (3-13), then we have
VecðX
a
ðtÞÞ ¼ ðI
m
AþB
T
I
n
ÞVecXðtÞþVecUðtÞ¼ðB
T
AÞVecXðtÞþVecUðtÞ:
This equation can be rewrite as follows:
x
a
ðtÞ¼HxðtÞþuðtÞ:xðt
0
Þ¼c;
where x
a
ðtÞ¼VecX
a
ðtÞ,H¼B
T
A,xðtÞ¼VecXðtÞ,uðtÞ¼VecUðtÞ, and c¼VecC.
Due to (3-2) and using properties of Kronecker products, we have:
VecXðtÞ¼xðtÞ¼E
a
ðHðtt
0
Þ
a
ÞcþZ
t
t
0
ðtsÞ
a
1
E
a
ðHðtsÞ
a
ÞuðsÞds
¼E
a
ððB
T
AÞðtt
0
Þ
a
ÞcþZ
t
t
0
ðtsÞ
a
1
E
a
ððB
T
AÞðtsÞ
a
ÞuðsÞds
¼ðE
a
ðB
T
ðtt
0
Þ
a
ÞE
a
ðAðtt
0
Þ
a
ÞÞcþZ
t
t
0
ðtsÞ
a
1
ðE
a
ðB
T
ðtsÞ
a
ÞE
a
ðAðtsÞ
a
ÞÞuðsÞds
¼ðE
a
ðB
T
ðtt
0
Þ
a
ÞE
a
ðAðtt
0
Þ
a
ÞÞVecC þZ
t
t
0
ðtsÞ
a
1
ðE
a
ðB
T
ðtsÞ
a
ÞE
a
ðAðtsÞ
a
ÞÞVecUðsÞds
¼VecðE
a
ðAðtt
0
Þ
a
ÞCE
a
ðBðtt
0
Þ
a
ÞÞ þ Z
t
t
0
ðtsÞ
a
1
VecðE
a
ðAðtsÞ
a
ÞUðsÞE
a
ðBðtsÞ
a
ÞÞds
¼VecðE
a
ðAðtt
0
Þ
a
ÞCE
a
ðBðtt
0
Þ
a
ÞþZ
t
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞE
a
ðBðtsÞ
a
ÞdsÞ:
This completes the proof of Theorem 1.h
If we set
a
¼1inTheorem 1, we get the following corollary.
Corollary 2. Let A 2M
n
,B2M
m
and C 2M
n;m
be given scalar matrices, UðtÞ2M
n;m
be a given matrix function and XðtÞ2M
n;m
be
the unknown matrix function to be solved. Then general solution of the following non-homogenous linear matrix ordinary differ-
ential equation:
X
=
ðtÞ¼AXðtÞþXðtÞBþUðtÞ;Xðt
0
Þ¼C;ð3-17Þ
is given by
XðtÞ¼e
Aðtt
0
Þ
Ce
Bðtt
0
Þ
þZ
t
t
0
e
AðtsÞ
UðsÞe
BðtsÞ
ds:ð3-18Þ
Theorem 2. Let A;B;C;D;E and F 2M
n
be scalar matrices such that AC ¼CA,BD ¼DB,and let XðtÞand YðtÞ2M
n
be the
unknown matrices to be solved. Then the general vector solutions of the following coupled linear matrix fractional differential equa-
tions of order 0<
a
<1:
X
a
ðtÞ¼AXðtÞBþCY ðtÞD;Y
a
ðtÞ¼CXðtÞDþAY ðtÞB:Xðt
0
Þ¼E;Yðt
0
Þ¼F;ð3-19Þ
is given by
VecXðtÞ¼E
a
ððB
T
AÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
ÞþE
a
ððD
T
CÞðtt
0
Þ
a
Þ
2
()
VecE
þE
a
ððB
T
AÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
Þ
2
()
VecF;ð3-20Þ
504 Z.A.A. Al-Zhour/ Applied Mathematics and Computation 232 (2014) 498–510

VecYðtÞ¼E
a
ððB
T
AÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
Þ
2
()
VecE
þE
a
ððB
T
AÞðtt
0
Þ
a
ÞE
a
ððD
T
CÞðtt
0
Þ
a
ÞþE
a
ððD
T
CÞðtt
0
Þ
a
Þ
2
()
VecF:ð3-21Þ
Proof. If we use the Vecð:Þ– notation, we have the following system:
VecX
a
ðtÞ
VecY
a
ðtÞ

¼B
T
AD
T
C
D
T
CB
T
A
"#
VecXðtÞ
VecYðtÞ

:ð3-22Þ
This system by using (3-8) has the following solution:
VecXðtÞ
VecYðtÞ

¼E
a
B
T
AD
T
C
D
T
CB
T
A
"#
ðtt
0
Þ
a
!
VecE
VecF

:ð3-23Þ
Now we will deal with E
a
B
T
AD
T
C
D
T
CB
T
A

.
Note that
B
T
A0
0B
T
A
"#
0D
T
C
D
T
C0
"#
¼0D
T
C
D
T
C0
"#
B
T
A0
0B
T
A
"#
:
Then
E
a
B
T
AD
T
C
D
T
CB
T
A
"# !
¼E
a
B
T
A0
0B
T
A
"#
þ0D
T
C
D
T
C0
"# !
¼E
a
B
T
A0
0B
T
A
"# !
E
a
0D
T
C
D
T
C0
"# !
:
But
E
a
B
T
A0
0B
T
A
"# !
¼E
a
ðB
T
AÞ0
0E
a
ðB
T
AÞ
"#
;
and
E
a
0D
T
C
D
T
C0
"# !
¼
E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
2
43
5:
So
E
a
B
T
AD
T
C
D
T
CB
T
A

¼E
a
ðB
T
AÞ0
0E
a
ðB
T
AÞ

E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
"#
¼E
a
ðB
T
AÞ
E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
no
E
a
ðB
T
AÞ
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
no
E
a
ðB
T
AÞ
E
a
ðD
T
CÞE
a
ðD
T
CÞ
2
no
E
a
ðB
T
AÞ
E
a
ðD
T
CÞþE
a
ðD
T
CÞ
2
no
2
43
5:ð3-24Þ
Now from (3-23) and (3-24), we get the results as in (3-20) and (3-21).h
If we set
a
¼1inTheorem 2, we get the following corollary.
Corollary 3. Let A;B;C;D;E and F 2M
n
be scalar matrices such that AC ¼CA, BD ¼DB, and let XðtÞand YðtÞ2M
n
be the
unknown matrices to be solved. Then the general vector solutions of the following coupled linear matrix ordinary differential
equations:
X
=
ðtÞ¼AXðtÞBþCY ðtÞD;Y
=
ðtÞ¼CXðtÞDþAY ðtÞB;Xðt
0
Þ¼E;Yðt
0
Þ¼F;ð3-25Þ
is given by
VecXðtÞ¼e
ðB
T
AÞðtt
0
Þ
f½coshðD
T
CÞðtt
0
ÞVecE þ½sinhðD
T
CÞðtt
0
ÞVecFg;ð3-26Þ
VecYðtÞ¼e
ðB
T
AÞðtt
0
Þ
f½sinhðD
T
CÞðtt
0
ÞVecE þ½coshðD
T
CÞðtt
0
ÞVecFg:ð3-27Þ
Corollary 4. Let E and F 2M
n
be scalar matrices, I 2M
n
2
be an identity matrix and XðtÞ,YðtÞ2M
n
be the unknown matrices to be
solved. Then the general vector solutions of the following coupled linear matrix fractional differential equations of order 0<
a
<1:
Z.A.A. Al-Zhour / Applied Mathematics and Computation 232 (2014) 498–510 505

X
a
ðtÞ¼XðtÞþYðtÞ;Y
a
ðtÞ¼XðtÞþYðtÞ;Xð0Þ¼E;Yð0Þ¼Fð3-28Þ
is given by
VecXðtÞ¼E
a
ðt
a
Þ
2diagðE
a
ðt
a
ÞþE
a
ðt
a
Þ;E
a
ðt
a
ÞþE
a
ðt
a
Þ;...;E
a
ðt
a
ÞþE
a
ðt
a
ÞÞVecE þE
a
ðt
a
Þ
2diagðE
a
ðt
a
Þ
E
a
ðt
a
Þ;E
a
ðt
a
ÞE
a
ðt
a
Þ;...;E
a
ðt
a
ÞE
a
ðt
a
ÞÞVecF;ð3-29Þ
VecYðtÞ¼E
a
ðt
a
Þ
2diagðE
a
ðt
a
ÞE
a
ðt
a
Þ;E
a
ðt
a
ÞE
a
ðt
a
Þ;...;E
a
ðt
a
ÞE
a
ðt
a
ÞÞVecE þE
a
ðt
a
Þ
2diagðE
a
ðt
a
Þ
þE
a
ðt
a
Þ;E
a
ðt
a
ÞþE
a
ðt
a
Þ;...;E
a
ðt
a
ÞþE
a
ðt
a
ÞÞVecF:ð3-30Þ
Proof. The proof is straightforward by applying Theorem 2 and by using the Mittag-Leffler matrix functions: E
a
ðIðtÞ
a
Þand
E
a
ðIðtÞ
a
Þwhich are given, respectively, as follow:
E
a
ðIðtÞ
a
Þ¼diagðE
a
ðt
a
Þ;E
a
ðt
a
Þ;...;E
a
ðt
a
ÞÞ 2 M
n
2
;ð3-31Þ
E
a
ðIðtÞ
a
Þ¼diagðE
a
ðt
a
Þ;E
a
ðt
a
Þ;...;E
a
ðt
a
ÞÞ 2 M
n
2
:ð3-32Þ
If we set
a
¼1inCorollary 4, we get the following corollary.
Corollary 5. Let E and F 2M
n
be scalar matrices, and let XðtÞand Y ðtÞ2M
n
be the unknown matrices to be solved. Then the gen-
eral vector solutions of the coupled linear matrix ordinary differential equations:
X
=
ðtÞ¼XðtÞþYðtÞ;Y
=
ðtÞ¼XðtÞþYðtÞ;Xð0Þ¼E;Yð0Þ¼F;ð3-33Þ
is given by
XðtÞ¼e
t
fEcoshðtÞþFsinhðtÞg;ð3-34Þ
YðtÞ¼e
t
fEsinhðtÞþFsinhðtÞg:ð3-35Þ
4. Some examples
In this section, we give some examples in order to illustrate our new approaches that obtained in Section 3which involves
sequential Caputo derivative.
Example 1. Consider the following matrix fractional differential equation:
X
a
ðtÞ¼AXðtÞ;Xð0Þ¼C;0<
a
<1;ð4-1Þ
with A¼10
32

and C¼11
33

.
Then the Mittag-Leffler matrix E
a
ðAt
a
Þcan be computed as follows:
E
a
ðAt
a
Þ¼ E
a
ðt
a
Þ0
3E
a
ðt
a
Þ3E
a
ð2t
a
ÞE
a
ð2t
a
Þ

:ð4-2Þ
Here the exact solution of (4-1) is given by:
XðtÞ¼E
a
ðAt
a
ÞC¼E
a
ðt
a
Þ0
3E
a
ðt
a
Þ3E
a
ð2t
a
ÞE
a
ð2t
a
Þ

11
33

¼E
a
ðt
a
ÞE
a
ðt
a
Þ
3E
a
ðt
a
Þ3E
a
ð2t
a
Þ

:ð4-3Þ
Example 2. Consider the following matrix fractional differential equation:
½X
a
ðtÞ¼AXðtÞþUðtÞ;Xð0Þ¼C;0<
a
<1;ð4-4Þ
with A¼10
32

,C¼11
33

and UðtÞ¼ fðtÞgðtÞ
hðtÞqðtÞ

.
Then the Mittag-Leffler matrix E
a
ðAðtsÞ
a
Þcan be computed as follows:
E
a
ðAðtsÞ
a
Þ¼ E
a
ððtsÞ
a
Þ0
3E
a
ððtsÞ
a
Þ3E
a
ð2ðtsÞ
a
ÞE
a
ð2ðtsÞ
a
Þ
"#
:ð4-5Þ
506 Z.A.A. Al-Zhour/ Applied Mathematics and Computation 232 (2014) 498–510

Here the exact solution of (4-4) is given by:
XðtÞ¼E
a
ðAt
a
ÞCþZ
t
0
ðtsÞ
a
1
E
a
ðAðtsÞ
a
ÞUðsÞds ¼E
a
ðt
a
ÞE
a
ðt
a
Þ
3E
a
ðt
a
Þ3E
a
ð2t
a
Þ

þaðtÞbðtÞ
cðtÞdðtÞ

;ð4-6Þ
where
aðtÞ¼Z
t
0
ðtsÞ
a
1
E
a
ððtsÞ
a
ÞfðsÞds;ð4-7Þ
bðtÞ¼Z
t
0
ðtsÞ
a
1
E
a
ððtsÞ
a
ÞgðsÞds;ð4-8Þ
cðtÞ¼Z
t
0
ðtsÞ
a
1
½ð3E
a
ððtsÞ
a
Þ3E
a
ð2ðtsÞ
a
ÞÞfðsÞþE
a
ð2ðtsÞ
a
ÞhðsÞds;ð4-9Þ
dðtÞ¼Z
t
0
ðtsÞ
a
1
½ð3E
a
ððtsÞ
a
Þ3E
a
ð2ðtsÞ
a
ÞÞgðsÞþE
a
ð2ðtsÞ
a
ÞqðsÞds:ð4-10Þ
Example 3. Consider the following matrix fractional differential equation:
X
a
ðtÞ¼AXðtÞþXðtÞA;Xð0Þ¼C;0<
a
<1;ð4-11Þ
with A¼10
32

and C¼11
33

:
Then the exact solution of (4-11) is given by:
XðtÞ¼E
a
ðAt
a
ÞCE
a
ðAt
a
Þ¼ E
a
ðt
a
ÞE
a
ðt
a
Þ
3E
a
ðt
a
Þ3E
a
ð2t
a
Þ

E
a
ðt
a
Þ0
3E
a
ðt
a
Þ3E
a
ð2t
a
ÞE
a
ð2t
a
Þ

¼wðtÞzðtÞ
rðtÞpðtÞ

;ð4-12Þ
where,
wðtÞ¼ðE
a
ðt
a
ÞÞ
2
3ðE
a
ðt
a
ÞÞ
2
þ3E
a
ðt
a
ÞE
a
ð2t
a
Þ;ð4-13Þ
zðtÞ¼E
a
ðt
a
ÞE
a
ð2t
a
Þ;ð4-14Þ
rðtÞ¼3ðE
a
ðt
a
ÞÞ
2
9E
a
ð2t
a
ÞE
a
ðt
a
Þþ9ðE
a
ð2t
a
ÞÞ
2
;ð4-15Þ
pðtÞ¼3ðE
a
ð2t
a
ÞÞ
2
:ð4-16Þ
Example 4. Consider the following coupled matrix fractional differential equation:
X
a
ðtÞ¼XðtÞþYðtÞ;Y
a
ðtÞ¼XðtÞþYðtÞ;Xð0Þ¼E;Yð0Þ¼F;0<
a
<1;ð4-17Þ
with E¼10
12

and F¼34
01

.
Then the vector exact solutions VecXðtÞand VecY ðtÞof (4-17) are given, respectively, by:
VecXðtÞ¼E
a
ðt
a
Þ
2
E
a
ðt
a
ÞþE
a
ðt
a
Þ000
0E
a
ðt
a
ÞþE
a
ðt
a
Þ00
00E
a
ðt
a
ÞþE
a
ðt
a
Þ0
000E
a
ðt
a
ÞþE
a
ðt
a
Þ
2
6
6
43
7
7
5
1
1
0
2
2
6
6
43
7
7
5
þE
a
ðt
a
Þ
2
E
a
ðt
a
ÞE
a
ðt
a
Þ000
0E
a
ðt
a
ÞE
a
ðt
a
Þ00
00E
a
ðt
a
ÞE
a
ðt
a
Þ0
000E
a
ðt
a
ÞE
a
ðt
a
Þ
2
6
6
43
7
7
5
3
0
4
1
2
6
6
43
7
7
5
¼E
a
ðt
a
Þ
2
4E
a
ðt
a
Þ2E
a
ðt
a
Þ
E
a
ðt
a
ÞE
a
ðt
a
Þ
4E
a
ðt
a
Þþ4E
a
ðt
a
Þ
E
a
ðt
a
Þ3E
a
ðt
a
Þ
2
6
6
43
7
7
5
;ð4-18Þ
Z.A.A. Al-Zhour / Applied Mathematics and Computation 232 (2014) 498–510 507

VecYðtÞ¼E
a
ðt
a
Þ
2
E
a
ðt
a
ÞE
a
ðt
a
Þ000
0E
a
ðt
a
ÞE
a
ðt
a
Þ00
00E
a
ðt
a
ÞE
a
ðt
a
Þ0
000E
a
ðt
a
ÞE
a
ðt
a
Þ
2
6
6
43
7
7
5
1
1
0
2
2
6
6
43
7
7
5
þE
a
ðt
a
Þ
2
E
a
ðt
a
ÞþE
a
ðt
a
Þ000
0E
a
ðt
a
ÞþE
a
ðt
a
Þ00
00E
a
ðt
a
ÞþE
a
ðt
a
Þ0
000E
a
ðt
a
ÞþE
a
ðt
a
Þ
2
6
6
43
7
7
5
3
0
4
1
2
6
6
43
7
7
5
¼E
a
ðt
a
Þ
2
4E
a
ðt
a
Þ2E
a
ðt
a
Þ
E
a
ðt
a
ÞE
a
ðt
a
Þ
4E
a
ðt
a
Þþ4E
a
ðt
a
Þ
E
a
ðt
a
Þ3E
a
ðt
a
Þ
2
6
6
43
7
7
5
:ð4-19Þ
Hence, the exact solutions XðtÞand YðtÞof (4-17) are given, respectively, by:
XðtÞ¼E
a
ðt
a
Þ
2E
a
ðt
a
ÞþE
a
ðt
a
Þ2E
a
ðt
a
Þ2E
a
ðt
a
Þ
E
a
ðt
a
ÞþE
a
ðt
a
Þ
2
E
a
ðt
a
Þþ3E
a
ðt
a
Þ
2
"#
;ð4-20Þ
YðtÞ¼E
a
ðt
a
Þ
2E
a
ðt
a
ÞE
a
ðt
a
Þ2E
a
ðt
a
Þþ2E
a
ðt
a
Þ
E
a
ðt
a
ÞE
a
ðt
a
Þ
2
E
a
ðt
a
Þ3E
a
ðt
a
Þ
2
"#
:ð4-21Þ
5. Conclusions
The general (vector) solutions of such non-homogeneous (coupled) linear matrix fractional differential equations are pre-
sented by using Kronecker structures and some important and interested special cases are also discussed. The comparison
between the fractional and integer order (vector) solutions and some examples are also given to illustrate our new ap-
proaches. How to extend this method to find the vector solutions of more general system of non-homogeneous linear matrix
fractional differential equations in Eq. (1-6) above require further research. Although the idea adopted can be easily extended
to study the coupled matrix non-linear fractional differential equations, e.g., coupled matrix Riccati fractional differential
equations.
Acknowledgments
The author expresses his sincere thanks to referees for careful reading and several helpful suggestions. The author also
gratefully acknowledges that this research was supported by the Deanship of Scientific Research, University of Dammam,
Saudi Arabia, under Grant No. 2013105.
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