Applied Mathematical Modelling 37 (2013) 4035–4050
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A computational matrix method for solving systems of high order fractional differential equations M.M. Khader a,⇑, Talaat S. El Danaf b, A.S. Hendy a a b
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Koom, Egypt
a r t i c l e
i n f o
Article history: Received 18 January 2012 Received in revised form 17 July 2012 Accepted 17 August 2012 Available online 17 September 2012 Keywords: Systems of fractional differential equations Caputo fractional derivatives Chebyshev polynomials Computational matrix method
a b s t r a c t In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution uðtÞ and the Chebyshev coefficient matrix AðmÞ of the fractional derivative uðmÞ . The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Fractional differential equations have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, biology, physics and engineering [1]. Consequently, considerable attention has been given to the solutions of fractional differential equations and integral equations of physical interest. Most of fractional differential equations do not have exact analytic solutions, so approximate and numerical techniques [2–11] must be used. Several numerical methods to solve FDEs have been given such as, variational iteration method [6,12], homotopy perturbation method [13], homotopy analysis method [14] and collocation method [15–24]. Representation of a function in terms of a series expansion using orthogonal polynomials is a fundamental concept in approximation theory and form the basis of the solution of differential equations [25–27]. In [15] Khader introduced an efficient numerical method for solving the fractional diffusion equation using the shifted Chebyshev polynomials and also introduced in Ref. [28] an operational matrix method for solving nonlinear multi-order fractional differential equations. In [29] two Chebyshev spectral methods for solving multi-term fractional orders differential equations are introduced. In [3] Chebyshev collocation method is used to solve the high order nonlinear ordinary differential equations. In this study, a new operational matrix method [30,31] is presented to find the approximate solutions of high order fractional differential equations in terms of shifted Chebyshev polynomials via Chebyshev collocation points in the interval ½0; L. The main characteristic of this new technique is that it gives a straight forward algorithm in converting FDEs to a system of
⇑ Corresponding author. E-mail addresses:
[email protected] (M.M. Khader),
[email protected] (T.S. El Danaf),
[email protected] (A.S. Hendy). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.08.009
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algebraic equations. This algorithm has several advantages such as being non-differentiable, non-integral and easily implemented on a computer, because its structure is dependent on matrix operations only. The aim of this study is to concern with the application of the proposed approach to obtain the approximate solution of the systems of FDEs of the linear form m X k X pnij ðxÞDbn uj ðxÞ ¼ fi ðxÞ;
i ¼ 1; 2; 3; . . . ; s;
ð1Þ
n¼0 j¼1
and the non-linear form m X k X qnij ðxÞ½Dan uj ðxÞ þ Dbn uj ðxÞDcn uj ðxÞ þ unj ðxÞ ¼ g i ðxÞ:
ð2Þ
n¼0 j¼1
The proceeding system forms (1) and (2) can be written into the following compact forms, respectively m X Pi ðxÞDbi uðxÞ ¼ fðxÞ;
ð3Þ
i¼0 m X Q i ðxÞ½Dai uðxÞ þ Dbi uðxÞDci uðxÞ þ ui ðxÞ ¼ gðxÞ;
ð4Þ
i¼0
subject to the mixed conditions m X ½aj udbj e1 ð0Þ þ bj udbj e1 ðLÞ þ cj udbj e1 ðcÞ ¼ kj ;
0 < c < L;
ð5Þ
j¼0
where
2
pi11 ðxÞ pi12 ðxÞ pi1s ðxÞ
3
2
7 6 i 6 p21 ðxÞ pi22 ðxÞ pi2s ðxÞ 7 7 6 ; Pi ðxÞ ¼ 6 . .. .. .. 7 6 .. . . . 7 5 4 pis1 ðxÞ pis2 ðxÞ piss ðxÞ 2
u1 ðxÞ
3
2
Dai u1 ðxÞ
3
7 6 i 6 q21 ðxÞ qi22 ðxÞ qi2s ðxÞ 7 7 6 Q i ðxÞ ¼ 6 . ; .. .. .. 7 6 .. . . . 7 5 4 qis1 ðxÞ qis2 ðxÞ qiss ðxÞ 3
2
6 Dai u ðxÞ 7 6 u2 ðxÞ 7 2 7 7 6 6 7; 7 Dai uðxÞ ¼ 6 uðxÞ ¼ 6 . .. 7 6 6 . 7 5 4 4 . 5 .
f1 ðxÞ
3
6 f ðxÞ 7 6 2 7 7 fðxÞ ¼ 6 6 .. 7; 4 . 5
Dai us ðxÞ
us ðxÞ
qi11 ðxÞ qi12 ðxÞ qi1s ðxÞ
fs ðxÞ
2
g 1 ðxÞ
3
6 g ðxÞ 7 6 2 7 7 gðxÞ ¼ 6 6 .. 7; 4 . 5 g s ðxÞ
m
here ui ðxÞ are unknown functions from C ½0; L, known functions pij ðxÞ and qij ðxÞ are defined on the interval ½0; L; i < ai ; bi ; ci 6 i þ 1 and aj ; bj ; cj are real valued matrices. The existence and the uniqueness for the solutions of systems of FDEs have been studied in [4]. The structure of this paper is arranged in the following way: In Section 2, we introduced some basic definitions about Caputo fractional derivatives and properties of the shifted Chebyshev polynomials. In Section 3, we introduce the fundamental relations for the new operational matrix method. In Sections 4 and 5, the procedure of solution for systems of FDEs of linear and non-linear forms are clarified, respectively. In Section 6, numerical examples are given to solve systems of FDEs and show the accuracy of the presented method. Finally, in Section 7, the report ends with a brief conclusion and some remarks. 2. Preliminaries and notations In this section, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory that will be required in the present paper. 2.1. The fractional derivative in the Caputo sense
Definition 1. The Caputo fractional derivative operator Dm of order
Dm f ðxÞ ¼
1 Cðm mÞ
Z
x 0
f ðmÞ ðnÞ ðx nÞmmþ1
where m 1 < m 6 m; m 2 N; x > 0.
dn;
m > 0;
m is defined in the following form:
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
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Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation
Dm ðkpðxÞ þ lqðxÞÞ ¼ kDm pðxÞ þ lDm qðxÞ;
ð6Þ
where k and l are constants. For the Caputo’s derivative we have
Dm C ¼ 0;
C is a constant;
( m n
D x ¼
for
0; Cðnþ1Þ nm ; Cðnþ1mÞ x
ð7Þ
for n 2 N0 and n < dme; for n 2 N0 and n P dme:
ð8Þ
We use the ceiling function dme to denote the smallest integer greater than or equal to m and N0 ¼ f0; 1; 2; . . .g. Recall that m 2 N, the Caputo differential operator coincides with the usual differential operator of integer order. For more details on fractional derivatives definitions and its properties see ([2–26,29–34]).
2.2. The definition and properties of the shifted Chebyshev polynomials The well known Chebyshev polynomials are defined on the interval ½1; 1 and can be determined with the aid of the following recurrence formula
T nþ1 ðzÞ ¼ 2zT n ðzÞ T n1 ðzÞ;
T 0 ðzÞ ¼ 1;
T 1 ðzÞ ¼ z;
n ¼ 1; 2; . . . :
n
It is well known that T n ð1Þ ¼ ð1Þ ; T n ð1Þ ¼ 1. The analytic form of the Chebyshev polynomials T n ðzÞ of degree n is given by
T n ðzÞ ¼
½n2 X nðn i 1Þ! n2i z ; ð1Þi 2n2i1 ðiÞ!ðn 2iÞ! i¼0
ð9Þ
where ½n=2 denotes the integer part of n=2. The orthogonality condition is
Z
1
1
8 > < p; for i ¼ j ¼ 0; T i ðzÞT j ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ p2 ; for i ¼ j – 0; > 1 z2 : 0; for i – j:
ð10Þ
In order to use these polynomials on the interval ½0; L, we define the so called shifted Chebyshev polynomials by introducing the change of variable z ¼ 2x 1. L The shifted Chebyshev polynomials are defined as
T n ðxÞ ¼ T n
2x 1 ; L
where T 0 ðxÞ ¼ 1; T 1 ðxÞ ¼
2x 1: L
The analytic form of the shifted Chebyshev polynomials T n ðxÞ of degree n is given by n X ðn þ k 1Þ!22k k T n ðxÞ ¼ n ð1Þnk x; ðn kÞ!ð2kÞ!Lk k¼0
ð11Þ
where T n ð0Þ ¼ ð1Þn ; T n ðLÞ ¼ 1. The orthogonality condition of these polynomials is
Z
0
L
T j ðxÞT k ðxÞwðxÞdx ¼ djk hk ;
ð12Þ
1 where the weight function wðxÞ ¼ pffiffiffiffiffiffiffiffiffi ; hk ¼ a2k p, with a0 ¼ 2; ak ¼ 1; k P 1. Lxx2 The function uðxÞ which belongs to the space of square integrable in ½0; L, may be expressed in terms of shifted Chebyshev polynomials as
uðxÞ ¼
1 X ci T i ðxÞ; i¼0
where the coefficients ci are given by
ci ¼
1 hi
Z
0
L
uðxÞT i ðxÞwðxÞdx;
i ¼ 0; 1; 2; . . . :
ð13Þ
3. Fundamental relations It is suggested that the solution ui ðxÞ which from C m ½0; L, it can be approximated in terms of the first ðN þ 1Þ-terms shifted Chebyshev polynomials only as
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ui ðxÞ ¼
N X cij T j ðxÞ;
i ¼ 1; 2; . . . ; s;
ð14Þ
j¼0
where N is any positive integer such that N P s and cij are unknown Chebyshev coefficients. To express the fractional derivative of the function ui ðxÞ in terms of shifted Chebyshev polynomials, we introduce the following theorem. Theorem 1. Let ui ðxÞ be approximated by shifted Chebyshev polynomials (14) and also suppose that fractional derivative can be written in the following form
Dm ðui ðxÞÞ ffi
2jð1Þjk ðj þ k 1Þ!Cðk m þ 12Þcij T r ðxÞ; 1 a L C ðk þ Þðj kÞ! C ðk m r þ 1Þ C ðk m þ r þ 1Þ r 2 r¼0 j¼dmek¼dme
j N X N X X
m > 0 then, its Caputo
ð15Þ
m
where j ¼ dme; dme þ 1; . . . ; N. Proof. Since the Caputo’s fractional differentiation is a linear operation we have
Dm ðui ðxÞÞ ¼
N X
cij Dm T j ðxÞ;
i ¼ 1; 2; . . . ; s:
ð16Þ
j¼0
Employing Eqs. (7) and (8) into (11) we have
Dm T j ðxÞ ¼ 0;
j ¼ 0; 1; . . . ; dme 1;
m > 0:
ð17Þ
Also, for j ¼ dme; . . . ; N, by using Eqs. (7) and (8) into (11) we get
Dm T j ðxÞ ¼ j
j j X X ðj þ k 1Þ!22k m k ðj þ k 1Þ!22k k! xkm : ð1Þjk D x ¼j ð1Þjk k ðj kÞ!ð2kÞ!L ðj kÞ!ð2kÞ!Lk Cðk m þ 1Þ k¼0 k¼dme
ð18Þ
From Eqs. (16)–(18), we obtain
Dm ðui ðxÞÞ ¼
j N X X jð1Þjk ðj þ k 1Þ!22k k!cij j¼dmek¼dme ðj
kÞ!ð2kÞ!Lk Cðk m þ 1Þ
xkm :
ð19Þ
Now, xkm can be expressed approximately in terms of shifted Chebyshev series, so we have
xkm ffi
N X bkr T r ðxÞ;
ð20Þ
r¼0
where bkr is obtained from (13) with uðxÞ ¼ xkm [29]. A combination of Eqs. (18) and (20) leads to the desired result. h The function ui ðxÞ defined in (14) can be written in the following matrix form
ui ðxÞ ¼ TðxÞAi ;
ð21Þ
where TðxÞ ¼ ½T 0 ðxÞT 1 ðxÞ . . . T N ðxÞ;
Ai ¼ ½ci0 ci1 . . . ciN T :
ð22Þ
Theorem 2. Let TðxÞ be a shifted Chebyshev vector defined in (22) and also suppose that m > 0 then, the matrix representation of Dm ui ðxÞ has the following form ðmÞ
Dm ui ðxÞ ¼ TðxÞAi ; ðmÞ Ai
where follows
ð23Þ
is the sðN þ 1Þ sðN þ 1Þ operational matrix of fractional derivatives of order
ðmÞ Ai ; Ai ¼ M
m in the Caputo sense and defined as ð24Þ
m
where
2 6 6 6 M¼6 6 m 6 4
M
0
0
M
.. . 0
.. . 0
m
m
0
3
7 07 7 7 .. .. 7 . . 7 5 M m
ss
;
ð25Þ
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M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
such that
2
0 6 60 6 6 6 6 M ¼ 60 m 6 6. 6. 6. 4 0
0
X
0 0
dme;k;0
dmeþ1;k;0
X
X
dme;k;1
X
dmeþ1;k;1
X
dme;k;2
X
dmeþ1;k;2
.. .
.. .
..
X
X
..
. . .. 0
dme;k;N
dmeþ1;k;N
3
X
N;k;0
7
X 7 7
N;k;1
7 7
X 7 7
;
N;k;2
7 .. 7 7 . 7 5
.
X
N;k;N
ðNþ1ÞðNþ1Þ
where
X ¼
j;k;N
2jð1Þjk ðj þ k 1Þ!Cðk m þ 12Þ ; a L Cðk þ 12Þðj kÞ!Cðk m N þ 1ÞCðk m þ N þ 1Þ k¼dme N j X
m
j ¼ dme; dme þ 1; . . . ; N:
Proof. Using Eq. (15), we can write the following relation:
2
j N X X
3
2jð1Þjk ðjþk1Þ!Cðkmþ12Þcij
7 6 ar Lm Cðkþ12ÞðjkÞ!Cðkmrþ1ÞCðkmþrþ1Þ 7 6 7 6 j¼dmek¼dme 7 6 7 6 N j 7 6XX jk 1 2jð1Þ ðjþk1Þ!Cðkmþ2Þcij 7 6 m 6 ar L Cðkþ12ÞðjkÞ!Cðkmrþ1ÞCðkmþrþ1Þ 7 m 7: D ui ðxÞ ffi ½ T 0 ðxÞ T 1 ðxÞ T N ðxÞ 6 j¼dmek¼dme 7 6 7 6 7 6 .. 7 6 . 7 6 7 6 N j 7 6XX jk 1Þc 2jð1Þ ðjþk1Þ! C ðk m þ 5 4 2 ij m j¼dmek¼dme
ar L Cðkþ12ÞðjkÞ!Cðkmrþ1ÞCðkmþrþ1Þ
Using this relation, we obtain sðN þ 1Þ sðN þ 1Þ operational matrix of fractional derivatives of order Then, substitute by (24) and Chebyshev collocation points into (23) gives
Ai ; Dm ui ðxn Þ ¼ Tðxn Þ M m
n ¼ 0; 1; . . . ; N;
m (24). h ð26Þ
or in the compact form
A; UðmÞ ¼ TðxÞ M
ð27Þ
m
where Uð0Þ ¼ U ¼ TðxÞA, such that
3
2
6 Tðx1 Þ 7 7 6 7 TðxÞ ¼ 6 6 .. 7; 4 . 5
6 6 TðxN Þ ¼ 6 6 4
2
Tðx0 Þ
TðxN Þ A ¼ ½ A1
0
3
0 .. .
TðxN Þ .. .. . .
0 .. .
7 7 7 7 5
0
0 As T ;
A2
TðxN Þ
As ¼ ½ cs0
0
TðxN Þ
ss
2 ;
ui ðx0 Þ
3
6 ui ðx1 Þ 7 7 6 7 U¼6 6 .. 7; 4 . 5 ui ðxN Þ
csN T :
cs1
To obtain the matrix representation of ur ðxÞ using Chebyshev collocation points we have
2
ur ðx0 Þ
3
2
6 ur ðx1 Þ 7 6 6 7 6 6 7 6 6 .. 7 ¼ 6 4 . 5 4 ur ðxN Þ
uðx0 Þ 0 .. . 0
0
3r1 2
uðx1 Þ .. .. . .
0 .. .
7 7 7 7 5
0
0
uðxN Þ
uðx0 Þ
3
6 uðx1 Þ 7 6 7 6 7 6 .. 7; 4 . 5
ð28Þ
uðxN Þ
which can be written in the following compact form
r1 U; Ur ¼ ðUÞ such that, ¼ LA where U
ð29Þ
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2 L
0
3
Tðx1 Þ .. .. . .
0 .. .
7 7 7; 7 5
Tðx0 Þ
6 6 ¼6 6 4
0
0 .. . 0
0
2
A
60 6 ¼6 . A 6. 4. 0
TðxN Þ
0
0
07 7 .. 7 7: .5
A .. . . . . 0
3
A
ci
bi
To obtain the matrix representation of D uðxÞD uðxÞ ¼ uðbi Þ ðxÞuðci Þ ðxÞ using Chebyshev collocation points we have
2
3 2 ðbi Þ u ðx0 Þ uðbi Þ ðx0 Þuðci Þ ðx0 Þ 0 6 uðbi Þ ðx Þuðci Þ ðx Þ 7 6 ðbi Þ 0 u ðx Þ 1 1 7 1 6 6 6 7¼6 .. .. .. .. 6 7 6 4 5 4 . . . . uðbi Þ ðxN Þuðci Þ ðxN Þ
0
32
3 uðci Þ ðx0 Þ 76 uðci Þ ðx Þ 7 1 7 76 76 7; .. 76 7 54 5 .
0 0 .. .
uðbi Þ ðxN Þ
0
uðci Þ ðxN Þ
which can be written in the following compact form
AT M A; ubi ðxÞuci ðxÞ ¼ L M bi
ð30Þ
ci
where
2 6 6 6 6 M¼6 ci 6 6 4
M
0
0
M
ci
.. .
0
0
2
7 07 7 7 .. .. 7 7 . . 7 5 M
ci
.. .
3
0
ci
6 6 6 6 M¼6 bi 6 6 4
;
3
2
7 07 7 7 .. .. 7 7 . . 7 5 M
6 6 6 6 M¼6 6 bi 6 6 4
M
0
0
M
.. . 0
.. . 0
bi
bi
0
bi
ss
;
M
0
0
M
bi
.. .
0
0
3
7 7 07 7 7 .. .. 7 7 . . 7 5 M
bi
.. .
0
bi
ss
:
ss
4. Procedure of solution for the linear form of FDEs To obtain the shifted Chebyshev solution of Eq. (1) under the mixed conditions (5), the following matrix method which based on computing Chebyshev coefficients is used. Firstly, we substitute by Chebyshev collocation points into Eq. (3) as follows m X Pi ðxn ÞDbi uðxn Þ ¼ fðxn Þ;
ð31Þ
i¼0
this system (31) can be written in the following matrix form m m X X A ¼ F; Pi Uðbi Þ ¼ Pi TðxÞM i¼0
ð32Þ
bi
i¼0
where
2
Pi ðx0 Þ
6 6 Pi ¼ 6 6 4
0 .. .
0
P i ðx1 Þ .. .. . .
0
0
0
3
0 .. .
7 7 7; 7 5
3 fi ðx0 Þ 6 f ðx Þ 7 6 i 1 7 7 F¼6 6 .. 7: 4 . 5
Pi ðxN Þ
2
fi ðxN Þ
This is the main matrix equation for the solution of Eq. (1), and can be written in the following compact form:
WA ¼ F;
W¼
m X ; Pi TðxÞM i¼0
bi
which corresponds to a system of sðN þ 1Þ linear algebraic equations with unknown Chebyshev coefficients cij . In addition, the matrix representation of the mixed conditions (5) has the following form m X A ¼ kj ; ½aj Tð0Þ þ bj TðLÞ þ cj TðcÞ M
ð33Þ
dbj e1
j¼0
which can be written in the following compact form
VA ¼ k;
V¼
m X ; ½aj Tð0Þ þ bj TðLÞ þ cj TðcÞ M j¼0
dbj e1
k¼
k0
k1
km
T :
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
4041
c b c ¼b Replacing the last rows of the augmented matrix ½W; F by rows of the matrix ½V; k, we have ½ W; F or WA F, which is a linear algebraic system, we obtain the unknown Chebyshev coefficients after solving it and so we can express the solution of Eq. (1) as a truncated series from the shifted Chebyshev polynomials (14). 5. Procedure of solution for the non-linear form of FDEs To obtain the shifted Chebyshev solution of Eq. (2) under the mixed conditions (5), the following matrix method which based on computing Chebyshev coefficients is used. Firstly, we substitute by Chebyshev collocation points into Eq. (4) as follows m X Q i ðxn Þ½Dai uðxn Þ þ Dbi uðxn ÞDci uðxn Þ þ ui ðxn Þ ¼ gðxn Þ;
ð34Þ
i¼0
this system (34) can be written in the following matrix form m m X X AT M i1 TA ¼ G; þ ðL þ LM Q i ½Uðai Þ þ Uðbi Þ Uðci Þ þ Ui ¼ Q i ½TðxÞM AÞ i¼0
ai
i¼0
bi
ci
ð35Þ
where
2
Q i ðx0 Þ
6 6 Qi ¼ 6 6 4
0
0 .. .
Q i ðx1 Þ .. .. . .
0
0
0
3
0 .. .
7 7 7; 7 5
3 g i ðx0 Þ 6 g ðx1 Þ 7 7 6 i 7 G¼6 6 .. 7: 4 . 5 2
g i ðxN Þ
Q i ðxN Þ
This is the main matrix equation for the solution of Eq. (2), and can be rewritten in the following compact form
WA ¼ F;
W¼
m X AT M i1 T; þ ðL þ LM AÞ Q i ½TðxÞM i¼0
ai
bi
ci
which corresponds to a system of sðN þ 1Þ non-linear algebraic equations with unknown Chebyshev coefficients cij . In addition, the matrix representation of the mixed conditions (5) has form (33), which can be written in the following compact form
VA ¼ k;
V¼
m X ; ½aj Tð0Þ þ bj TðLÞ þ cj TðcÞ M dbj e1
j¼0
k ¼ ½ k0
k1
km T :
c b c ¼ F, b which is a Replacing the last rows of the augmented matrix ½W; F by rows of the matrix ½V; k, we have ½ W; F or WA non-linear algebraic system, we obtain the unknown Chebyshev coefficients after solving it and so we can express the solution of Eq. (2) as a truncated series from the shifted Chebyshev polynomials (14). 6. Numerical simulation In order to illustrate the effectiveness and accuracy of the proposed method, we implement it to solve the following systems of fractional differential equations. Problem 1. In this problem, the pollution problem of three lakes with interconnecting channels has been modeled in [35] and studied numerically in [36]. As monitoring pollution is the first step toward planning to save the environment then, we apply the proposed collocation method to obtain the approximate solutions for the fractional model of this problem. We consider the fractional model of the system in [36]
F 13 F 31 F 21 x3 ðtÞ x1 ðtÞ x1 ðtÞ þ PðtÞ; V3 V1 V1 F 21 F 32 x1 ðtÞ x2 ðtÞ; Da2 x2 ðtÞ ¼ V1 V2 F 31 F 32 F 13 x1 ðtÞ þ x2 ðtÞ x3 ðtÞ; Da3 x3 ðtÞ ¼ V1 V2 V3
Da1 x1 ðtÞ ¼
ð36Þ
with the following initial conditions
x1 ð0Þ ¼ k1 ;
x2 ð0Þ ¼ k2 ;
x3 ð0Þ ¼ k3 ;
where 0 < t 6 b; 0 < ai 6 1; i ¼ 1; 2; 3; x1 ðtÞ; x2 ðtÞ; x3 ðtÞ are the unknown functions, the known function PðtÞ is defined on interval 0 < t 6 b and also F 13 ; F 21 ; F 31 ; F 32 ; V 1 ; V 2 and V 3 are appropriate constants. Let us consider an example would be a
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M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
manufacturing plant dumping waste, producing more output during the day than at night because of the hour of operation; hence a periodic input. The concentration of the lake eventually converges to the average input concentration of the contaminant. For model (36), we assume PðtÞ ¼ 1 þ sinðtÞ and the parameter values have been fixed to 3 3 3 3 3 3 3 V 1 ¼ 2900mi ; V 2 ¼ 850mi ; V 3 ¼ 1180mi ; F 13 ¼ 38mi =year; F 31 ¼ 20mi =year; F 21 ¼ 18mi =year; F 32 ¼ 18mi =year. So the system (36) becomes
38 20 18 x3 ðtÞ x1 ðtÞ x1 ðtÞ þ ð1 þ sinðtÞÞ; 1180 2900 2900 18 18 x1 ðtÞ x2 ðtÞ; Da2 x2 ðtÞ ¼ 2900 850 20 18 38 x1 ðtÞ þ x2 ðtÞ x3 ðtÞ; Da3 x3 ðtÞ ¼ 2900 850 1180
Da1 x1 ðtÞ ¼
ð37Þ
with initial conditions
x1 ð0Þ ¼ 0;
x2 ð0Þ ¼ 0;
x3 ð0Þ ¼ 0:
We apply the suggested method with N ¼ 3 and approximate the solutions xi ðtÞ; i ¼ 1; 2; 3 as follows
xi ðtÞ ffi
3 X cij T j ðtÞ:
ð38Þ
j¼0
Eq. (38) can be written in the following matrix form:
U ¼ TA;
ð39Þ
where
T ¼ ½ Tðx0 Þ Tðx3 Þ T ;
A ¼ ½ c10
c11
c12
c13
c20
c21
c22
c23
c30
c31
c32
c33 T :
Using the procedure in Section 5, the main matrix equation for this problem is
þP2 T M þP3 T M P0 TA ¼ F: ½P1 T M ða 1 Þ
ða2 Þ
ð40Þ
ða3 Þ
And the matrix equations for the conditions of this problem are:
Tð0ÞA1 ¼ 0;
Tð0ÞA2 ¼ 0;
Tð0ÞA3 ¼ 0:
ð41Þ
We obtain a system of 12 algebraic equations, the last three of them are from initial conditions. After solving this system using conjugate gradient method at ai ¼ 1; i ¼ 1; 2; 3, we obtain
c10 ¼ 0:67379;
c11 ¼ 0:72891;
c12 ¼ 0:05331;
c13 ¼ 0:00181;
c20 ¼ 0:00146;
c21 ¼ 0:00199;
c22 ¼ 0:00056;
c23 ¼ 0:00003;
c30 ¼ 0:00163;
c31 ¼ 0:00223;
c32 ¼ 0:00063;
c33 ¼ 0:00003:
Therefore, the approximate solutions of the system (37) take the following form
x1 ðtÞ ffi 0:99870t þ 0:51349t 2 0:058009t3 ; 2
ð42Þ
3
x2 ðtÞ ffi 0:00315t þ 0:00091t ;
ð43Þ
x3 ðtÞ ffi 0:003496t 2 þ 0:0010219t3 ;
ð44Þ
which have an excellent agreement with the approximate solutions obtained in [36]. Also, after solving this system using conjugate gradient method at ai ¼ 0:95; i ¼ 1; 2; 3, we obtain
c10 ¼ 0:70991;
c11 ¼ 0:75407;
c12 ¼ 0:05211;
c13 ¼ 0:00791;
c20 ¼ 0:00164;
c21 ¼ 0:00220;
c22 ¼ 0:00057;
c23 ¼ 0:00001;
c30 ¼ 0:00183;
c31 ¼ 0:00246;
c32 ¼ 0:00064;
c33 ¼ 0:00001:
Therefore, the approximate solutions of the system (37) take the following form:
x1 ðtÞ ffi 1:23395t þ 0:03700t2 þ 0:25299t 3 ; 2
3
ð45Þ
x2 ðtÞ ffi 0:00004t þ 0:00403t þ 0:00036t ;
ð46Þ
x3 ðtÞ ffi 0:00004t þ 0:00448t 2 þ 0:00041t 3 :
ð47Þ
The approximate solutions of the system (37) for different values of a are illustrated in Figs. 1–3.
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
4043
Problem 2. In this problem, we assume the space distribution of vibrations, the system is simplified to a system of two second-order differential equations. The planar non-linear vibration of a simply supported, initially straight slender rod experiencing large deformation and subjected to an axially compressive force at two ends has a number of special models [37,38]. We consider the fractional model of the coupled system in [38] ð2Þ ð2Þ Da1 xðtÞ þ mx3 ðtÞ yðtÞ ¼ f1 ðxðtÞ; Dð1Þ t xðtÞ; Dt xðtÞ; yðtÞ; Dt yðtÞÞ;
ð48Þ
ð2Þ ð2Þ Da2 yðtÞ þ px5 ðtÞ axðtÞ ¼ f2 ðxðtÞ; Dð1Þ t xðtÞ; Dt xðtÞ; yðtÞ; Dt yðtÞÞ;
with initial conditions
xð0Þ ¼ x0 ;
x0 ð0Þ ¼ x00 ;
yð0Þ ¼ y0 ;
y0 ð0Þ ¼ y00 ;
where 1 < ai 6 2; i ¼ 1; 2; 1 and m; p and a are the parameters of the system. It is a system of two coupled fractional differential equations with a fifth-order strong non-linearity. In the equations also the small non-linear functions f1 and f2 exist. Let us consider the system of Eqs. (48) for the parameter values a ¼ 1; m ¼ 1, ¼ 0 and p ¼ 0:54. So, it has the following form
Da1 xðtÞ þ x3 ðtÞ yðtÞ ¼ 0;
ð49Þ
Da2 yðtÞ þ 0:54x5 ðtÞ xðtÞ ¼ 0; with initial conditions
xð0Þ ¼ 1:2;
x0 ð0Þ ¼ 0;
yð0Þ ¼ 1:6412;
y0 ð0Þ ¼ 0:
We apply the suggested method with N ¼ 3 and approximate the solutions xðtÞ and yðtÞ as follows
xðtÞ ffi
3 X c1j T j ðtÞ;
yðtÞ ffi
j¼0
3 X c2j T j ðtÞ:
ð50Þ
j¼0
Eq. (50) can be written in the following matrix form
U ¼ TA;
ð51Þ T
T
where T ¼ ½ Tðx0 Þ Tðx3 Þ , A ¼ ½ c10 c11 c12 c13 c20 c21 c22 c23 . Using the procedure in Section 5, the main matrix equation for this problem is
2 T þ P5 ðL 4 TA ¼ F: þP2 T M þP3 T þ P4 ðL AÞ AÞ ½P1 T M ða1 Þ
ð52Þ
ða2 Þ
And the matrix equations for the conditions of this problem are:
Tð0ÞA1 ¼ 1:2;
A1 ¼ 0; Tð0Þ M ð1Þ
Tð0ÞA2 ¼ 1:6412;
A2 ¼ 0: Tð0Þ M ð1Þ
ð53Þ
We obtain a system of 8 algebraic equations, the last four of them are from initial conditions. After solving this system using Newton iteration method at ai ¼ 1; i ¼ 1; 2, we obtain:
Fig. 1. The approximate solution x1 ðtÞ for different values of a.
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Fig. 2. The approximate solution x2 ðtÞ for different values of a.
Fig. 3. The approximate solution x3 ðtÞ for different values of a.
c10 ¼ 1:73680;
c11 ¼ 0:97176;
c12 ¼ 0:58850;
c13 ¼ 0:15360;
c20 ¼ 2:14762;
c21 ¼ 0:61666;
c22 ¼ 0:07512;
c23 ¼ 0:03513:
Therefore, the approximate solutions of the system (49) take the following form
xðtÞ ffi
3 X c1j T j ðtÞ ¼ 1:2 2:66457t2 þ 4:9153t 3 ;
ð54Þ
j¼0
yðtÞ ffi
3 X c2j T j ðtÞ ¼ 2:28734t 2 1:12428t 3 :
ð55Þ
j¼0
Also, after solving this system using Newton iteration method at ai ¼ 0:95; i ¼ 1; 2, we obtain
c10 ¼ 1:86669;
c11 ¼ 1:20279;
c12 ¼ 0:72442;
c13 ¼ 0:18832;
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
c20 ¼ 2:18121;
c21 ¼ 0:66009;
c22 ¼ 0:08412;
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c23 ¼ 0:03591:
Therefore, the approximate solutions of the system (49) take the following form
xðtÞ ffi
3 X c1j T j ðtÞ ¼ 1:2 3:244t 2 þ 6:02622t 3 ;
ð56Þ
j¼0
yðtÞ ffi
3 X c2j T j ðtÞ ¼ 1:64120 þ 2:39886t 2 1:15060t 3 :
ð57Þ
j¼0
The approximate solutions of the system (49) for different values of a are illustrated in Figs. 4 and 5. In Problem 3, which introduced by Akyuz and Sezer [39] at a ¼ 1 is an application to show the accuracy of their approach on systems of high order differential equations with variable coefficients. We consider here their fractional models in order to show the accuracy of the proposed implementation on fractional differential equations with variable coefficients. Problem 3. Consider the following linear system of FDEs with variable coefficients
"
0 4t 3 0
0
"
# u
ða4 Þ
0 þ 4 3 t 3
# " t2 ða3 Þ 4t2 þ 1 2 u 0 t 2
# 2t 0 uða2 Þ þ 2t t
0 0 4t uða1 Þ þ u¼ ; 1 1 0 6t 0
ð58Þ
such that, i 1 < ai 6 i; i ¼ 1; 2; 3; 4, and with the following conditions
u1 ð1Þ ¼ 8;
u001 ð0Þ ¼ 2;
u2 ð1Þ ¼ 3;
u2 ð0Þ ¼ 2;
u02 ð0Þ ¼ 1:
The exact solution to this example at ai ¼ i is
u1 ðtÞ ¼ 4t 3 þ t 2 þ 2t þ 1;
u2 ðtÞ ¼ t4 þ 2t 3 þ 3t2 t 2:
ð59Þ
We apply the suggested method with N ¼ 4 and approximate the solutions ui ðtÞ; i ¼ 1; 2 as follows
ui ðtÞ ffi
4 X cij T j ðtÞ:
ð60Þ
j¼0
Eq. (60) can be written in the following matrix form
U ¼ TA;
ð61Þ T
T
where T ¼ ½ Tðx0 Þ Tðx4 Þ ; A ¼ ½ c10 c11 c12 c13 c14 c20 c21 c22 c23 c24 . Using the procedure in Section 6, the main matrix equation for this problem is
þP3 T M þP2 T M þP1 T M þP0 TA ¼ F: ½P4 T M ða4 Þ
ða3 Þ
ða2 Þ
ða1 Þ
And the matrix equations for the conditions of this problem are
Fig. 4. The approximate solution xðtÞ for different values of a.
ð62Þ
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M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
Fig. 5. The approximate solution yðtÞ for different values of a.
Tð1ÞA1 ¼ 8;
Tð0Þ M A1 ¼ 2;
Tð1ÞA2 ¼ 3;
ð2Þ
Tð0ÞA2 ¼ 2;
Tð0Þ M A2 ¼ 1: ð1Þ
ð63Þ
We obtain a system of 10 algebraic equations, the last five of them are from the conditions. After solving using conjugate gradient method at ai ¼ i; i ¼ 1; 2; 3; 4, we obtain:
c10 ¼ 3:62500; c20 ¼ 0:47656;
c11 ¼ 3:3750; c21 ¼ 2:375;
c12 ¼ 0:875;
c13 ¼ 0:125;
c22 ¼ 0:96875;
c14 ¼ 0;
c23 ¼ 0:12500;
c24 ¼ 0:00781:
Therefore, the approximate solutions of the system (58) take the following form
u1 ðtÞ ffi
4 X c1j T j ðtÞ ¼ 4t 3 þ t 2 þ 2t þ 1;
ð64Þ
j¼0
u2 ðtÞ ffi
4 X c2j T j ðtÞ ¼ t 4 þ 2t 3 þ 3t 2 t 2; j¼0
which coincides with the exact solution for this example at ai ¼ i; i ¼ 1; 2; 3; 4.
Fig. 6. The approximate solution u1 ðtÞ for different values of a.
ð65Þ
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
4047
The approximate solutions of the system (58) for different values of a are illustrated in Figs. 6 and 7. Problem 4. In this problem, which introduced by Momani and Al-Khaled [40] as a fractional model to show the accuracy of their approach. For the special case when a ¼ 1, this system represents a non-linear reaction and was found in [41]. Consider the following non-linear system of FDEs:
Da1 u1 ¼ u1 ; Da2 u2 ¼ u1 u22 ;
ð66Þ
Da3 u3 ¼ u22 ; where 0 < ai 6 1; i ¼ 1; 2; 3 with the following initial conditions u1 ð0Þ ¼ 1; u2 ð0Þ ¼ 0; u3 ð0Þ ¼ 0. We apply the suggested method with N ¼ 3, and approximate the solutions ui ðtÞ as follows
ui ðtÞ ffi
3 X cij T j ðtÞ;
i ¼ 1; 2; 3:
ð67Þ
j¼0
Eq. (67) can be written in the matrix form U ¼ TA. Using the procedure in Section 5, the main matrix equation for this problem is
þP2 T M þP3 T M þP4 T þ P5 ðL AÞTA ½P1 T M ¼ F: ða1 Þ
ða2 Þ
ða3 Þ
ð68Þ
And the matrix equations for initial conditions for this problem are
Tð0ÞA1 ¼ 1;
Tð0ÞA2 ¼ 0;
Tð0ÞA3 ¼ 0:
ð69Þ
From Eqs. (68) and (69), we obtain a system of 12 algebraic equations, the last three of them are from initial conditions. After solving at ai ¼ 1; i ¼ 1; 2; 3, we obtain the approximate solutions of the system (66) for different values of a and illustrated in Figs. 8–10. From the previous introduced examples in this section, it is clear that when N increased the approximate solution improved as the errors are decreased which is the main advantage of the proposed matrix method. The algorithms introduced in this paper can be suited for handling linear and non-linear fractional order differential equations with boundary conditions. This approach can also reformulated using Legendre polynomials and Jacobi polynomials in the future work. Remark. The method can also be extended to be applicable to systems of fractional partial differential equations in the future work by considering that the solution in two dimensions can be written in this form
uðx; yÞ ¼
N X N X cr;s T r;s ðx; yÞ; r¼0 s¼0
Fig. 7. The approximate solution u2 ðtÞ for different values of a.
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Fig. 8. The approximate solution u1 ðtÞ for different values of a.
Fig. 9. The approximate solution u2 ðtÞ for different values of a.
such that, N is any positive integer with N P s and cr;s ; r ¼ 0; 1; . . . ; N; s ¼ 0; 1; . . . ; N are unknown Chebyshev coefficients to be determined. Or in the matrix form 0 1 uðx; yÞ ¼ T x AT y ; c0;0 c1;0 cN;0 B c0;1 c1;1 cN;1 C B C T where, T x ¼ ½ T 0 ðxÞ T 1 ðxÞ T N ðxÞ , T y ¼ ½ T 0 ðyÞ T 1 ðyÞ T N ðyÞ and A ¼ B . .. .. C. @ .. . . A c0;N c1;N cN;N 7. Conclusion and remarks In this paper, an accurate Chebyshev approximation method has been presented for the solution of systems of higher order fractional differential equations. These systems of equations are transformed to systems of algebraic equations which provided a matrix representation. This new proposed method is non-differentiable, non-integral, straightforward and well adapted to the computer implementation. The solution is expressed as a truncated Chebyshev series and so it can be easily evaluated for arbitrary values of time using any computer program without any computational effort. From illustrative examples, we can conclude that this matrix approach can obtain very accurate and satisfactory results. An important feature
M.M. Khader et al. / Applied Mathematical Modelling 37 (2013) 4035–4050
4049
Fig. 10. The approximate solution u3 ðtÞ for different values of a.
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