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Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations
ARTICLE IN PRESS
Journal of the Franklin Institute 342 (2005) 688–701 www.elsevier.com/locate/jfranklin
Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations Ays-egu¨l Akyu¨z-Das-cıog˘lua, , Mehmet Sezerb a
Department of Mathematics, Faculty of Science, Pamukkale University, Denizli, Turkey b Department of Mathematics, Faculty of Science, Mug˘la University, Mug˘la, Turkey
Abstract A Chebyshev collocation method, an expansion method, has been proposed in order to solve the systems of higher-order linear integro-differential equations. This method transforms the IDE system and the given conditions into the matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the systems of differential and integral equations. r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Chebyshev polynomials and series; System of integro-differential equations; Fredholm and Volterra systems
1. Introduction Most of the equations in practice have a fairly simple form, and they can usually be reduced to a system of integral equations. Since few of these equations can be solved explicitly, it is often necessary to resort to numerical techniques which are Corresponding author.
E-mail address: [email protected] (A. Akyu¨z-Das-cıog˘lu). 0016-0032/$30.00 r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.04.001
ARTICLE IN PRESS A. Akyu¨z-Das- cıog˘lu, M. Sezer / Journal of the Franklin Institute 342 (2005) 688–701
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appropriate combinations of numerical integration and interpolation [1,2]. Furthermore, there are also expansion methods for integro-differential equations such as El-gendi’s, Wolfe’s and Galerkin methods [3]. Conversely, the solution of the first-order IDE system which occurs in physics [4], biology [5] and engineering [6,7] is based on the numerical integration methods such as Euler–Chebyshev [8] and Runge–Kutta [9] methods, and also in a recent research, the first-order linear Fredholm IDE system is solved by using rationalized Haar functions method [10] and by Galerkin methods with hybrid functions [11]. Nonetheless, there is not a research on the solution methods of the higher-order IDE systems. Thus, the presented method which is an expansion method has been proposed to obtain approximate solution and also analytical solution of the systems of higher-order linear differential, integral and integro-differential equations. In this paper, we will consider systems of k linear IDEs of Fredholm–Volterra type in the form Z 1X Z x X m X k k k X pnij ðxÞyðnÞ ðxÞ ¼ g ðxÞ þ F ðx; tÞy ðtÞ dt þ K ij ðx; tÞyj ðtÞ dt, ij i j j 1 j¼1
n¼0 j¼1
1 j¼1
i ¼ 1; 2; . . . ; k; 1pxp1
ð1Þ
under the mixed conditions m 1 X
n ðnÞ n ðnÞ anj yðnÞ j ð1Þ þ bj yj ð1Þ þ cj yj ðcÞ ¼ kj ;
j ¼ 1; 2; . . . ; k; 1oco1,
(2)
n¼0
where kj , anj , bnj and cnj are real-valued column matrices with m 1 dimension and yðnÞ indicates the nth-order derivative and yð0Þ j ðxÞ ¼ yj ðxÞ. There is not a general method to find analytical or numerical solutions of this system. Therefore, a Chebyshev collocation method [12], which is given for the solution of the linear integro-differential equations, is developed for the system of differential equations by Akyu¨z and Sezer [13] and the system of integral equations by Akyu¨z-Das- cıog˘lu [14] and it is, thereafter, applied to the mentioned Fredholm–Volterra IDE system. The aim of this study is to get solution as truncated Chebyshev series defined by yj ðxÞ ¼
N X
ajr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 1pxp1,
(3)
r¼0
where T r ðxÞ denotes the Chebyshev polynomials of the first kind, ajr are unknown Chebyshev coefficients, and N is chosen any positive integer such that NXm. The systems (1) and (2) can always be written as a matrix equation, respectively Z 1 Z x m X pn ðxÞyðnÞ ðxÞ ¼ gðxÞ þ Fðx; tÞyðtÞ dt þ Kðx; tÞyðtÞ dt, (4) n¼0 m1 X n¼0
1
an yðnÞ ð1Þ þ bn yðnÞ ð1Þ þ cn yðnÞ ðcÞ ¼ k.
1
(5)
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This is accomplished by introducing k-dimensional column matrix ðnÞ ðnÞ T yðnÞ ðxÞ ¼ ½yðnÞ 1 ðxÞ; y2 ðxÞ; . . . ; yk ðxÞ
and an , bn and cn are diagonal matrices with entries anj , bnj and cnj , j ¼ 1; 2; . . . ; k and the other matrices are as follows: k ¼ ½kj ;
gðxÞ ¼ ½gi ðxÞ;
pn ðxÞ ¼ ½pni;j ðxÞ;
Fðx; tÞ ¼ ½F i;j ðx; tÞ;
Kðx; tÞ ¼ ½K i;j ðx; tÞ,
n ¼ 0; 1; 2; . . . ; m; i; j ¼ 1; 2; . . . ; k.
To obtain the Chebyshev polynomial solution of IDE systems, it is assumed that pn ðxÞ; gðxÞ; Fðx; tÞ and Kðx; tÞ are defined on ½1; 1. If the integrals are bounded in the range ½0; 1, then solution can be obtained by means of the shifted Chebyshev polynomials T r ðxÞ. However, it is not considered in this study since any finite interval can be transformed to interval ½1; 1, which is domain of the Chebyshev polynomials of the first kind.
2. Fundamental relations It is supposed that the solution yj ðxÞ and its derivatives have truncated Chebyshev series expansions of the form yðnÞ j ðxÞ ¼
N X
aðnÞ jr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 1pxp1.
(6)
r¼0
Then functions defined in relation (3) can be written in the matrix form yj ðxÞ ¼ TðxÞAj , where TðxÞ ¼ ½T 0 ðxÞ; T 1 ðxÞ; . . . ; T N ðxÞ;
Aj ¼ ½aj0 ; aj1 ; . . . ; ajN T .
Similarly, the matrix representations of functions (6) become ðnÞ yðnÞ j ðxÞ ¼ TðxÞAj
or using the relation between the Chebyshev coefficient matrices Aj and AðnÞ j [15], n n yðnÞ j ðxÞ ¼ 2 TðxÞM Aj
ARTICLE IN PRESS A. Akyu¨z-Das- cıog˘lu, M. Sezer / Journal of the Franklin Institute 342 (2005) 688–701
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so that 2
N 2
3
0
3 2
0
5 2
60 6 6 60 6 M¼6. 6 .. 6 6 40
1 2
0 0 .. . 0
2 0 .. . 0
0 4 3 0 .. .. . . 0 0
0 5 .. . 0
07 7 7 N7 7 .. 7 . 7 7 7 N5
0
0
0
0 0
0
0
2
0
0
60 6 6 60 6 M¼6. 6 .. 6 6 40 0
1 2
0
3 2
0
5 2
0
2
0 4
0
0 .. .
0 .. .
3 0 .. .. . .
5 .. .
0 0
0 0
0 0 0 0
0 0
0
for odd N,
ðNþ1ÞðNþ1Þ
3
N7 7 7 07 7 .. 7 . 7 7 7 N5 0
for even N.
ðNþ1ÞðNþ1Þ
ðnÞ
In this case, the matrix y ðxÞ is reduced to yðnÞ ðxÞ ¼ 2n TðxÞMn A;
n ¼ 0; 1; 2; . . . ; m,
(7)
where 2
TðxÞ
0
6 6 0 TðxÞ 6 TðxÞ ¼ 6 .. 6 .. 6 . . 4 0 0 2 3 A1 6 7 6 A2 7 6 7 7 A¼6 6 .. 7 . 6 . 7 4 5 Ak k1
..
.
3
2
7 0 7 7 7 .. 7 . 7 5 TðxÞ
6 6 6 M ¼6 6 4
0
;
n
kk
Mn
0
0
3
0 .. .
Mn .. .
.. .
0 .. .
7 7 7 7 7 5
0
0
Mn
,
kk
Similarly, it is assumed that the kernel functions F ij ðx; tÞ and K ij ðx; tÞ can be expanded to univariate Chebyshev series in the forms F ij ðxs ; tÞ ¼
N X 00
f ijr ðxs ÞT r ðtÞ;
K ij ðxs ; tÞ ¼
r¼0
N X 00
kijr ðxs ÞT r ðtÞ
r¼0
in which a summation symbol with double primes denotes a sum with first and last terms halved, xs are the Chebyshev collocation points defined by xs ¼ cosðsp=NÞ;
s ¼ 0; 1; . . . ; N,
ARTICLE IN PRESS A. Akyu¨z-Das- cıog˘lu, M. Sezer / Journal of the Franklin Institute 342 (2005) 688–701
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which are turning points of the polynomial of degree N. Chebyshev polynomial T N ðxÞ takes the extreme values 1 at these points. The Chebyshev coefficients f ijr ðxs Þ and kijr ðxs Þ are determined by Clenshaw and Curtis [16]. Then the matrix representations of F ij ðxs ; tÞ and K ij ðxs ; tÞ become F ij ðxs ; tÞ ¼ F ij ðxs ÞTðtÞT ;
K ij ðxs ; tÞ ¼ K ij ðxs ÞTðtÞT ,
(8)
where F ij ðxs Þ ¼ K ij ðxs Þ ¼
h
ij ij 1 ij 1 ij 2f 0 ðxs Þ; f 1 ðxs Þ; . . . ; f N1 ðxs Þ; 2f N ðxs Þ
h
i
ij ij 1 ij 1 ij 2k0 ðxs Þ; k 1 ðxs Þ; . . . ; k N1 ðxs Þ; 2kN ðxs Þ
,
i .
3. The method of the solution To obtain the Chebyshev polynomial solution of the system (1) under the mixed conditions (2), it is used in the following matrix method. This method is based on computing the Chebyshev coefficients by means of the Chebyshev collocation points. Substituting the Chebyshev collocation points into Eq. (4) it is obtained a new matrix equation in the form m X
Pn YðnÞ ¼ G þ I þ J
(9)
n¼0
in which I and J denote the Fredholm and Volterra integral parts of Eq. (4), respectively and 3 3 3 3 2 2 2 2 ðnÞ Iðx0 Þ Jðx0 Þ gðx0 Þ y ðx0 Þ 7 7 7 7 6 6 6 6 ðnÞ 6 gðx1 Þ 7 6 Iðx1 Þ 7 6 Jðx1 Þ 7 6 y ðx1 Þ 7 7 7 7 7 6 6 6 6 ðnÞ Y ¼6 7; G ¼ 6 .. 7; I ¼ 6 .. 7; J ¼ 6 .. 7, .. 7 6 . 7 6 . 7 6 . 7 6 . 5 5 5 5 4 4 4 4 gðxN Þ IðxN Þ JðxN Þ yðnÞ ðxN Þ 2 6 6 6 Pn ¼ 6 6 4
pn ðx0 Þ
0
0
0 .. .
pn ðx1 Þ .. .
.. .
0 .. .
0
0
...
pn ðxN Þ
3 7 7 7 7. 7 5
Now, let us seek the representations of Y, I and J matrices in terms of Chebyshev coefficient matrix A. Firstly, substituting the Chebyshev collocation points into relation (7) we have YðnÞ ¼ 2n TMn A,
(10)
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where T ¼ ½Tðx0 Þ; Tðx1 Þ; . . . ; TðxN ÞT . Similarly, putting the Chebyshev collocation points into Fredholm and Volterra integral parts I i ðxÞ and J i ðxÞ of the system (1), and then using the relation (8) and also the following relations: Z 1 Z 1 T TðtÞ TðtÞ dt ¼ T q ðtÞT r ðtÞ dt ¼ ½zqr ; q; r ¼ 0; 1; . . . ; N, Z¼ 1
1
whose entries are given by Fox and Parker [17], and Z xs Z xs Zðxs Þ ¼ TðtÞT TðtÞ dt ¼ T q ðtÞT r ðtÞ dt ¼ ½zqr ðxs Þ; 1
q; r; s ¼ 0; 1; . . . ; N,
1
whose entries are computed by Akyu¨z [18], we obtain I i ðxs Þ ¼
k X
F ij ðxs ÞZAj ;
J i ðxs Þ ¼
j¼1
k X
K ij ðxs ÞZðxs ÞAj ;
i ¼ 1; 2; . . . ; k
j¼1
or compact notation Iðxs Þ ¼ Fðxs ÞZA;
Jðxs Þ ¼ Kðxs ÞZðxs ÞA;
s ¼ 0; 1; . . . ; N
(11)
in which Iðxs Þ and Jðxs Þ are k 1 matrices with entries I i ðxs Þ and J i ðxs Þ, respectively, and others can be written by the blocked matrices 3 2 2 3 F 11 ðxs Þ F 12 ðxs Þ F 1k ðxs Þ Z 0
0 7 6 6 0 Z
0 7 6 F 21 ðxs Þ F 22 ðxs Þ F 2k ðxs Þ 7 6 7 7 6 6 . . . 7 , ; Z ¼ Fðxs Þ ¼ 6 7 . .. .. .. 6 . . .. 7 7 6 . . . 4 .. .. 5 5 4 0 0 Z kk F k1 ðxs Þ F k2 ðxs Þ F kk ðxs Þ 2
K 11 ðxs Þ
K 12 ðxs Þ
K 1k ðxs Þ
3
7 6 6 K 21 ðxs Þ K 22 ðxs Þ K 2k ðxs Þ 7 7 6 7, Kðxs Þ ¼ 6 .. .. .. 7 6 7 6 . . . 5 4 K k1 ðxs Þ K k2 ðxs Þ K kk ðxs Þ 3 2 0
0 Zðxs Þ 7 6 6 0 Zðxs Þ
0 7 7 6 7 Zðxs Þ ¼ 6 .. 7 . .. .. 6 .. 7 6 . . . . 5 4 0 0
Zðxs Þ kk
In this way, the matrices I and J are acquired in the forms
I = FZA, J = KZA ,
(12)
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0 .. .
K ( x1 ) .. .
F( x N )
0
0
.. .
F( x1 ) , K= .. .
0 0
.
0
Z( x 0 ) , Z=
..
F=
K ( x0 )
.. .
F( x0 )
.. .
where
K(xN )
Z( x1) . .. . Z( x N )
After substitution of relations (10) and (12) into Eq. (9), the expression
n =0
A=G
(13)
Σ 2 n Pn T M n − FZ − KZ
m
is obtained. This is the fundamental matrix equation for the solution of Fredholm–Volterra IDE systems. Briefly this equation can also be written in the form (14)
WA ¼ G,
which corresponds to a system of kðN þ 1Þ linear algebraic equations with the unknown Chebyshev coefficients so that
Σ 2 n Pn T M n − FZ − KZ
n =0
m
W=
[ ]
= wi , j , i, j = 1, 2,... , k ( N + 1) .
Finally, by using the relation (7) at points 1, 1 and c, the matrix representation of mixed conditions (5) which depends on the Chebyshev coefficient matrix A is formed m1 X
2n ðan Tð1Þ þ bn Tð1Þ þ cn TðcÞÞMn A ¼ k
n¼0
and defining V¼
m 1 X
2n ðan Tð1Þ þ bn Tð1Þ þ cn TðcÞÞMn ,
n¼0
the fundamental matrix form of the conditions becomes VA ¼ k.
(15)
Consequently, replacing the rows of the matrices V and k by the rows of the matrices W and G, respectively, we have ~ ~ ¼ G. WA
(16)
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For convenience, last rows of the matrix are replaced. If the number of conditions is mk, the augmented matrix of the above system is as follows: 3 2 w1;2 ... w1;kðNþ1Þ ; g1 ðx0 Þ w1;1 6 w2;1 w2;2 ... w2;kðNþ1Þ ; g2 ðx0 Þ 7 7 6 7 6 7 6 .. .. .. .. .. 7 6 . . . . . 7 6 7 6 wk;1 wk;2 ... wk;kðNþ1Þ ; gk ðx0 Þ 7 6 7 6 6 wkþ1;1 wkþ1;2 ... wkþ1;kðNþ1Þ ; g1 ðx1 Þ 7 7 6 7 6 . . . 7 6 ~ ¼6 ~ G .. .. .. ½W; 7. 7 6 7 6w 6 kðNmþ1Þ;1 wkðNmþ1Þ;2 . . . wkðNmþ1Þ;kðNþ1Þ ; gk ðxNm Þ 7 7 6 7 6 v1;1 v1;2 ... v1;kðNþ1Þ ; l1 7 6 7 6 v2;1 v2;2 ... v2;kðNþ1Þ ; l2 7 6 7 6 7 6 .. .. .. .. .. 7 6 . . . . . 5 4 vmk;1 vmk;2 ... vmk;kðNþ1Þ ; lmk However, we do not have to replace the last rows. For example, if the matrix W is singular, then the rows that have the same factor or all zero are replaced. Solving this system by usual methods, the Chebyshev coefficients are determined, and therefore, the finite Chebyshev series approach of IDE system under the mixed conditions is obtained. The number of the conditions must be at most mk to have the unique solution, since the integro-differential system has mth-order derivative at most and k unknown function. On the other hand, the unique solution can sometimes be obtained using fewer conditions to the problem. Note that, if pi ðxÞ ¼ 0; i ¼ 1; 2; . . . ; m, Eq. (4) is reduced to the system of linear integral equations, and then only Eq. (14) is used to find the solution of IE systems. When the domain of the problem is [0, 1], the solution is sought in terms of shifted Chebyshev polynomials T r ðxÞ in the form yj ðxÞ ¼
N X
a jr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 0pxp1,
r¼0
where T r ðxÞ ¼ T r ð2x 1Þ. Following the previous procedure and using the collocation points defined by
sp 1 1 þ cos ; s ¼ 0; 1; . . . ; N xs ¼ 2 N we obtain the fundamental matrix equation for the FVIDE system as m
n =0
Σ 4 n Pn T* M n − F * Z* − K * Z *
A* = G
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and the matrix forms of the conditions become m1 X
4n ðan T ð0Þ þ bn T ð1Þ þ cn T ðcÞÞMn A ¼ k.
n¼0
4. Illustrations In this section, we will consider five examples of the systems of Fredholm, Volterra and Fredholm–Volterra type to illustrate the usage of the presented method. All results were computed using a program written in Mathcad 2001 Professional. Example 1. Consider the system of second order Fredholm IDEs: Z 1 00 2 0 0 3 2 y1 þ x y1 þ 3y1 þ y2 4y2 ¼ 2x 2x 5x 18 þ ð3y1 þ 2xy2 Þ dt, 20 y01 2y1 þ y002 2y02 þ y2 ¼ 2x2 x 5 þ 3
Z
1 1
ððt xÞy1 þ 5t2 y2 Þ dt 1
with conditions y01 ð1Þ ¼ 3; y1 ð1Þ ¼ 0; y2 ð1Þ ¼ 2; y2 ð0Þ ¼ 4. Fundamental matrix equation in Section 3 for this system is ð4P2 TM2 þ 2P1 TM þ P0 T FZÞA ¼ G. When approximated the solution by truncated Chebyshev series for NX2, Chebyshev coefficients are found as a10 ¼ 12; a20 ¼ 4;
a11 ¼ 1;
a12 ¼ 12;
a21 ¼ 2;
a22 ¼ 0;
a3;4;... ¼ 0, a3;4;... ¼ 0.
Therefore the solution of this Fredholm system is y1 ðxÞ ¼ x2 þ x;
y2 ðxÞ ¼ 2x þ 4,
which is the exact solution. If we take three of the conditions, we also obtain the same result for small N. For greater value of N, the Chebyshev coefficients are getting worse and worse. Furthermore, if no condition is used, we obtain the same result for small N. For example, taking N ¼ 10, the Chebyshev coefficients are found without error by using four conditions, with 1014 error by using three conditions and with 1013 error by using no conditions.
Example 2. Let us consider a system of linear Volterra IDEs with two unknowns: Z x 1 3 0 4 2 ðy1 3xy2 Þ dt, y1 xy1 þ y2 ¼ x 9x x þ 1 þ 2 2 1 Z x x2 y1 þ y02 xy2 ¼ 4x þ ðð2x þ tÞy1 þ 3t2 y2 Þ dt 1
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under the conditions y1 ð0Þ ¼ 0, y2 ð0Þ ¼ 1. The fundamental matrix equation of this system is ð2P1 TM þ P0 T KÞA ¼ G. After required operations have been done for Eq. (16), Chebyshev coefficients a10 ¼ 0;
a11 ¼ 0;
a12 ¼ 0;
a13 ¼ 1,
a20 ¼ 0;
a21 ¼ 0;
a22 ¼ 1;
a23 ¼ 0
and thereby the solution y1 ðxÞ ¼ 4x3 3x;
y2 ðxÞ ¼ 2x2 1
are obtained by taking N ¼ 3. Besides, the same result is acquired for N43.
Example 3. Consider a system of third-order linear Volterra IDEs in the interval [0, 1]: ð2Þ x ð2Þ yð3Þ 1 þ e y1 þ y2 þ 2y1 þ y2 ¼ 1 þ coshðxÞ þ x sinðxÞ
y1ð2Þ
cosðxÞyð2Þ 2
y1 ð1Þ ¼ e1 ; yð1Þ 2 ð0Þ ¼ 2;
þ
y1ð1Þ
Z
ðext y1 þ cosðx tÞy2 Þ dt,
0
Z
2
2x
x
xe y1 ¼ x sin ðxÞ þ sinð2xÞ
x
ðexþt y1 þ x cosðtÞy2 Þ dt,
0
y2 ð1Þ ¼ 2 sinð1Þ; y1 ð0Þ ¼ 1;
ð2Þ 1 1 yð1Þ 1 ð2Þ þ y1 ð2Þ ¼ 0,
y2 ð0Þ ¼ 0.
This system has y1 ðxÞ ¼ ex ; y2 ðxÞ ¼ 2 sinðxÞ the exact solution. The fundamental matrix equation of this system is
ð64P3 T M3 þ 16P2 T M2 þ 4P1 T M þ P0 T K Z ÞA ¼ G. Replacing the first two and last four rows of the augmented matrix bW ; Gc obtained above equation by the rows of the matrices for conditions at point x ¼ 1 and the ~ The results obtained from the solution of this ~ ; Gc. others, respectively, we have bW system are given in Tables 1 and 2.
Example 4. Consider a first-order system of Fredholm–Volterra IDEs: y01
þ
xy02
1 þ 3y2 ¼ 20x þ 2x þ x 3 þ 3
x2 y01 y02 þ xy1 þ y2
3
2
Z
1 2
Z
x
ððx tÞy1 þ x y2 Þ dt þ 1
2 ¼ 5x3 15x2 8x þ þ 3
ð4ty1 y2 Þ dt, 1
Z
1
ð3ty1 þ ðt2 4xÞy2 Þ dt þ 1
Z
x
6y1 dt. 1
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Table 1 Comparison of the absolute errors of y1 xi
N ¼3
N¼5
N ¼6
N¼8
N ¼ 10
N ¼ 11
N ¼ 12
0 0.25 0.5 0.75 1
0.00 1.92e3 1.05e3 2.49e4 0.00
0.00 5.09e6 2.98e6 1.74e6 0.00
0.00 2.71e7 3.37e7 2.56e7 0.00
0.00 8.46e10 8.16e10 5.11e10 0.00
0.00 4.86e13 5.90e13 3.81e13 0.00
0.00 2.00e15 1.22e15 0.00 0.00
0.00 0.00 0.00 0.00 0.00
Table 2 Comparison of the absolute errors of y2 xi
N ¼3
N¼5
N ¼6
N¼8
N ¼ 10
N ¼ 11
N ¼ 12
0 0.25 0.5 0.75 1
0.00 1.21e3 2.35e3 1.39e3 0.00
0.00 1.38e6 3.21e6 1.94e6 0.00
0.00 5.68e7 5.82e7 4.46e7 0.00
0.00 4.82e10 7.29e10 6.22e10 0.00
0.00 6.86e13 7.99e13 5.28e13 0.00
0.00 2.39e15 1.78e15 0.00 0.00
0.00 0.00 0.00 0.00 0.00
Let us approximate the solution by Chebyshev series of degree three. Then, Chebyshev collocation points become x1 ¼ 12;
x0 ¼ 1;
x2 ¼ 12;
x3 ¼ 1.
Following the previous procedure, the augmented matrix for Eq. (14) is found as 2
2 1 6 11 0 6 6 6 0:5 0:1667 6 6 6 8:5 0:75 ½W; G ¼ 6 6 2:5 0:5 6 6 6 3:5 0:75 6 6 2 1:6667 4 1
0
4:6667
10:2
3
4
7
12
;
9 2:7083 4:75
11:2 0:8 0:425
8:3333 4 4:3333
0 1:625 0:5
5:8 1:0833 3:9667
8 2:8125 1
; ; ;
1:9583 0:75 4:6667
0 0:425 8:6
3 3:6667 1
2:375 1:5 4
0:25 2:7 7:6667
3:1875 1 12
; ; ;
5
11:2
7:6667
2
7:5333
10
;
58=3
3
52=3 7 7 7 1=6 7 7 7 155=24 7 7. 31=6 7 7 7 7=24 7 7 64=3 7 5 34=3
Solving this system, we have the solution y1 ðxÞ ¼ x2 þ 3x;
y2 ðxÞ ¼ 4x3 1.
Note that any conditions are not used. If some conditions are given such that y01 ð0Þ y1 ð1Þ ¼ 1; 2y2 ð1Þ þ y2 ð1Þ ¼ 1, augmented matrix forms of these conditions are obtained as " # 1 0 1 4 0 0 0 0 ; 1 ½V; k ¼ 0 0 0 0 3 1 3 1 ; 1
ARTICLE IN PRESS A. Akyu¨z-Das- cıog˘lu, M. Sezer / Journal of the Franklin Institute 342 (2005) 688–701
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and replaced the last two rows of the above augmented matrix. Then, it is obtained the same solution by solving the new system.
Example 5. Finally, consider a system of Fredholm–Volterra IDEs: ð2Þ ð1Þ x x x yð3Þ þ 1 þ y2 þ y1 þ e y2 ¼ 1 xe 2e
Z
Z
1
x
ext y2 dt,
ðsinhðtÞy1 þ coshðtÞy2 Þ dt þ 1
ð1Þ x ð3Þ x yð3Þ þ xexþ2 þ 2 e y1 y2 þ xy2 þ y1 ¼ 6 x xe
1
Z
1
ð3et y1 xexþ1 y2 Þ dt þ
Z
1
x
ðty1 þ et y2 Þ dt 1
with the mixed conditions y1 ð1Þ ¼ e1 ;
y2 ð1Þ ¼ e;
y1 ð0Þ ¼ 1;
y2 ð0Þ ¼ 1;
y1 ð1Þ ¼ e;
y2 ð1Þ ¼ e.
The exact solution of this system is y1 ðxÞ ¼ ex , y2 ðxÞ ¼ ex . The computational results, which are obtained by replacing the first two and the last four rows of the augmented matrix, are given in Table 3 for different N. Besides, the absolute errors are all zero for N ¼ 14. However, if the last six rows of the augmented matrix are replaced, the absolute errors are found almost 1014 for N ¼ 14.
5. Conclusions An interesting feature of this method is that when IDE system has linearly independent polynomial solution of degree N or less than N, the method can be used for finding the analytical solution. Besides, it is seen that when the truncation limit N is increased, there exists a solution, which is closer to the exact solution. Problems defined in a finite range [a,b] can be solved by the Chebyshev collocation method since by means of the linear transformation, any finite range can be
Table 3 The numerical results of Example 5 xi
1 0.75 0.5 0.25 0 0.25 0.5 0.75 1
Absolute errors of y1
Absolute errors of y2
N¼5
N ¼ 10
N ¼ 13
N¼5
N ¼ 10
N ¼ 13
0.00 6.88e4 3.66e4 8.99e5 0.00 3.27e5 8.35e5 9.63e5 0.00
0.00 3.72e10 3.17e10 1.93e11 0.00 1.21e10 1.18e10 7.95e11 0.00
0.00 2.66e14 1.09e14 7.11e15 0.00 2.89e15 3.89e15 3.89e15 0.00
0.00 7.25e4 3.87e4 8.61e5 0.00 3.24e5 1.15e4 1.47e4 0.00
0.00 2.61e10 2.07e10 2.16e11 0.00 1.29e10 1.01e10 9.18e11 0.00
0.00 2.80e14 1.21e14 8.22e15 0.00 4.00e15 5.11e15 5.77e15 0.00
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transformed into the range ½1; 1, which is the domain of the Chebyshev polynomials T r ðxÞ. If the range is ½0; 1, then, it can be assumed that the solution functions can be expressed by shifted Chebyshev polynomials T r ðxÞ, but some modifications are required for the fundamental matrix equations. The presented method is based on computing the coefficients in the Chebyshev expansion of solution of a linear IDE system, and is valid when the matrix functions pn ðxÞ, gðxÞ, Fðx; tÞ and Kðx; tÞ are defined in ½1; 1 and the kernel functions have a Chebyshev series expansion in this range. Moreover, if the kernel functions F ij ðx; tÞ and K ij ðx; tÞ converge rapidly, it can be obtained better results. A considerable advantage of the method is that it allows us to make use of the computer because this Chebyshev method transforms the problem into the matrix equation, which is a linear algebraic system. Therefore, Chebyshev coefficients of the solution are found very easily by using the computer programs such as Qbasic, Cþþ , etc. Furthermore, the values of the solution at the collocation points are evaluated with the aid of the computer programs without any computational effort.
References [1] C.T.H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math. 125 (2000) 217–249. [2] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, PA, 1985. [3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985. [4] F. Bloom, Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory, J. Math. Anal. Appl. 73 (1980) 524–542. [5] K. Holmaker, Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal. 24 (1) (1993) 116–128. [6] M.A. Abdou, Fredholm–Volterra integral equation of the first kind and contact problem, Appl. Math. Comput. 125 (2002) 177–193. [7] L.K. Forbes, S. Crozier, D.M. Doddrell, Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math. 57 (2) (1997) 401–425. [8] P.J. van der Houwen, B.P. Sommeijer, Euler–Chebyshev methods for integro-differential equations, Appl. Numer. Math. 24 (1997) 203–218. [9] W.H. Enright, M. Hu, Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay, Appl. Numer. Math. 24 (1997) 175–190. [10] K. Maleknejad, F. Mirzae, S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput. 155 (2004) 317–328. [11] K. Maleknejad, M. Tavassoli Kajani, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159 (2004) 603–612. [12] A. Akyu¨z, M. Sezer, A Chebyshev collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 72 (4) (1999) 491–507. [13] A. Akyu¨z, M. Sezer, Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput. 144 (2003) 237–247. [14] A. Akyu¨z-Das-cıog˘lu, Chebyshev polynomial solutions of systems of linear integral equations, Appl. Math. Comput. 151 (2004) 221–232. [15] M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol. 27 (4) (1996) 607–618.
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[16] C.W. Clenshaw, A.R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205. [17] L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968. [18] A. Akyu¨z, Chebyshev Collocation method for solution of linear integro-differential equations, M.Sc. Thesis, Dokuz Eylu¨l University, Graduate School of Natural and Applied Sciences, 1997.
Journal of the Franklin Institute 342 (2005) 688–701 www.elsevier.com/locate/jfranklin
Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations Ays-egu¨l Akyu¨z-Das-cıog˘lua, , Mehmet Sezerb a
Department of Mathematics, Faculty of Science, Pamukkale University, Denizli, Turkey b Department of Mathematics, Faculty of Science, Mug˘la University, Mug˘la, Turkey
Abstract A Chebyshev collocation method, an expansion method, has been proposed in order to solve the systems of higher-order linear integro-differential equations. This method transforms the IDE system and the given conditions into the matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the systems of differential and integral equations. r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Chebyshev polynomials and series; System of integro-differential equations; Fredholm and Volterra systems
1. Introduction Most of the equations in practice have a fairly simple form, and they can usually be reduced to a system of integral equations. Since few of these equations can be solved explicitly, it is often necessary to resort to numerical techniques which are Corresponding author.
E-mail address: [email protected] (A. Akyu¨z-Das-cıog˘lu). 0016-0032/$30.00 r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.04.001
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appropriate combinations of numerical integration and interpolation [1,2]. Furthermore, there are also expansion methods for integro-differential equations such as El-gendi’s, Wolfe’s and Galerkin methods [3]. Conversely, the solution of the first-order IDE system which occurs in physics [4], biology [5] and engineering [6,7] is based on the numerical integration methods such as Euler–Chebyshev [8] and Runge–Kutta [9] methods, and also in a recent research, the first-order linear Fredholm IDE system is solved by using rationalized Haar functions method [10] and by Galerkin methods with hybrid functions [11]. Nonetheless, there is not a research on the solution methods of the higher-order IDE systems. Thus, the presented method which is an expansion method has been proposed to obtain approximate solution and also analytical solution of the systems of higher-order linear differential, integral and integro-differential equations. In this paper, we will consider systems of k linear IDEs of Fredholm–Volterra type in the form Z 1X Z x X m X k k k X pnij ðxÞyðnÞ ðxÞ ¼ g ðxÞ þ F ðx; tÞy ðtÞ dt þ K ij ðx; tÞyj ðtÞ dt, ij i j j 1 j¼1
n¼0 j¼1
1 j¼1
i ¼ 1; 2; . . . ; k; 1pxp1
ð1Þ
under the mixed conditions m 1 X
n ðnÞ n ðnÞ anj yðnÞ j ð1Þ þ bj yj ð1Þ þ cj yj ðcÞ ¼ kj ;
j ¼ 1; 2; . . . ; k; 1oco1,
(2)
n¼0
where kj , anj , bnj and cnj are real-valued column matrices with m 1 dimension and yðnÞ indicates the nth-order derivative and yð0Þ j ðxÞ ¼ yj ðxÞ. There is not a general method to find analytical or numerical solutions of this system. Therefore, a Chebyshev collocation method [12], which is given for the solution of the linear integro-differential equations, is developed for the system of differential equations by Akyu¨z and Sezer [13] and the system of integral equations by Akyu¨z-Das- cıog˘lu [14] and it is, thereafter, applied to the mentioned Fredholm–Volterra IDE system. The aim of this study is to get solution as truncated Chebyshev series defined by yj ðxÞ ¼
N X
ajr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 1pxp1,
(3)
r¼0
where T r ðxÞ denotes the Chebyshev polynomials of the first kind, ajr are unknown Chebyshev coefficients, and N is chosen any positive integer such that NXm. The systems (1) and (2) can always be written as a matrix equation, respectively Z 1 Z x m X pn ðxÞyðnÞ ðxÞ ¼ gðxÞ þ Fðx; tÞyðtÞ dt þ Kðx; tÞyðtÞ dt, (4) n¼0 m1 X n¼0
1
an yðnÞ ð1Þ þ bn yðnÞ ð1Þ þ cn yðnÞ ðcÞ ¼ k.
1
(5)
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This is accomplished by introducing k-dimensional column matrix ðnÞ ðnÞ T yðnÞ ðxÞ ¼ ½yðnÞ 1 ðxÞ; y2 ðxÞ; . . . ; yk ðxÞ
and an , bn and cn are diagonal matrices with entries anj , bnj and cnj , j ¼ 1; 2; . . . ; k and the other matrices are as follows: k ¼ ½kj ;
gðxÞ ¼ ½gi ðxÞ;
pn ðxÞ ¼ ½pni;j ðxÞ;
Fðx; tÞ ¼ ½F i;j ðx; tÞ;
Kðx; tÞ ¼ ½K i;j ðx; tÞ,
n ¼ 0; 1; 2; . . . ; m; i; j ¼ 1; 2; . . . ; k.
To obtain the Chebyshev polynomial solution of IDE systems, it is assumed that pn ðxÞ; gðxÞ; Fðx; tÞ and Kðx; tÞ are defined on ½1; 1. If the integrals are bounded in the range ½0; 1, then solution can be obtained by means of the shifted Chebyshev polynomials T r ðxÞ. However, it is not considered in this study since any finite interval can be transformed to interval ½1; 1, which is domain of the Chebyshev polynomials of the first kind.
2. Fundamental relations It is supposed that the solution yj ðxÞ and its derivatives have truncated Chebyshev series expansions of the form yðnÞ j ðxÞ ¼
N X
aðnÞ jr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 1pxp1.
(6)
r¼0
Then functions defined in relation (3) can be written in the matrix form yj ðxÞ ¼ TðxÞAj , where TðxÞ ¼ ½T 0 ðxÞ; T 1 ðxÞ; . . . ; T N ðxÞ;
Aj ¼ ½aj0 ; aj1 ; . . . ; ajN T .
Similarly, the matrix representations of functions (6) become ðnÞ yðnÞ j ðxÞ ¼ TðxÞAj
or using the relation between the Chebyshev coefficient matrices Aj and AðnÞ j [15], n n yðnÞ j ðxÞ ¼ 2 TðxÞM Aj
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so that 2
N 2
3
0
3 2
0
5 2
60 6 6 60 6 M¼6. 6 .. 6 6 40
1 2
0 0 .. . 0
2 0 .. . 0
0 4 3 0 .. .. . . 0 0
0 5 .. . 0
07 7 7 N7 7 .. 7 . 7 7 7 N5
0
0
0
0 0
0
0
2
0
0
60 6 6 60 6 M¼6. 6 .. 6 6 40 0
1 2
0
3 2
0
5 2
0
2
0 4
0
0 .. .
0 .. .
3 0 .. .. . .
5 .. .
0 0
0 0
0 0 0 0
0 0
0
for odd N,
ðNþ1ÞðNþ1Þ
3
N7 7 7 07 7 .. 7 . 7 7 7 N5 0
for even N.
ðNþ1ÞðNþ1Þ
ðnÞ
In this case, the matrix y ðxÞ is reduced to yðnÞ ðxÞ ¼ 2n TðxÞMn A;
n ¼ 0; 1; 2; . . . ; m,
(7)
where 2
TðxÞ
0
6 6 0 TðxÞ 6 TðxÞ ¼ 6 .. 6 .. 6 . . 4 0 0 2 3 A1 6 7 6 A2 7 6 7 7 A¼6 6 .. 7 . 6 . 7 4 5 Ak k1
..
.
3
2
7 0 7 7 7 .. 7 . 7 5 TðxÞ
6 6 6 M ¼6 6 4
0
;
n
kk
Mn
0
0
3
0 .. .
Mn .. .
.. .
0 .. .
7 7 7 7 7 5
0
0
Mn
,
kk
Similarly, it is assumed that the kernel functions F ij ðx; tÞ and K ij ðx; tÞ can be expanded to univariate Chebyshev series in the forms F ij ðxs ; tÞ ¼
N X 00
f ijr ðxs ÞT r ðtÞ;
K ij ðxs ; tÞ ¼
r¼0
N X 00
kijr ðxs ÞT r ðtÞ
r¼0
in which a summation symbol with double primes denotes a sum with first and last terms halved, xs are the Chebyshev collocation points defined by xs ¼ cosðsp=NÞ;
s ¼ 0; 1; . . . ; N,
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which are turning points of the polynomial of degree N. Chebyshev polynomial T N ðxÞ takes the extreme values 1 at these points. The Chebyshev coefficients f ijr ðxs Þ and kijr ðxs Þ are determined by Clenshaw and Curtis [16]. Then the matrix representations of F ij ðxs ; tÞ and K ij ðxs ; tÞ become F ij ðxs ; tÞ ¼ F ij ðxs ÞTðtÞT ;
K ij ðxs ; tÞ ¼ K ij ðxs ÞTðtÞT ,
(8)
where F ij ðxs Þ ¼ K ij ðxs Þ ¼
h
ij ij 1 ij 1 ij 2f 0 ðxs Þ; f 1 ðxs Þ; . . . ; f N1 ðxs Þ; 2f N ðxs Þ
h
i
ij ij 1 ij 1 ij 2k0 ðxs Þ; k 1 ðxs Þ; . . . ; k N1 ðxs Þ; 2kN ðxs Þ
,
i .
3. The method of the solution To obtain the Chebyshev polynomial solution of the system (1) under the mixed conditions (2), it is used in the following matrix method. This method is based on computing the Chebyshev coefficients by means of the Chebyshev collocation points. Substituting the Chebyshev collocation points into Eq. (4) it is obtained a new matrix equation in the form m X
Pn YðnÞ ¼ G þ I þ J
(9)
n¼0
in which I and J denote the Fredholm and Volterra integral parts of Eq. (4), respectively and 3 3 3 3 2 2 2 2 ðnÞ Iðx0 Þ Jðx0 Þ gðx0 Þ y ðx0 Þ 7 7 7 7 6 6 6 6 ðnÞ 6 gðx1 Þ 7 6 Iðx1 Þ 7 6 Jðx1 Þ 7 6 y ðx1 Þ 7 7 7 7 7 6 6 6 6 ðnÞ Y ¼6 7; G ¼ 6 .. 7; I ¼ 6 .. 7; J ¼ 6 .. 7, .. 7 6 . 7 6 . 7 6 . 7 6 . 5 5 5 5 4 4 4 4 gðxN Þ IðxN Þ JðxN Þ yðnÞ ðxN Þ 2 6 6 6 Pn ¼ 6 6 4
pn ðx0 Þ
0
0
0 .. .
pn ðx1 Þ .. .
.. .
0 .. .
0
0
...
pn ðxN Þ
3 7 7 7 7. 7 5
Now, let us seek the representations of Y, I and J matrices in terms of Chebyshev coefficient matrix A. Firstly, substituting the Chebyshev collocation points into relation (7) we have YðnÞ ¼ 2n TMn A,
(10)
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where T ¼ ½Tðx0 Þ; Tðx1 Þ; . . . ; TðxN ÞT . Similarly, putting the Chebyshev collocation points into Fredholm and Volterra integral parts I i ðxÞ and J i ðxÞ of the system (1), and then using the relation (8) and also the following relations: Z 1 Z 1 T TðtÞ TðtÞ dt ¼ T q ðtÞT r ðtÞ dt ¼ ½zqr ; q; r ¼ 0; 1; . . . ; N, Z¼ 1
1
whose entries are given by Fox and Parker [17], and Z xs Z xs Zðxs Þ ¼ TðtÞT TðtÞ dt ¼ T q ðtÞT r ðtÞ dt ¼ ½zqr ðxs Þ; 1
q; r; s ¼ 0; 1; . . . ; N,
1
whose entries are computed by Akyu¨z [18], we obtain I i ðxs Þ ¼
k X
F ij ðxs ÞZAj ;
J i ðxs Þ ¼
j¼1
k X
K ij ðxs ÞZðxs ÞAj ;
i ¼ 1; 2; . . . ; k
j¼1
or compact notation Iðxs Þ ¼ Fðxs ÞZA;
Jðxs Þ ¼ Kðxs ÞZðxs ÞA;
s ¼ 0; 1; . . . ; N
(11)
in which Iðxs Þ and Jðxs Þ are k 1 matrices with entries I i ðxs Þ and J i ðxs Þ, respectively, and others can be written by the blocked matrices 3 2 2 3 F 11 ðxs Þ F 12 ðxs Þ F 1k ðxs Þ Z 0
0 7 6 6 0 Z
0 7 6 F 21 ðxs Þ F 22 ðxs Þ F 2k ðxs Þ 7 6 7 7 6 6 . . . 7 , ; Z ¼ Fðxs Þ ¼ 6 7 . .. .. .. 6 . . .. 7 7 6 . . . 4 .. .. 5 5 4 0 0 Z kk F k1 ðxs Þ F k2 ðxs Þ F kk ðxs Þ 2
K 11 ðxs Þ
K 12 ðxs Þ
K 1k ðxs Þ
3
7 6 6 K 21 ðxs Þ K 22 ðxs Þ K 2k ðxs Þ 7 7 6 7, Kðxs Þ ¼ 6 .. .. .. 7 6 7 6 . . . 5 4 K k1 ðxs Þ K k2 ðxs Þ K kk ðxs Þ 3 2 0
0 Zðxs Þ 7 6 6 0 Zðxs Þ
0 7 7 6 7 Zðxs Þ ¼ 6 .. 7 . .. .. 6 .. 7 6 . . . . 5 4 0 0
Zðxs Þ kk
In this way, the matrices I and J are acquired in the forms
I = FZA, J = KZA ,
(12)
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0 .. .
K ( x1 ) .. .
F( x N )
0
0
.. .
F( x1 ) , K= .. .
0 0
.
0
Z( x 0 ) , Z=
..
F=
K ( x0 )
.. .
F( x0 )
.. .
where
K(xN )
Z( x1) . .. . Z( x N )
After substitution of relations (10) and (12) into Eq. (9), the expression
n =0
A=G
(13)
Σ 2 n Pn T M n − FZ − KZ
m
is obtained. This is the fundamental matrix equation for the solution of Fredholm–Volterra IDE systems. Briefly this equation can also be written in the form (14)
WA ¼ G,
which corresponds to a system of kðN þ 1Þ linear algebraic equations with the unknown Chebyshev coefficients so that
Σ 2 n Pn T M n − FZ − KZ
n =0
m
W=
[ ]
= wi , j , i, j = 1, 2,... , k ( N + 1) .
Finally, by using the relation (7) at points 1, 1 and c, the matrix representation of mixed conditions (5) which depends on the Chebyshev coefficient matrix A is formed m1 X
2n ðan Tð1Þ þ bn Tð1Þ þ cn TðcÞÞMn A ¼ k
n¼0
and defining V¼
m 1 X
2n ðan Tð1Þ þ bn Tð1Þ þ cn TðcÞÞMn ,
n¼0
the fundamental matrix form of the conditions becomes VA ¼ k.
(15)
Consequently, replacing the rows of the matrices V and k by the rows of the matrices W and G, respectively, we have ~ ~ ¼ G. WA
(16)
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For convenience, last rows of the matrix are replaced. If the number of conditions is mk, the augmented matrix of the above system is as follows: 3 2 w1;2 ... w1;kðNþ1Þ ; g1 ðx0 Þ w1;1 6 w2;1 w2;2 ... w2;kðNþ1Þ ; g2 ðx0 Þ 7 7 6 7 6 7 6 .. .. .. .. .. 7 6 . . . . . 7 6 7 6 wk;1 wk;2 ... wk;kðNþ1Þ ; gk ðx0 Þ 7 6 7 6 6 wkþ1;1 wkþ1;2 ... wkþ1;kðNþ1Þ ; g1 ðx1 Þ 7 7 6 7 6 . . . 7 6 ~ ¼6 ~ G .. .. .. ½W; 7. 7 6 7 6w 6 kðNmþ1Þ;1 wkðNmþ1Þ;2 . . . wkðNmþ1Þ;kðNþ1Þ ; gk ðxNm Þ 7 7 6 7 6 v1;1 v1;2 ... v1;kðNþ1Þ ; l1 7 6 7 6 v2;1 v2;2 ... v2;kðNþ1Þ ; l2 7 6 7 6 7 6 .. .. .. .. .. 7 6 . . . . . 5 4 vmk;1 vmk;2 ... vmk;kðNþ1Þ ; lmk However, we do not have to replace the last rows. For example, if the matrix W is singular, then the rows that have the same factor or all zero are replaced. Solving this system by usual methods, the Chebyshev coefficients are determined, and therefore, the finite Chebyshev series approach of IDE system under the mixed conditions is obtained. The number of the conditions must be at most mk to have the unique solution, since the integro-differential system has mth-order derivative at most and k unknown function. On the other hand, the unique solution can sometimes be obtained using fewer conditions to the problem. Note that, if pi ðxÞ ¼ 0; i ¼ 1; 2; . . . ; m, Eq. (4) is reduced to the system of linear integral equations, and then only Eq. (14) is used to find the solution of IE systems. When the domain of the problem is [0, 1], the solution is sought in terms of shifted Chebyshev polynomials T r ðxÞ in the form yj ðxÞ ¼
N X
a jr T r ðxÞ;
j ¼ 1; 2; . . . ; k; 0pxp1,
r¼0
where T r ðxÞ ¼ T r ð2x 1Þ. Following the previous procedure and using the collocation points defined by
sp 1 1 þ cos ; s ¼ 0; 1; . . . ; N xs ¼ 2 N we obtain the fundamental matrix equation for the FVIDE system as m
n =0
Σ 4 n Pn T* M n − F * Z* − K * Z *
A* = G
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and the matrix forms of the conditions become m1 X
4n ðan T ð0Þ þ bn T ð1Þ þ cn T ðcÞÞMn A ¼ k.
n¼0
4. Illustrations In this section, we will consider five examples of the systems of Fredholm, Volterra and Fredholm–Volterra type to illustrate the usage of the presented method. All results were computed using a program written in Mathcad 2001 Professional. Example 1. Consider the system of second order Fredholm IDEs: Z 1 00 2 0 0 3 2 y1 þ x y1 þ 3y1 þ y2 4y2 ¼ 2x 2x 5x 18 þ ð3y1 þ 2xy2 Þ dt, 20 y01 2y1 þ y002 2y02 þ y2 ¼ 2x2 x 5 þ 3
Z
1 1
ððt xÞy1 þ 5t2 y2 Þ dt 1
with conditions y01 ð1Þ ¼ 3; y1 ð1Þ ¼ 0; y2 ð1Þ ¼ 2; y2 ð0Þ ¼ 4. Fundamental matrix equation in Section 3 for this system is ð4P2 TM2 þ 2P1 TM þ P0 T FZÞA ¼ G. When approximated the solution by truncated Chebyshev series for NX2, Chebyshev coefficients are found as a10 ¼ 12; a20 ¼ 4;
a11 ¼ 1;
a12 ¼ 12;
a21 ¼ 2;
a22 ¼ 0;
a3;4;... ¼ 0, a3;4;... ¼ 0.
Therefore the solution of this Fredholm system is y1 ðxÞ ¼ x2 þ x;
y2 ðxÞ ¼ 2x þ 4,
which is the exact solution. If we take three of the conditions, we also obtain the same result for small N. For greater value of N, the Chebyshev coefficients are getting worse and worse. Furthermore, if no condition is used, we obtain the same result for small N. For example, taking N ¼ 10, the Chebyshev coefficients are found without error by using four conditions, with 1014 error by using three conditions and with 1013 error by using no conditions.
Example 2. Let us consider a system of linear Volterra IDEs with two unknowns: Z x 1 3 0 4 2 ðy1 3xy2 Þ dt, y1 xy1 þ y2 ¼ x 9x x þ 1 þ 2 2 1 Z x x2 y1 þ y02 xy2 ¼ 4x þ ðð2x þ tÞy1 þ 3t2 y2 Þ dt 1
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under the conditions y1 ð0Þ ¼ 0, y2 ð0Þ ¼ 1. The fundamental matrix equation of this system is ð2P1 TM þ P0 T KÞA ¼ G. After required operations have been done for Eq. (16), Chebyshev coefficients a10 ¼ 0;
a11 ¼ 0;
a12 ¼ 0;
a13 ¼ 1,
a20 ¼ 0;
a21 ¼ 0;
a22 ¼ 1;
a23 ¼ 0
and thereby the solution y1 ðxÞ ¼ 4x3 3x;
y2 ðxÞ ¼ 2x2 1
are obtained by taking N ¼ 3. Besides, the same result is acquired for N43.
Example 3. Consider a system of third-order linear Volterra IDEs in the interval [0, 1]: ð2Þ x ð2Þ yð3Þ 1 þ e y1 þ y2 þ 2y1 þ y2 ¼ 1 þ coshðxÞ þ x sinðxÞ
y1ð2Þ
cosðxÞyð2Þ 2
y1 ð1Þ ¼ e1 ; yð1Þ 2 ð0Þ ¼ 2;
þ
y1ð1Þ
Z
ðext y1 þ cosðx tÞy2 Þ dt,
0
Z
2
2x
x
xe y1 ¼ x sin ðxÞ þ sinð2xÞ
x
ðexþt y1 þ x cosðtÞy2 Þ dt,
0
y2 ð1Þ ¼ 2 sinð1Þ; y1 ð0Þ ¼ 1;
ð2Þ 1 1 yð1Þ 1 ð2Þ þ y1 ð2Þ ¼ 0,
y2 ð0Þ ¼ 0.
This system has y1 ðxÞ ¼ ex ; y2 ðxÞ ¼ 2 sinðxÞ the exact solution. The fundamental matrix equation of this system is
ð64P3 T M3 þ 16P2 T M2 þ 4P1 T M þ P0 T K Z ÞA ¼ G. Replacing the first two and last four rows of the augmented matrix bW ; Gc obtained above equation by the rows of the matrices for conditions at point x ¼ 1 and the ~ The results obtained from the solution of this ~ ; Gc. others, respectively, we have bW system are given in Tables 1 and 2.
Example 4. Consider a first-order system of Fredholm–Volterra IDEs: y01
þ
xy02
1 þ 3y2 ¼ 20x þ 2x þ x 3 þ 3
x2 y01 y02 þ xy1 þ y2
3
2
Z
1 2
Z
x
ððx tÞy1 þ x y2 Þ dt þ 1
2 ¼ 5x3 15x2 8x þ þ 3
ð4ty1 y2 Þ dt, 1
Z
1
ð3ty1 þ ðt2 4xÞy2 Þ dt þ 1
Z
x
6y1 dt. 1
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Table 1 Comparison of the absolute errors of y1 xi
N ¼3
N¼5
N ¼6
N¼8
N ¼ 10
N ¼ 11
N ¼ 12
0 0.25 0.5 0.75 1
0.00 1.92e3 1.05e3 2.49e4 0.00
0.00 5.09e6 2.98e6 1.74e6 0.00
0.00 2.71e7 3.37e7 2.56e7 0.00
0.00 8.46e10 8.16e10 5.11e10 0.00
0.00 4.86e13 5.90e13 3.81e13 0.00
0.00 2.00e15 1.22e15 0.00 0.00
0.00 0.00 0.00 0.00 0.00
Table 2 Comparison of the absolute errors of y2 xi
N ¼3
N¼5
N ¼6
N¼8
N ¼ 10
N ¼ 11
N ¼ 12
0 0.25 0.5 0.75 1
0.00 1.21e3 2.35e3 1.39e3 0.00
0.00 1.38e6 3.21e6 1.94e6 0.00
0.00 5.68e7 5.82e7 4.46e7 0.00
0.00 4.82e10 7.29e10 6.22e10 0.00
0.00 6.86e13 7.99e13 5.28e13 0.00
0.00 2.39e15 1.78e15 0.00 0.00
0.00 0.00 0.00 0.00 0.00
Let us approximate the solution by Chebyshev series of degree three. Then, Chebyshev collocation points become x1 ¼ 12;
x0 ¼ 1;
x2 ¼ 12;
x3 ¼ 1.
Following the previous procedure, the augmented matrix for Eq. (14) is found as 2
2 1 6 11 0 6 6 6 0:5 0:1667 6 6 6 8:5 0:75 ½W; G ¼ 6 6 2:5 0:5 6 6 6 3:5 0:75 6 6 2 1:6667 4 1
0
4:6667
10:2
3
4
7
12
;
9 2:7083 4:75
11:2 0:8 0:425
8:3333 4 4:3333
0 1:625 0:5
5:8 1:0833 3:9667
8 2:8125 1
; ; ;
1:9583 0:75 4:6667
0 0:425 8:6
3 3:6667 1
2:375 1:5 4
0:25 2:7 7:6667
3:1875 1 12
; ; ;
5
11:2
7:6667
2
7:5333
10
;
58=3
3
52=3 7 7 7 1=6 7 7 7 155=24 7 7. 31=6 7 7 7 7=24 7 7 64=3 7 5 34=3
Solving this system, we have the solution y1 ðxÞ ¼ x2 þ 3x;
y2 ðxÞ ¼ 4x3 1.
Note that any conditions are not used. If some conditions are given such that y01 ð0Þ y1 ð1Þ ¼ 1; 2y2 ð1Þ þ y2 ð1Þ ¼ 1, augmented matrix forms of these conditions are obtained as " # 1 0 1 4 0 0 0 0 ; 1 ½V; k ¼ 0 0 0 0 3 1 3 1 ; 1
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and replaced the last two rows of the above augmented matrix. Then, it is obtained the same solution by solving the new system.
Example 5. Finally, consider a system of Fredholm–Volterra IDEs: ð2Þ ð1Þ x x x yð3Þ þ 1 þ y2 þ y1 þ e y2 ¼ 1 xe 2e
Z
Z
1
x
ext y2 dt,
ðsinhðtÞy1 þ coshðtÞy2 Þ dt þ 1
ð1Þ x ð3Þ x yð3Þ þ xexþ2 þ 2 e y1 y2 þ xy2 þ y1 ¼ 6 x xe
1
Z
1
ð3et y1 xexþ1 y2 Þ dt þ
Z
1
x
ðty1 þ et y2 Þ dt 1
with the mixed conditions y1 ð1Þ ¼ e1 ;
y2 ð1Þ ¼ e;
y1 ð0Þ ¼ 1;
y2 ð0Þ ¼ 1;
y1 ð1Þ ¼ e;
y2 ð1Þ ¼ e.
The exact solution of this system is y1 ðxÞ ¼ ex , y2 ðxÞ ¼ ex . The computational results, which are obtained by replacing the first two and the last four rows of the augmented matrix, are given in Table 3 for different N. Besides, the absolute errors are all zero for N ¼ 14. However, if the last six rows of the augmented matrix are replaced, the absolute errors are found almost 1014 for N ¼ 14.
5. Conclusions An interesting feature of this method is that when IDE system has linearly independent polynomial solution of degree N or less than N, the method can be used for finding the analytical solution. Besides, it is seen that when the truncation limit N is increased, there exists a solution, which is closer to the exact solution. Problems defined in a finite range [a,b] can be solved by the Chebyshev collocation method since by means of the linear transformation, any finite range can be
Table 3 The numerical results of Example 5 xi
1 0.75 0.5 0.25 0 0.25 0.5 0.75 1
Absolute errors of y1
Absolute errors of y2
N¼5
N ¼ 10
N ¼ 13
N¼5
N ¼ 10
N ¼ 13
0.00 6.88e4 3.66e4 8.99e5 0.00 3.27e5 8.35e5 9.63e5 0.00
0.00 3.72e10 3.17e10 1.93e11 0.00 1.21e10 1.18e10 7.95e11 0.00
0.00 2.66e14 1.09e14 7.11e15 0.00 2.89e15 3.89e15 3.89e15 0.00
0.00 7.25e4 3.87e4 8.61e5 0.00 3.24e5 1.15e4 1.47e4 0.00
0.00 2.61e10 2.07e10 2.16e11 0.00 1.29e10 1.01e10 9.18e11 0.00
0.00 2.80e14 1.21e14 8.22e15 0.00 4.00e15 5.11e15 5.77e15 0.00
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transformed into the range ½1; 1, which is the domain of the Chebyshev polynomials T r ðxÞ. If the range is ½0; 1, then, it can be assumed that the solution functions can be expressed by shifted Chebyshev polynomials T r ðxÞ, but some modifications are required for the fundamental matrix equations. The presented method is based on computing the coefficients in the Chebyshev expansion of solution of a linear IDE system, and is valid when the matrix functions pn ðxÞ, gðxÞ, Fðx; tÞ and Kðx; tÞ are defined in ½1; 1 and the kernel functions have a Chebyshev series expansion in this range. Moreover, if the kernel functions F ij ðx; tÞ and K ij ðx; tÞ converge rapidly, it can be obtained better results. A considerable advantage of the method is that it allows us to make use of the computer because this Chebyshev method transforms the problem into the matrix equation, which is a linear algebraic system. Therefore, Chebyshev coefficients of the solution are found very easily by using the computer programs such as Qbasic, Cþþ , etc. Furthermore, the values of the solution at the collocation points are evaluated with the aid of the computer programs without any computational effort.
References [1] C.T.H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math. 125 (2000) 217–249. [2] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, PA, 1985. [3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985. [4] F. Bloom, Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory, J. Math. Anal. Appl. 73 (1980) 524–542. [5] K. Holmaker, Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal. 24 (1) (1993) 116–128. [6] M.A. Abdou, Fredholm–Volterra integral equation of the first kind and contact problem, Appl. Math. Comput. 125 (2002) 177–193. [7] L.K. Forbes, S. Crozier, D.M. Doddrell, Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math. 57 (2) (1997) 401–425. [8] P.J. van der Houwen, B.P. Sommeijer, Euler–Chebyshev methods for integro-differential equations, Appl. Numer. Math. 24 (1997) 203–218. [9] W.H. Enright, M. Hu, Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay, Appl. Numer. Math. 24 (1997) 175–190. [10] K. Maleknejad, F. Mirzae, S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput. 155 (2004) 317–328. [11] K. Maleknejad, M. Tavassoli Kajani, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159 (2004) 603–612. [12] A. Akyu¨z, M. Sezer, A Chebyshev collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 72 (4) (1999) 491–507. [13] A. Akyu¨z, M. Sezer, Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput. 144 (2003) 237–247. [14] A. Akyu¨z-Das-cıog˘lu, Chebyshev polynomial solutions of systems of linear integral equations, Appl. Math. Comput. 151 (2004) 221–232. [15] M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol. 27 (4) (1996) 607–618.
ARTICLE IN PRESS A. Akyu¨z-Das- cıog˘lu, M. Sezer / Journal of the Franklin Institute 342 (2005) 688–701
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[16] C.W. Clenshaw, A.R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205. [17] L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968. [18] A. Akyu¨z, Chebyshev Collocation method for solution of linear integro-differential equations, M.Sc. Thesis, Dokuz Eylu¨l University, Graduate School of Natural and Applied Sciences, 1997.