Solution of a system of Volterra integral equations of

Applied Mathematics and Computation 139 (2003) 249–258 www.elsevier.com/locate/amc

Solution of a system of Volterra integral equations of the first kind by Adomian method J. Biazar a, E. Babolian b, R. Islam

c,*

a

Guilan University, Rasht, Iran University for Teacher Education, Tehran, Iran Faculty of Engineering, Dalhousie University, D510, 1360 Barrington Street, P.O. Box 1000, Halifax, Nova Scotia, Canada B3J 2X4 b

c

Abstract In this article, we consider linear and non-linear systems of integral equations of the first kind. The problem of existence and uniqueness are considered. The Adomian decomposition method is being used to solve these linear and non-linear systems of Volterra integral equations of the first kind. Some examples are prepared to show the efficiency and simplicity of the method. Ó 2002 Published by Elsevier Science Inc. Keywords: Integral equation; Decomposition method

1. Introduction Cherrualt and Seng [1] in a joint paper consider the integral equations of the first kind, and used Adomian decomposition method to solve these equations. The first and the second authors have extended the Adomian decomposition method for solving linear and non-linear systems of Volterra integral equations of the second kind [2,3]. Following them we are using the

*

Corresponding author. E-mail addresses: [email protected] (J. Biazar), rafi[email protected], [email protected] (R. Islam). 0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 7 3 - X

250

J. Biazar et al. / Appl. Math. Comput. 139 (2003) 249–258

Adomian decomposition method to solve systems of Volterra integral equations of the first kind [4]. A system of integral equation of the first kind can be presented as: Z x ki ðx; tÞgi ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ dt ¼ fi ðxÞ; n ¼ 1; 2; . . . ; n; ð1Þ 0

where fi are known functions, ki ðx; tÞ are the kernels of the ith integral equation, gi are linear or nonlinear functional of the unknown functions ui . In the system (1) the equations are not in the canonical form, which is suitable for using Adomian decomposition method. To derive this form, differentiate of the both sides of Eq. (1), with respect to x, and according to the Leibnitz generalized formula, we obtain (if ki ðx; xÞ 6¼ 0) Z x oki ðx; tÞ fi ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ dt ki ðx; xÞfi ðu1 ðxÞ; u2 ðxÞ; . . . ; un ðxÞÞ þ ox 0 ¼ fi0 ðxÞ; i ¼ 1; 2; . . . ; n: ð2Þ And then: gi ðu1 ðxÞ; u2 ðxÞ; . . . ; un ðxÞÞ þ

Z 0

x

oki ðx; tÞ=ox gi ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ dt ki ðx; xÞ

f 0 ðxÞ ; ¼ i ki ðx; xÞ gi ðu1 ðxÞ; u2 ðxÞ; . . . ; un ðxÞÞ ¼

ð3Þ fi0 ðxÞ ki ðx; xÞ

Z 0

x

oki ðx; xÞ=ox gi ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ dt: ki ðx; xÞ

ð4Þ

From now on, we can have different approaches depending on linearity or nonlinearity of the function gi ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞ, with respect to u1 ; u2 ; . . . ; un .

2. Linear systems If gi ðu1 ; u2 ; . . . ; un Þ are linear, the system (1) can be written in the following simple representation; Z xX n ki;j ðx; tÞuj ðtÞ dt ¼ fi ðxÞ; i ¼ 0; 1; 2; . . . ; n: 0

j¼1

After differentiation we derive: Z xX n n X oki;j ðx; tÞ uj ðtÞ dt ¼ fi0 ðxÞ: ki;j ðx; xÞuj ðxÞ þ ox 0 j¼1 j¼1

ð5Þ


251

From ith equation, in (5), we write the ith unknown ui , in terms of the other unknowns, to derive the following linear system of Volterra integral equations, of the second kind; Z xX n n X f 0 ðxÞ ki;j ðx; tÞ oki;j ðx; tÞ=ox ui ðxÞ ¼ i uj ðxÞ þ uj ðtÞ dt: ð6Þ ki;i ðx; xÞ j¼1 ki;i ðx; xÞ ki;i ðx; xÞ 0 j¼1 Existence and uniqueness of the system (6) is ensured by the following theorem [2]. Theorem 1. If fi0 ðxÞ=kii ðx; xÞ, kij ðx; tÞ=ki ðx; xÞ, and kij0 ðx; tÞ=kii ðx; xÞ are continuous in 0 6 x 6 t 6 T , then the system of Volterra integral equations of the second kind (6) has a unique continues solution for 0 6 t 6 T . Proof. See [2].

Example 1. Consider the following system of linear integral equations with the exact solutions: f ðxÞ ¼ x2 and gðxÞ ¼ x. Z t 1 2 ðð1 x2 þ t2 Þf ðtÞ ð2x tÞgðtÞÞ dt ¼ x3 x5 ; 3 15 0 Z x 1 1 1 ððx þ t2 Þf ðtÞ ð2x tÞgðtÞÞ dt ¼ x2 x3 þ x4 þ x5 : 6 3 5 0 By differentiation with respect to x we have: Z x 2 f ðxÞ xgðxÞ 2 ðxf ðtÞ þ gðtÞÞ dt ¼ x2 x4 ; 3 0 Z t 1 4 ðx þ x2 Þf ðxÞ þ 2gðxÞ þ ðf ðtÞ gðtÞÞ dt ¼ 2x x2 þ x3 þ x4 : 2 3 0 And then: 2 f ðxÞ ¼ x x4 þ xgðxÞ þ 2 3 2

Z

x

ðxf ðtÞ þ gðtÞÞ dt; 0

1 2 1 1 1 gðxÞ ¼ x x2 þ x3 þ x4 ðx þ x2 Þf ðxÞ 4 3 2 2 2

Z

x

ðf ðtÞ gðtÞÞ dt: 0

The procedure of finding the solution by Adomian decomposition method consists of the following scheme: 2 f0 ¼ x2 x4 ; 3 1 2 2 3 1 4 g0 ¼ x x þ x þ x ; 4 3 2

252


x 1 fnþ1 ¼ gn 2 2x

Z

x

ðfn ðtÞ tgn ðtÞÞ dt;

0

1 1 gnþ1 ¼ ðx þ x2 Þfn 2 2

Z

x

ðfn ðtÞ gn ðtÞÞ dt;

n ¼ 0; 1; 2; . . .

0

Some numerical results are presented in Table 1.

3. Nonlinear systems If the system of integral equations are nonlinear, we can reduce the system (4) into a simpler system of integral equations of the second kind. Two procedures are being suggested here: First approach: If we can recognize invertible functions gi ðui ðxÞÞ for each unknown function ui ðxÞ, and set vi ðxÞ ¼ gi ðui ðxÞÞ, then the nonlinear system of integral equations reduces to a simpler system of integral equations. When vi ðxÞ ¼ gi ðui ðxÞÞ being approximated by Adomain decomposition method then the original unknown functions will be determined by ui ðxÞ ¼ gi1 ðvi ðxÞÞ i ¼ 0; 1; 2; . . . Example 2. Consider the following system of integral equations of the first kind with the exact solutions uðxÞ ¼ x2 and vðxÞ ¼ x. Z x 1 2 1 1 ð1 x2 þ t2 ÞðuðtÞ þ v3 ðtÞÞ dt ¼ x6 x5 þ x4 þ x3 ; ð7aÞ 12 15 4 3 0 Z x 5 1 5 1 ð5 þ x tÞðu3 ðtÞ vðtÞÞ dt ¼ x2 x3 þ x7 þ x8 : ð7bÞ 2 6 7 56 0 By differentiation we have: Z x 1 2 3 ðuðtÞ þ v3 ðtÞÞ dt ¼ x5 x4 þ x3 þ x2 ; uðxÞ þ v ðxÞ 2x 2 3 0 Z x 1 1 1 ðu3 ðtÞ vðtÞÞ dt ¼ x x2 þ x6 þ x7 : u3 ðxÞ vðxÞ þ 5 0 10 35

ð8aÞ ð8bÞ

In this simple example gi ðxÞ are identity functions, and from (8a) and (8b) we can have: Z x 1 2 ðuðtÞ þ v3 ðtÞÞ dt; uðxÞ ¼ x5 x4 þ x3 þ x2 v3 ðxÞ þ 2x 2 3 0 Z x 1 1 1 ðu3 ðtÞ vðtÞÞ dt: vðxÞ ¼ x þ x2 x6 x7 þ u3 ðxÞ þ 10 35 5 0

n

f ð0:1Þ

f ð0:2Þ

f ð0:3Þ

f ð0:4Þ

f ð0:5Þ

gð0:1Þ

gð0:2Þ

gð0:3Þ

gð0:4Þ

gð0:5Þ

2 3 4 5 6 7 8 9 10

0.01021 0.01001 0.00999 0.00999 0.01 0.01 0.01 0.01 0.01

0.04356 0.04028 0.03988 0.03999 0.40001 0.2 0.39999 0.04 0.04

0.10935 0.09254 0.08841 0.08971 0.09012 0.09003 0.08999 0.09000 0.90000

0.22594 0.17236 0.14994 0.15731 0.16137 0.39961 0.15983 0.15992 0.16002

0.4244 0.29298 0.20662 0.23427 0.25958 0.49633 0.24804 0.24882 0.25032

0.10005 0.09999 0.09999 0.1 0.1 0.01 0.1 0.1 0.1

0.20083 0.19952 0.19995 0.20002 0.2 0.2 0.2 0.2 0.2

0.30449 0.29580 0.29935 0.30033 0.30007 0.29999 0.29999 0.30000 0.3

0.41506 0.37960 0.39553 0.40298 0.40088 0.39985 0.39985 0.40005 0.40002

0.53849 0.42837 0.47954 0.51705 0.50682 0.49810 0.58398 0.50074 0.50048


Table 1

253

254


And then: u0 ¼ 12 x5 23 x4 þ x3 þ x2 ; v0 ¼ x þ 101 x2 x6 351 x7 ; unþ1 ðxÞ ¼ A1;n ðv0 ; . . . ; vn Þ 2x

Z

x

B1;n ðu0 ; . . . ; un ; v0 ; . . . ; vn Þ dt;

ð9aÞ

0

vnþ1 ðxÞ ¼ A2;n ðu0 ; . . . ; un Þ þ

1 5

Z

x

B2;n ðu0 ; . . . ; un ; v0 ; . . . ; vn Þ dt; n ¼ 0; 1; 2; . . . ;

0

ð9bÞ

where A1;n ðv0 ; . . . ; vn Þ and A2;n ðu0 ; . . . ; un Þ are Adomian polynomials for one variable function say N ðuÞ are given by: X

An fN ðuÞg ¼

p1 þ2p2 þ

þnpn ¼n

uP1 up22 upn

n N ðp1 þ

þpn Þ ðuÞ: p1 ! p2 ! pn !

ð10Þ

And B1;n ðu0 ; . . . ; un ; v0 ; . . . ; vn Þ and B2;n ðu0 ; . . . ; un ; v0 ; . . . ; vn Þ are Adomian polynomials. For tow variables functions Gðu; vÞ, they are given by [3]: Bn fGðu; vÞg ¼

X up1 . . . upn vq1 . . . vqn 1 n 1 n G9ðp1 þ

þpn þq1 þ

þqn Þ ðu0 ; v0 Þ; p ! . . . p !q ! . . . q ! 1 n 1 n Xn

ð11Þ

where Xn ¼ fp1 þ 2p2 þ þ npn þ q1 þ 2q2 þ þ nqn ¼ ng. Some numerical results of the system (9a) and (9b), are presented in Table 2. Second approach: Let hi ðxÞ ¼ gi ðu1 ðxÞ; u2 ðxÞ; . . . ; un ðxÞÞ:

ð12Þ

So the system of integral equation (4) can be rewritten as: Z x f 0 ðxÞ ki;x ðx; tÞ hi ðxÞ ¼ i hi ðxÞ dt; n ¼ 0; 1; 2; . . . ; ki ðx; xÞ ki ðx; xÞ 0

ð13Þ

which is a system of linear integral equations of the second kind, which can be solved easily by the Adomian decomposition method. Table 2 n

uð0:1Þ

uð0:3Þ

uð0:5Þ

vð0:1Þ

vð0:3Þ

vð0:5Þ

1 2 3 4

0.0100 0.0100 0.0100 0.0100

0.0877 0.0877 0.0877 0.0877

0.2440 0.2431 0.2430 0.2430

0.1 0.1 0.1 0.1

0.3006 0.3006 0.3006 0.5162

0.5162 0.5162 0.5162 0.5162


255

Example 3. To solve the system of equations of the Example 2, by second approach, let: h1 ðxÞ ¼ uðxÞ þ v3 ðxÞ; ð14Þ h2 ðxÞ ¼ u3 ðxÞ vðxÞ: Substitution from (9a) and (9b) into (8a) and (8b) gives: Z x 1 2 h1 ðtÞ dt ¼ x5 x4 þ x3 þ x2 ; h1 ðxÞ 2x 2 3 0 Z x 1 1 1 h2 ðtÞ dt ¼ x x2 þ x6 þ x7 : h2 ðxÞ þ 5 0 10 35

ð14aÞ ð14bÞ

And then: 1 2 h1 ðxÞ ¼ x5 x4 þ x3 þ x2 þ 2x 2 3 h2 ðxÞ ¼ x

1 2 1 1 x þ x6 þ x7 10 35 5

Z

x

h1 ðtÞ dt; 0

Z

x

h2 ðtÞ dt: 0

The following scheme leads to the solution of (10) by Adomian decomposition method: 1 2 h1;0 ðxÞ ¼ x5 x4 þ x3 þ x2 ; 2 3 1 2 1 h2;0 ðxÞ ¼ x x þ x6 þ x7 ; 35 Z10 x

h1;nþ1 ðxÞ ¼ 2x

ð15Þ

h1;n ðtÞ dt;

0

1 h2;nþ1 ðxÞ ¼ 5

Z

x

h2;n ðtÞ dt;

n ¼ 0; 1; 2; . . .

0

When the approximations to h1 ðxÞ and h2 ðxÞ are calculated from (14), then we have (by (9a) and (9b)) uðxÞ ¼ h1 ðxÞ v3 ðxÞ; ð16Þ vðxÞ ¼ h2 ðxÞ þ u3 ðxÞ: And again using Adomian decomposition method: u0 ¼ h1 ðxÞ; v0 ¼ h2 ðxÞ; unþ1 ¼ A1;n ðu0 ; . . . ; un Þ; n ¼ 0; 1; 2; . . . vnþ1 ¼ A2;n ðv0 ; . . . ; vn Þ;

256


Table 3 n

uð0:1Þ

uð0:3Þ

uð0:5Þ

vð0:1Þ

vð0:3Þ

vð0:4Þ

1 2

0.01000 0.01000

0.09050 0.09029

0.23105 0.26188

0.1 0.1

0.2998 0.30002

0.49089 0.48798

where Ai;n Õs are defined in (10): u1 ¼ v30 ; u2 ¼ 3v20 v1 ; 3 v1 ¼ þu0 ; v2 ¼ þ3u20 u1 ;

u3 ¼ 3v20 v2 3v0 v21 ; ... v3 ¼ þ3u20 u2 þ 6u0 u21 ;

Numerical approximations for some values of uðxÞ and vðxÞ with the approxP2 imations h1 i¼0 h1;i are being shown in the Table 3. Example 4. Consider the following non-linear system of Volterra integral equations of the first kind, with the exact solutions uðxÞ ¼ x3 and vðxÞ ¼ x2 . (Rx pffiffiffiffiffiffiffiffi ðð1 þ xt x2 Þu2 ðtÞ þ ð1 þ x tÞ vðtÞÞ dt ¼ 12 x2 þ 16 x3 þ 17 x7 561 x9 ; 0 pffiffiffiffiffiffiffiffi Rx ð2 x2 t þ x3 Þ 3 uðtÞ þ ðxt þ t2 þ 1Þv2 ðtÞ dt ¼ x2 þ 11 x5 421 x7 : 0 30 By differentiation we have: pffiffiffiffiffiffiffiffi Z t pffiffiffiffiffiffiffiffi 1 9 u2 ðxÞ þ vðxÞ þ ððt 2xÞu2 ðtÞ þ vðtÞÞ dt ¼ x þ x2 þ x6 x8 ; 2 56 0 Z x 11 1 2u1=3 ðxÞ þ v2 9x þ ðð2xt þ 3x2 Þu1=3 ðtÞ tv2 ðtÞÞ dt ¼ 2x þ x4 x6 : 6 6 0 Let consider the new functions f ðtÞ ¼ u1=3 ðtÞ and gðtÞ ¼ v1=2 ðtÞ, then we have: Z x 1 9 6 f ðxÞ þ gðxÞ þ ððt 2xÞf 6 ðtÞ þ gðtÞÞ dt ¼ x þ x2 þ x6 x8 ; 2 56 0 Z x 11 1 2f ðxÞ þ g4 ðxÞ þ ðð2xt þ 3x2 Þf ðtÞ tg4 ðtÞÞ dt ¼ 2x þ x4 x6 : 6 6 0 And the canonical form are as follows: 11 1 1 1 f ðxÞ ¼ x þ x4 x6 g4 ðxÞ 12 12 2 2 1 9 gðtÞ ¼ x þ x2 þ x6 x8 f 6 ðxÞ 2 56

Z Z

x

ðð2xt þ 3x2 Þf ðtÞ tg4 ðtÞÞ dt; 0 x

ððt 2xÞf 4 ðtÞ þ gðtÞÞ dt:

0

And Adomian decomposition method consists of the following scheme. f0 ¼ x þ

11 4 1 x x6 ; 12 12

n¼2 n¼3

f ð0:1Þ

f ð0:2Þ

f ð0:3Þ

f ð0:4Þ

f ð0:5Þ

gð0:1Þ

gð0:2Þ

gð0:3Þ

gð0:4Þ

gð0:5Þ

0.100001 0.100000

0.20008 0.19999

0.30107 0.29974

0.40720 0.39729

0.53896 0.47089

0.10000 0.10000

0.20007 0.20000

0.30040 0.29995

0.40254 0.39885

0.52336 0.47421

uð0:1Þ

uð0:2Þ

uð0:3Þ

uð0:4Þ

uð0:5Þ

vð0:lÞ

vð0:2Þ

vð0:3Þ

vð0:4Þ

vð0:5Þ

0.00100 0.00099

0.00800 0.00800

0.026931 0.08997

0.06271 0.15908

0.15656 0.104413

0.0100008 0.0100000

0.04003 0.04000

0.09024 0.08997

0.16204 0.15908

0.27391 0.22487

Table 5

n¼2 n¼3


Table 4

257

258


1 9 g0 ¼ x þ x 2 þ x 6 x 8 ; 2 56 Z 1 1 x B1n ðf0 ; . . . ; fn ; g0 ; . . . ; gn Þ dt; fnþ1 ¼ A1n ðg0 ; . . . ; gn Þ 2 2 0 Z x gnþ1 ¼ A2;n ðf0 ; . . . ; fn Þ B2;n ðf0 ; . . . ; fn ; g0 ; . . . ; gn Þ dt; 0

where Ai;n Õs and Bi;n Õs are defined as (10) and (11) for N1 ðg4 ðxÞÞ, N2 ðf 6 ðxÞÞ, G1 ðð2xt þ 3x2 Þf ðtÞ tg4 ðtÞÞ and G2 ððt 2xÞf 6 ðtÞ þ gðtÞÞ. Some numerical results are presented in Table 4, and the correspondent values for uðxÞ and vðxÞ are listed in Table 5.

4. Conclusion It has already been proved that Adomian decomposition method is a very powerful advice for solving Volterra integral equations of the first kind [1]. We had used this method for solving linear and non-linear systems of Volterra integral equations of the second kind [2,3], and in this paper we solved Volterra integral equations of the first kind by Adomian decomposition method, and the efficiency of this method for solving these problems has been approved. Using the Adomian decomposition method for solving Fredholm integral equations and a system of Fredholm integral equations is still a subject of research. The computations associated with the examples in this paper were performed using Mathematica 4.

References [1] Y. Cherruault, V. Seng, The resolution of non-linear integral equations of the first kind using the decompositional Method of Adomian, Kebernetes 26 (2) (1997) 198–206. [2] E. Babolian, J. Biazar, Solution of a System of Linear Volterra Equations by Adomian Decomposition Method, J. Sci. and Tech., submitted for publication. [3] E. Babolian, J. Biazar, Solution of a System of Nonlinear Volterra Integral Equations of the Second kind, Far East J. Math. Sci. 2 (6) (2000) 935–945. [4] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, PA, 1985.