- Email: [email protected]
Restarted Adomian method for integral equations
Applied Mathematics and Computation 153 (2004) 353–359 www.elsevier.com/locate/amc
Restarted Adomian method for integral equations E. Babolian a, Sh. Javadi a
a,*
, H. Sadeghi
b
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, 599 Taleghani Avenue, Tehran 15614, Iran b Department of Mathematics, Azad University of Kerman, Kerman, Iran
Abstract In this paper we introduce a new method, based on Adomian method, for solving nonlinear integral equations. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Restarted Adomian method; Modified Adomian method; Integral equation
1. Introduction The Adomian decomposition scheme is a method for solving a wide range of problems whose mathematical models yield equation or system of equations involving algebraic, differential, integral and integro-differential (for example see [1,2]). The main algorithm of AdomianÕs decomposition method applies to a general nonlinear equation of the form u N ðuÞ ¼ f ;
ð1:1Þ
where N is a nonlinear operator on a Hilbert space H , f is a known element of H and we are seeking u 2 H satisfying (1.1). We assume that for every f 2 H , Eq. (1.1) has a unique solution. The decomposition technique consists of representing the solution of (1.1) as a series
*
Corresponding author. E-mail address: [email protected] (Sh. Javadi).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00636-2
354
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
u¼
1 X
ð1:2Þ
un
n¼0
and the nonlinear operator N is decomposed as N ðuÞ ¼
1 X
ð1:3Þ
An ;
n¼0
where the An Õs are Adomian polynomials of u0 ; . . . ; un given by " !# 1 X 1 dn i N k ui ; n ¼ 0; 1; 2; . . . An ¼ n! dkn i¼0
ð1:4Þ
k¼0
upon substitution of Eqs. (1.2) and (1.3) into the functional equation (1.1) we obtain: 1 X
un
1 X
n¼0
An ¼ f ;
ð1:5Þ
n¼0
the convergence of the series in (1.5) will yield u0 ¼ f ; u 1 ¼ A0 ; .. . un ¼ An1 : In this paper, we present a new algorithm based on Adomian method. In the new algorithm, we use the modified Adomian method [8] which proposed a slight variation only on the components u0 and u1 , and restarted Adomian method applied to algebraic equations [4]. We will show by some examples that convergence rate of the new method is more accelerate than standard Adomian method.
2. Description of the restarted Adomian method Various formulas exist for calculating Adomian polynomials. The following theorem gives a formula for An as a function of u0 . Theorem 1 An ¼ where
X
ca1 ...an ðN ðu0 ÞÞ
nþ1a1
ðN 0 ðu0 ÞÞ
a1 a2
ðN ðn1Þ ðu0 ÞÞ
an1 an
a
ÞðN ðnÞ ðu0 ÞÞ n ;
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
ca1 ...an ¼
355
n! ða1 a2 Þ! ðan1 an Þ!an !ð1!Þ 1 ðn þ 1 a1 Þ!
Proof. See [3].
ða1a2Þ
½ðn 1Þ!
ðan1 an Þ
This theorem shows the dependence of Adomian polynomials (and correspondingly Adomian method) on u0 . In the new algorithm we update u0 in each step. This can be done by adding a proper function to two sides of Eq. (1.1). Let g be the proper function which will be determined next, then u N ðuÞ þ g ¼ f þ g: ð2:1Þ To solve the problem (2.1), we use the modified Adomian method. Let u and N ðuÞ be decomposed as (1.2) and (1.3). Then we can obtain ui as u0 ¼ f ; u 1 ¼ f g þ A0 ; u 2 ¼ A1 ; .. . un ¼ An1 ; and introduce the following algorithm named ‘‘restarted Adomian method’’. 2.1. The algorithm Choose small natural numbers m, n. Step 1: Apply the Adomian method to Eq. (1.1) and calculate u0 ; u1 ; . . . ; un : Set /1 ¼ u0 þ u1 þ þ un : Step 2: For i ¼ 2 : m, g ¼ /i1 ; u0 ¼ g; u 1 ¼ f g þ A0 ; u 2 ¼ A1 ; .. . un ¼ An1 : Set /i ¼ u0 þ u1 þ þ un . end of for.
356
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
Remark 1. /m can be considered as the approximate solution of Eq. (2.1). Remark 2. In the Adomian method usually the sum of the first few terms gives an approximation of u. In the new algorithm to obtain an accurate approximation of the exact solution, we do not calculate the Adomian polynomials for large indexes, i.e. m, n are considered small, say m ¼ 3, n ¼ 2.
3. Numerical results Example 1. Consider the nonlinear Fredholm integral equation:
uðxÞ ¼
Z
1
ðx tÞu3 ðtÞ dt þ
0
3x 1 þ ; 4 5
ð3:1Þ
with the exact P1solution uðxÞ ¼ x (see [6]). Let u ¼ i¼0 uP i be a solution for (3.1) where ui is specified by Adomian i method and si ¼ j¼0 uj . By applying the standard Adomian method on (3.1), we get: 3x 1 þ ; 4 5 Z 1 ui ðxÞ ¼ ðx tÞAi1 dt;
u0 ðxÞ ¼
i ¼ 1; 2; . . .
0
The first nine terms and corresponding partial sums are given the following table:
i
ui
si
0 1 2 3 4 5 6 7 8
0:2 þ 0:75x 0:2027500000 þ 0:2709687500x 0:004733515625 0:01775068359x 0:01096026963 0:007714847995x 0:003482967564 þ 0:005681606709x 0:0006986036410 0:0006525609164x 0:0009340732109 0:0009331769228x 0:0001614637427 þ 0:0004346747149x 0:0001456621164 þ 0:00004623896704x
0:2 þ 0:75x 1:012930896x 0:06130888154 1:003218066x 0:00748351562 0:9955032184x þ :003476754005 1:001184825x 0:000006213559 1:000532264x 0:0007048172000 0:9995990873x þ 0:0002292560109 1:000033762x þ 0:0000677922682 1:000080001x 0:0000778698482
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
357
By applying the new algorithm on (3.1), with n ¼ 2 and m ¼ 3, we obtain: Step 1 u0 ¼ 0:2 þ 0:75x; u1 ¼ 0:2027500000 þ 0:2709687500x; u2 ¼ 0:004733515625 0:01775068359x; /1 ¼ u0 þ u1 þ u2 ¼ 1:003218066x 0:007483515625: Step 2 u0 ¼ 1:003218066x 0:007483515625; u1 ¼ 0:0111393006 0:0082446435x; u2 ¼ 0:003355718426 þ 0:004861851276x; /2 ¼ u0 þ u1 þ u2 ¼ 0:9998352735x þ 0:000300066549: Step 3 u0 ¼ 0:9998352735x þ 0:000300066549; u1 ¼ 0:0004263127 þ 0:0003413047x; u2 ¼ 0:0001150160701 0:0001704570404x; /3 ¼ u0 þ u1 þ u2 ¼ 1:000006121x 0:0000112300809: Obviously /3 is more accurate than the approximation obtained with standard Adomian method. Note that in standard Adomian method we used A0 ; A1 ; A2 ; . . . ; A7 where as in the new method we obtained a better approximation by calculating only A0 , A1 , three times. Example 2. Consider the following nonlinear Volterra integral equation (see [5]): uðxÞ ¼
Z 0
x
1 þ u2 ðtÞ dt; 1 þ t2
ð3:2Þ
with the exact solution uðxÞ ¼ x. We use standard and restarted Adomian method. For different values of x, we show the errors of standard Adomian method (ESA) and restarted Adomian method (ERA) in the the following table.
358
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
x
ESA (four iterations)
ERA (m; n ¼ 2)
ESA (six iterations)
ERA (m ¼ 3, n ¼ 2)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.85800e)13 0.15967e)9 0.11834e)7 0.22795e)6 0.20666e)5 0.11544e)4 0.460341e)4 0.143445e)3 0.370789e)3 0.828933e)3
0.21190e)13 0.39889e)10 0.30123e)8 0.59510e)7 0.556476e)6 0.322075e)5 0.133524e)4 0.433488e)4 0.116874e)3 0.27259e)3
0.1391e)17 0.39819e)13 0.14026e)10 0.784880e)9 0.15686e)7 0.16173e)6 0.105192e)5 0.48837e)5 0.17564e)4 0.51808e)4
0.44e)19 0.12600e)14 0.45993e)12 0.26989e)10 0.57129e)9 0.62878e)8 0.43914e)7 0.21978e)6 0.85409e)6 0.27250e)5
Example 3. Consider the following nonlinear Volterra integro-differential equation: Z t u0 ðtÞ ¼ t5 =5 ðu2 ðsÞ 2Þds; uð0Þ ¼ 0; ð3:3Þ 0
with the exact solution uðtÞ ¼ t2 . Proceeding as before, we apply both of the methods on problem (3.3). We summarize the errors of both methods for various values of x in the following table.
x
ERA (m; n ¼ 2)
ESA (four iterations)
ERA (m ¼ 3, n ¼ 2)
ESA (six iterations)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.142e)18 0.23339e)14 0.68133e)12 0.38238e)10 0.86939e)9 0.111616e)7 0.96590e)7 0.62624e)6 0.32565e)5 0.1422847e)4
0.74074e)13 0.75849e)10 0.43733e)8 0.77634e)7 0.72250e)6 0.44678e)5 0.20827e)4 0.78906e)4 0.25499e)3 0.726293e)3
0.4e)21 0.1e)20 0.3e)20 0.72e)18 0.9640e)16 0.530867e)14 0.15761e)12 0.29739e)11 0.39690e)10 0.40309e)9
0.142e)18 0.23338e)14 0.68124e)12 0.38221e)10 0.86847e)9 0.11137e)7 0.96197e)7 0.62188e)6 0.32201e)5 0.13985e)4
Example 4. Consider the nonlinear Volterra integro-differential equation (see [7])
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
k
du ¼ uðtÞ u2 ðtÞ uðtÞ dt
Z
359
t
ð3:4Þ
uðxÞ dx; 0
where uðtÞ is the scaled population of identical individuals at a time t and k is a prescribed parameter. The exact values of umax were evaluated by using k umax ¼ 1 þ k ln : 1 þ k u0 We apply the restarted Adomian method (RA) and standard Adomian method (SA) on problem (3.4) for k ¼ 0:1, 0.2, 0.5. The following table shows the approximate values of umax for different values of k. k
Critical t
SA (12 iterations)
RA (n ¼ 2, m ¼ 4)
Exact umax
0.1 0.2 0.5
0.4644767322 0.8168581189 1.6267110031
)4.168187974 )0.4299144851 0.4202784599
0.771005464 0.6591036567 0.4851910375
0.76974144907 0.6590503816 0.4851902914
4. Conclusion In this paper, we applied the restarted Adomian method to nonlinear integral equations and nonlinear integro-differential equations and showed that the new algorithm gives better approximate solutions than the standard Adomian method. References [1] G. Adomian, Solving Frontier Problems of Physics. The Decomposition Method, Kluwer, Boston, MA, 1994. [2] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501–544. [3] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition method, Comput. Math. Appl. 29 (1995) 103–108. [4] E. Babolian, SH. Javadi, Restarted Adomian method for algebraic equations, Appl. Math. Comput., in press. [5] T. Badradine, K. Abbaoui, Y. Cherruault, Convergence of AdomianÕs method applied to integral equations, Kybernetes 28 (1999) 557–564. [6] Y. Cherruault, G. Saccomandi, B. Some, New results for convergence of AdomianÕs method applied to integral equations, Math. Comput. Model. 16 (1992) 85–93. [7] A.M. Wazwaz, Analytical approximations and Pade approximants for VolterraÕs population model, Appl. Math. Comput. 100 (1999) 13–25. [8] A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999) 77–86.
Restarted Adomian method for integral equations E. Babolian a, Sh. Javadi a
a,*
, H. Sadeghi
b
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, 599 Taleghani Avenue, Tehran 15614, Iran b Department of Mathematics, Azad University of Kerman, Kerman, Iran
Abstract In this paper we introduce a new method, based on Adomian method, for solving nonlinear integral equations. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Restarted Adomian method; Modified Adomian method; Integral equation
1. Introduction The Adomian decomposition scheme is a method for solving a wide range of problems whose mathematical models yield equation or system of equations involving algebraic, differential, integral and integro-differential (for example see [1,2]). The main algorithm of AdomianÕs decomposition method applies to a general nonlinear equation of the form u N ðuÞ ¼ f ;
ð1:1Þ
where N is a nonlinear operator on a Hilbert space H , f is a known element of H and we are seeking u 2 H satisfying (1.1). We assume that for every f 2 H , Eq. (1.1) has a unique solution. The decomposition technique consists of representing the solution of (1.1) as a series
*
Corresponding author. E-mail address: [email protected] (Sh. Javadi).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00636-2
354
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
u¼
1 X
ð1:2Þ
un
n¼0
and the nonlinear operator N is decomposed as N ðuÞ ¼
1 X
ð1:3Þ
An ;
n¼0
where the An Õs are Adomian polynomials of u0 ; . . . ; un given by " !# 1 X 1 dn i N k ui ; n ¼ 0; 1; 2; . . . An ¼ n! dkn i¼0
ð1:4Þ
k¼0
upon substitution of Eqs. (1.2) and (1.3) into the functional equation (1.1) we obtain: 1 X
un
1 X
n¼0
An ¼ f ;
ð1:5Þ
n¼0
the convergence of the series in (1.5) will yield u0 ¼ f ; u 1 ¼ A0 ; .. . un ¼ An1 : In this paper, we present a new algorithm based on Adomian method. In the new algorithm, we use the modified Adomian method [8] which proposed a slight variation only on the components u0 and u1 , and restarted Adomian method applied to algebraic equations [4]. We will show by some examples that convergence rate of the new method is more accelerate than standard Adomian method.
2. Description of the restarted Adomian method Various formulas exist for calculating Adomian polynomials. The following theorem gives a formula for An as a function of u0 . Theorem 1 An ¼ where
X
ca1 ...an ðN ðu0 ÞÞ
nþ1a1
ðN 0 ðu0 ÞÞ
a1 a2
ðN ðn1Þ ðu0 ÞÞ
an1 an
a
ÞðN ðnÞ ðu0 ÞÞ n ;
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
ca1 ...an ¼
355
n! ða1 a2 Þ! ðan1 an Þ!an !ð1!Þ 1 ðn þ 1 a1 Þ!
Proof. See [3].
ða1a2Þ
½ðn 1Þ!
ðan1 an Þ
This theorem shows the dependence of Adomian polynomials (and correspondingly Adomian method) on u0 . In the new algorithm we update u0 in each step. This can be done by adding a proper function to two sides of Eq. (1.1). Let g be the proper function which will be determined next, then u N ðuÞ þ g ¼ f þ g: ð2:1Þ To solve the problem (2.1), we use the modified Adomian method. Let u and N ðuÞ be decomposed as (1.2) and (1.3). Then we can obtain ui as u0 ¼ f ; u 1 ¼ f g þ A0 ; u 2 ¼ A1 ; .. . un ¼ An1 ; and introduce the following algorithm named ‘‘restarted Adomian method’’. 2.1. The algorithm Choose small natural numbers m, n. Step 1: Apply the Adomian method to Eq. (1.1) and calculate u0 ; u1 ; . . . ; un : Set /1 ¼ u0 þ u1 þ þ un : Step 2: For i ¼ 2 : m, g ¼ /i1 ; u0 ¼ g; u 1 ¼ f g þ A0 ; u 2 ¼ A1 ; .. . un ¼ An1 : Set /i ¼ u0 þ u1 þ þ un . end of for.
356
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
Remark 1. /m can be considered as the approximate solution of Eq. (2.1). Remark 2. In the Adomian method usually the sum of the first few terms gives an approximation of u. In the new algorithm to obtain an accurate approximation of the exact solution, we do not calculate the Adomian polynomials for large indexes, i.e. m, n are considered small, say m ¼ 3, n ¼ 2.
3. Numerical results Example 1. Consider the nonlinear Fredholm integral equation:
uðxÞ ¼
Z
1
ðx tÞu3 ðtÞ dt þ
0
3x 1 þ ; 4 5
ð3:1Þ
with the exact P1solution uðxÞ ¼ x (see [6]). Let u ¼ i¼0 uP i be a solution for (3.1) where ui is specified by Adomian i method and si ¼ j¼0 uj . By applying the standard Adomian method on (3.1), we get: 3x 1 þ ; 4 5 Z 1 ui ðxÞ ¼ ðx tÞAi1 dt;
u0 ðxÞ ¼
i ¼ 1; 2; . . .
0
The first nine terms and corresponding partial sums are given the following table:
i
ui
si
0 1 2 3 4 5 6 7 8
0:2 þ 0:75x 0:2027500000 þ 0:2709687500x 0:004733515625 0:01775068359x 0:01096026963 0:007714847995x 0:003482967564 þ 0:005681606709x 0:0006986036410 0:0006525609164x 0:0009340732109 0:0009331769228x 0:0001614637427 þ 0:0004346747149x 0:0001456621164 þ 0:00004623896704x
0:2 þ 0:75x 1:012930896x 0:06130888154 1:003218066x 0:00748351562 0:9955032184x þ :003476754005 1:001184825x 0:000006213559 1:000532264x 0:0007048172000 0:9995990873x þ 0:0002292560109 1:000033762x þ 0:0000677922682 1:000080001x 0:0000778698482
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
357
By applying the new algorithm on (3.1), with n ¼ 2 and m ¼ 3, we obtain: Step 1 u0 ¼ 0:2 þ 0:75x; u1 ¼ 0:2027500000 þ 0:2709687500x; u2 ¼ 0:004733515625 0:01775068359x; /1 ¼ u0 þ u1 þ u2 ¼ 1:003218066x 0:007483515625: Step 2 u0 ¼ 1:003218066x 0:007483515625; u1 ¼ 0:0111393006 0:0082446435x; u2 ¼ 0:003355718426 þ 0:004861851276x; /2 ¼ u0 þ u1 þ u2 ¼ 0:9998352735x þ 0:000300066549: Step 3 u0 ¼ 0:9998352735x þ 0:000300066549; u1 ¼ 0:0004263127 þ 0:0003413047x; u2 ¼ 0:0001150160701 0:0001704570404x; /3 ¼ u0 þ u1 þ u2 ¼ 1:000006121x 0:0000112300809: Obviously /3 is more accurate than the approximation obtained with standard Adomian method. Note that in standard Adomian method we used A0 ; A1 ; A2 ; . . . ; A7 where as in the new method we obtained a better approximation by calculating only A0 , A1 , three times. Example 2. Consider the following nonlinear Volterra integral equation (see [5]): uðxÞ ¼
Z 0
x
1 þ u2 ðtÞ dt; 1 þ t2
ð3:2Þ
with the exact solution uðxÞ ¼ x. We use standard and restarted Adomian method. For different values of x, we show the errors of standard Adomian method (ESA) and restarted Adomian method (ERA) in the the following table.
358
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
x
ESA (four iterations)
ERA (m; n ¼ 2)
ESA (six iterations)
ERA (m ¼ 3, n ¼ 2)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.85800e)13 0.15967e)9 0.11834e)7 0.22795e)6 0.20666e)5 0.11544e)4 0.460341e)4 0.143445e)3 0.370789e)3 0.828933e)3
0.21190e)13 0.39889e)10 0.30123e)8 0.59510e)7 0.556476e)6 0.322075e)5 0.133524e)4 0.433488e)4 0.116874e)3 0.27259e)3
0.1391e)17 0.39819e)13 0.14026e)10 0.784880e)9 0.15686e)7 0.16173e)6 0.105192e)5 0.48837e)5 0.17564e)4 0.51808e)4
0.44e)19 0.12600e)14 0.45993e)12 0.26989e)10 0.57129e)9 0.62878e)8 0.43914e)7 0.21978e)6 0.85409e)6 0.27250e)5
Example 3. Consider the following nonlinear Volterra integro-differential equation: Z t u0 ðtÞ ¼ t5 =5 ðu2 ðsÞ 2Þds; uð0Þ ¼ 0; ð3:3Þ 0
with the exact solution uðtÞ ¼ t2 . Proceeding as before, we apply both of the methods on problem (3.3). We summarize the errors of both methods for various values of x in the following table.
x
ERA (m; n ¼ 2)
ESA (four iterations)
ERA (m ¼ 3, n ¼ 2)
ESA (six iterations)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.142e)18 0.23339e)14 0.68133e)12 0.38238e)10 0.86939e)9 0.111616e)7 0.96590e)7 0.62624e)6 0.32565e)5 0.1422847e)4
0.74074e)13 0.75849e)10 0.43733e)8 0.77634e)7 0.72250e)6 0.44678e)5 0.20827e)4 0.78906e)4 0.25499e)3 0.726293e)3
0.4e)21 0.1e)20 0.3e)20 0.72e)18 0.9640e)16 0.530867e)14 0.15761e)12 0.29739e)11 0.39690e)10 0.40309e)9
0.142e)18 0.23338e)14 0.68124e)12 0.38221e)10 0.86847e)9 0.11137e)7 0.96197e)7 0.62188e)6 0.32201e)5 0.13985e)4
Example 4. Consider the nonlinear Volterra integro-differential equation (see [7])
E. Babolian et al. / Appl. Math. Comput. 153 (2004) 353–359
k
du ¼ uðtÞ u2 ðtÞ uðtÞ dt
Z
359
t
ð3:4Þ
uðxÞ dx; 0
where uðtÞ is the scaled population of identical individuals at a time t and k is a prescribed parameter. The exact values of umax were evaluated by using k umax ¼ 1 þ k ln : 1 þ k u0 We apply the restarted Adomian method (RA) and standard Adomian method (SA) on problem (3.4) for k ¼ 0:1, 0.2, 0.5. The following table shows the approximate values of umax for different values of k. k
Critical t
SA (12 iterations)
RA (n ¼ 2, m ¼ 4)
Exact umax
0.1 0.2 0.5
0.4644767322 0.8168581189 1.6267110031
)4.168187974 )0.4299144851 0.4202784599
0.771005464 0.6591036567 0.4851910375
0.76974144907 0.6590503816 0.4851902914
4. Conclusion In this paper, we applied the restarted Adomian method to nonlinear integral equations and nonlinear integro-differential equations and showed that the new algorithm gives better approximate solutions than the standard Adomian method. References [1] G. Adomian, Solving Frontier Problems of Physics. The Decomposition Method, Kluwer, Boston, MA, 1994. [2] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501–544. [3] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition method, Comput. Math. Appl. 29 (1995) 103–108. [4] E. Babolian, SH. Javadi, Restarted Adomian method for algebraic equations, Appl. Math. Comput., in press. [5] T. Badradine, K. Abbaoui, Y. Cherruault, Convergence of AdomianÕs method applied to integral equations, Kybernetes 28 (1999) 557–564. [6] Y. Cherruault, G. Saccomandi, B. Some, New results for convergence of AdomianÕs method applied to integral equations, Math. Comput. Model. 16 (1992) 85–93. [7] A.M. Wazwaz, Analytical approximations and Pade approximants for VolterraÕs population model, Appl. Math. Comput. 100 (1999) 13–25. [8] A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999) 77–86.