Restarted Adomian method for algebraic equations

Applied Mathematics and Computation 146 (2003) 533–541 www.elsevier.com/locate/amc

Restarted Adomian method for algebraic equations E. Babolian *, Sh. Javadi Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, 599 Taleghani Avenue, Tehran 15614, Iran

Abstract In this paper we introduce a new algorithm based on Adomian method to solve algebraic equations. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Decomposition method; Algebraic equation

1. Introduction The Adomian method is a technique for solving nonlinear equations of various kinds. In many papers, e.g. [1] and [4], authors consider a nonlinear equation: F ðxÞ ¼ 0; which can be transformed to x ¼ F0 ðxÞ þ c0 ;

ð1:1Þ

where F0 is a nonlinear function and c0 is a constant. The Adomian method consists of calculating x as a series: x¼

1 X

xi :

ð1:2Þ

i¼0

*

Corresponding author. E-mail addresses: [email protected] (E. Babolian), [email protected] (Sh. Javadi).

0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00603-3

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The nonlinear function is decomposed as: 1 X Ai ; F ðxÞ ¼

ð1:3Þ

i¼0

where the Ai Õs are Adomian polynomials defined by X   x i ki  : An ðx0 ; . . . ; xn Þ ¼ ð1=n!Þðdn =dkn ÞF k¼0

Putting (1.2) and (1.3) into (1.1) leads to 1 X i¼0

xi ¼

1 X

A i þ c0 ;

i¼0

P each term of the series x ¼ 1 i¼0 xi ; according to the Adomian method, can be calculated from the relations: x0 ¼ c0 x 1 ¼ A0 x 2 ¼ A1 .. . xn ¼ An1 : In computing x using any software, as n increases the number of terms in the expression for An increases and this causes propagation of round off errors, on the other hand, the factor 1=n! in the formula of An makes it very small, P so that its contribution to x is negligible, hence, the first few terms of the series 1 i¼0 xi determine the accuracy of the approximate solution. By attention to this, we introduce a new algorithm based on Adomian method to improve the accuracy dramatically (See Examples 2.1 and 2.2.). 2. Modified Adomian method P In Ref. [3], when An is absolutely convergent, the following equation was proved: X X ðx  x0 Þn An ¼ F ðnÞ ðx0 Þ; ð2:1Þ n! Then by Eq. (2.1), if x0 be close to the exact solution, the truncated series P n i¼0 xi is a good approximation, for a moderate value of n. So, we modify the Eq. (1.1) so that x0 be a better starting point. Suppose that we have the nonlinear equation (1.1) with the exact solution x . Let c1 be a point close to x such that: jx  c1 j < jx  c0 j:

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535

Now add ðc1  c0 Þ to both sides of (1.1) to obtain a new nonlinear equation: x  F1 ðxÞ ¼ c1 ; ð2:2Þ where, F1 ðxÞ ¼ F0 ðxÞ  ðc1  c0 Þ: It is clear that x is the exact solution of (2.2). Now, we can solve the Eq. (2.2) with the Adomian method instead of the Eq. (1.1). Therefore we suggest the following algorithm named ‘‘the restarted Adomian method’’, for solving x  F0 ðxÞ ¼ c0 : 2.1. Restarted Adomian algorithm Choose small positive number  and small natural number m. For i ¼ 0; 1; 2; . . . ; do, (step 1) x0 ¼ ci x 1 ¼ A0 x 2 ¼ A1 .. . xm ¼ Am1 (step 2) xðiÞ ¼ ciþ1 :¼ x0 þ x1 þ þ xm ; if jciþ1  ci j <  stop, (step 3) Fnew ¼ F0  ðciþ1  c0 Þ; end(for). Remark 2.1. If one chooses m ¼ 1, the fixed point method is obtained. Remark 2.2. Usually we choose m small, say 2 6 m 6 5, so we calculate An for small values of n. Example 2.1. Consider the equation x ¼ 2 þ expðxÞ;

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with the exact solution x ¼ 2:12002823898764ð14DÞ. When we solve the above equation with standard Adomian method, we obtain: x ¼ x0 þ x1 þ þ x9 ¼ 2:12009404275892; where, x0 ¼ x1 þ x2 þ x3 þ x4 þ x5 ¼ 2:12010236353753; and x6 ¼ 8:75550260349273  106 ; x7 ¼ þ4:51566290513189  107 ; x8 ¼ 1:73715011769016  108 ; x9 ¼ þ5:29196468808098  1010 : Clearly, comparing x , x and x0 , terms x6 up to x9 have no significant contribution to the first few wrong digits of x0 . Hence, if we calculate other terms, the approximate solution does not improve significantly. Now, we apply the new algorithm with m ¼ 5. Passing steps (1) and (2) xð1Þ ¼ x0 þ x1 þ þ x5 ¼ 2:12010236353753; after modifying the last equation in step (3) and return to step (1) we have: xð2Þ ¼ x0 þ x1 þ þ x5 ¼ 2:12002823714773; which is closer to the exact solution x . Example 2.2 x3 þ 4x2 þ 8x þ 8 ¼ 0 with the exact solution x ¼ 2. Eq. (1.1) involves x¼

1 X i¼0

1 1 xi ¼ 1  x2  x3 : 2 8

The approximate solution using standard Adomian method is: x ¼ x0 þ x1 þ þ x9 ¼ 1:5961570739746; where, x7 , x8 and x9 are numerically zero. Now, we apply the new algorithm to solve the above equation with m ¼ 5. Steps (1) and (2) give: xð1Þ ¼ x0 þ x1 þ þ x5 ¼ 1:60882568359375; in step (3) we modify the old equation. The algorithm returns to step (1) and the new approximation is:

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537

xð2Þ ¼ x0 þ x1 þ þ x5 ¼ 1:92852548353598; with repeating the new algorithm, we have xð3Þ ¼ x0 þ x1 þ þ x5 ¼ 1:99564152220937; xð4Þ ¼ x0 þ x1 þ þ x5 ¼ 1:99985578998793; xð5Þ ¼ x0 þ x1 þ þ x5 ¼ 1:99999548466382; xð6Þ ¼ x0 þ x1 þ þ x5 ¼ 1:99999985888714: Obviously xðiÞ tends to the root as i increases.

3. Restarted Adomian algorithm and solving modified equation In Ref. [2], authors took functional equation s ¼ F ðx0 þ sÞ;

ð3:1Þ

which can be solved with the Adomian method. They modified Eq. (3.1) as following: s ¼ Fl ðx0 þ sÞ ¼

ls þ F ðx0 þ sÞ ; 1þl

ð3:2Þ

where l 6¼ 1 is a scalar such that Fl0 ðx0 þ sÞ ¼ 0. Now, we use this strategy to solve x ¼ F0 ðxÞ þ c0 , we obtain: x¼

F0 ðxÞ  F00 ðx Þx þ c0 ; 1  F00 ðx Þ

ð3:3Þ

where x is the exact solution (1.1). Since x in not available, we can start the restarted Adomian method with new equation and x0 instead of x , then after calculating find better solutions, we substitute it in Eq. (3.3) and continue this process to obtain a good approximate solution. Concerning above statement, we present following algorithm to solve equation x ¼ F1 ðxÞ þ c1 , where, F1 ðxÞ ¼

F0 ðxÞ  F00 ðc0 Þx þ c0  c0 ; 1  F00 ðc0 Þ

c1 ¼ c0 : 3.1. Algorithm Choose small positive number  and small natural number m. For i ¼ 1; 2; . . . ; do,

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(step 1) x0 ¼ ci x 1 ¼ A0 x 2 ¼ A1 .. . xm ¼ Am1 (step 2) xðiÞ ¼ ciþ1 :¼ x0 þ x1 þ þ xm if jciþ1  ci j <  stop, (step 3) Fnew ¼

F0 ðxÞ  F00 ðciþ1 Þx þ c0  ciþ1 1  F00 ðciþ1 Þ

end(for). Remark 3.1. Since 0 Fnew ðciþ1 Þ ¼

F00 ðciþ1 Þ  F00 ðciþ1 Þ ¼ 0; 1  F00 ðciþ1 Þ

then we have x2 ¼ 0, for i ¼ 1; 2; . . . Example 3.1. Consider again the equation x ¼ 2 þ expðxÞ; which can be transformed to x¼

expðxÞ þ expðc0 Þx þ 2 : 1 þ expðc0 Þ

The following approximate solution was obtained in [2] after four iterations: x 2 þ s1 þ s2 þ s3 þ s4 ¼ 2:120016168: We apply the Algorithm 3.1 with m ¼ 3. i

xðiÞ ’ x0 þ x1 þ x3

1 2

2.12006139548564 2.12002823898764

Now, we apply Algorithm 2.1 with m ¼ 2 on changed equation in [2], we get:

E. Babolian, Sh. Javadi / Appl. Math. Comput. 146 (2002) 533–541

i

xðiÞ ’ x0 þ s1 þ s2

1 2 3

2.11920292202212 2.12002805439380 2.12002823895408

539

This shows that Algorithm 3.1 is working much better.

Example 3.2. Consider the equation 5

x2  ð1  xÞ ¼ 0

or

x ¼ 0:2 þ 1:8x2  2x3 þ x4  0:2x5 ;

with the exact solution x ¼ 0:34595481584825ð14DÞ: When we apply standard Adomian method, after 10 iterations, we obtain: x x0 þ þ x10 ¼ 0:341469: The modified equation is: s¼

3125s5  12500s4 þ 20000s3  12875s2  899 : 7650

The obtained solution is: x 0:2 þ s1 þ s2 þ s3 þ s4 þ s5 ¼ 0:346021. By applying restarted Adomian algorithm 2.1 with m ¼ 2, we obtain: i

xðiÞ ’ x0 þ s1 þ s2

1 2 3 4 5

0.34256006171720 0.34555420265661 0.34590752257473 0.34594923241099 0.34595415666318

Now, by applying Algorithm 3.1 with m ¼ 3, F1 ðxÞ ¼

ð1:8x2  2x3 þ x4  0:2x5 Þ  ð3:6c0  6c20 þ 4c30  c40 Þx þ 0:2 ; 1  ð3:6c0  6c20 þ 4c30  c40 Þ

and c1 ¼ 0:2:

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We get: i

xðiÞ ’ x0 þ x1 þ x3

1 2 3

0.34075881183065 0.34595482569796 0.34595481584824

Example 3.3. Consider the equation, ex  3x2 ¼ 0; and its solution in (0,1) which is approximately 0:910007572488701ð15DÞ. Rewrite it in canonical form: rffiffiffiffi ex x¼ ¼ F ðxÞ: 3

equal

to

ð3:4Þ

By the standard Adomian technique we get x x0 þ þ x10 ¼ 0:905950: In [2], Eq. (3.4) modified to formula: pffiffiffi 41060 3es=2 11853 ; s¼  29207 87621 and five terms approximation was given as: x 0 þ s1 þ þ s5 ¼ 0:90850: By applying Algorithm 3.1 with m ¼ 3, we get i

xi ’ x0 þ x1 þ x3

1 2 3

0.87849294897851 0.91000393030657 0.91000757248871

which is a closer approximation to the exact solution x : All calculations have been carried out using Matlab 5.3.

4. Conclusion In this work, we presented a new algorithm based on the Adomian technique. This new algorithm calculates better approximations to the exact solution of algebraic equations, in comparison with the standard Adomian method.

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References [1] K. Abboui, Y. Cherrualt, Convergence of AdomianÕs method applied to nonlinear equations, Math. Comp. Modell. 20 (1994) 69–73. [2] E. Babolian, J. Biazar, Solution of nonlinear equations by modified Adomian method, Appl. Math. Comput. 132 (2002) 167–172. [3] L. Gabet, The theoretical foundation of Adomian method, Comp. Math. Appl. 27 (12) (1994) 41–52. [4] R.Z. Ouedraogo, Y. Cherruault, K. Abbaoui, Convergence of AdomianÕs method applied to algebraic equations, Kybernetes 29 (9/10) (2000) 1298–1305.