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A reliable modification of Adomian decomposition method. (English) Zbl 0928.65083
The purpose of this paper is to show that, although the modified technique needs only a slight variation from the standard Adomian method, the results are improved and the convergence of the series solution is accelerated. Some illustrative examples are treated proving the performance of the modified algorithms. This interesting paper has some relationships with results previously published by {\it K. Abbaoui} and {\it Y. Cherruault} [cf. Comput. Math. Appl. 28, No. 5, 103-109 (1995; Zbl 0809.65073)]. These works proved that when the Adomian method did not converge a change in the first term of the series solution could involve the convergence of the technique. Furthermore, it has also be proved that the convergence was accelerated by modifying the choice of the first term. The author also asserts that his method minimizes the size of calculation needed. This is possible but not proved in this paper. May be that a good choice of the decomposition $f= f_1+ f_2$ could minimize the calculations size. It would be an interesting following of the present paper.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
Full Text: DOI
[1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994 · Zbl 0802.65122
[2] Adomian, G.: A review of the decomposition method and some recent results for nonlinear equation. Math. comput. Model. 13, No. 7, 17-43 (1992) · Zbl 0713.65051
[3] Adomian, G.; Rach, R.: Noise terms in decomposition series solution. Comput. math. Appl. 24, No. 11, 61-64 (1992) · Zbl 0777.35018
[4] Cherruault, Y.; Saccomandi, G.; Some, B.: New results for convergence of Adomian’s method applied to integral equations. Math. comput. Model. 16, No. 2, 85-93 (1992) · Zbl 0756.65083
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[9] E. Yee, Application of the decomposition method to the solution of the reaction-convection-diffusion equation, Appl. Math. Comput. 56 (1993) 1--27 · Zbl 0773.76055