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A convenient computational form for the Adomian polynomials. (English) Zbl 0552.60061

Recent important generalizations by G. Adomian [Stochastic systems (1983; Zbl 0523.60056)] have extended the scope of his decomposition method for nonlinear stochastic operator equations (see also iterative method, inverse operator method, symmetrized method, or stochastic Green’s function method) very considerably so that they are now applicable to differential, partial differential, delay, and coupled equations which may be strongly nonlinear and/or strongly stochastic (or linear or deterministic as subcases). Thus, for equations modeling physical problems, solutions are obtained rapidly, easily, and accurately. The methodology involves an analytic parametrization in which certain polynomials \(A_ n\), dependent on the nonlinearity, are derived. This paper establishes simple symmetry rules which yield Adomian’s polynomials quickly to high orders.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)

Citations:

Zbl 0523.60056
Full Text: DOI

References:

[1] Adomian, G., Stochastic Systems (1983), Academic Press: Academic Press New York · Zbl 0504.60066
[2] Adomian, G., The solution of general linear and nonlinear stochastic system, (Rose, J., Norbert Wiener Memorial Volume. Norbert Wiener Memorial Volume, Modern Trends in Cybernetics and Systems (1976), Editura Technica: Editura Technica Bucharest), 160-170
[3] Adomian, G., Nonlinear stochastic differential equations, J. Math. Anal. Appl., 55, 441-542 (1976) · Zbl 0351.60053
[4] Adomian, G., On the modeling and Analysis of nonlinear stochastic systems, (Avula; Bellman; Luke; Rigler, Proceedings, International Conference on Mathematical Modeling, Vol. 1 (1980), Univ. of Missouri: Univ. of Missouri Rolla), 29-40 · Zbl 0546.60062
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[6] Adomian, G., Solution of nonlinear stochastic physical problems, (Rend. Sem. Mat. Univ. Politec.. Rend. Sem. Mat. Univ. Politec., Torino (1982)), 1-22, Numero Speciale · Zbl 0825.35055
[7] G. Adomian; G. Adomian · Zbl 0659.93003
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[10] Adomian, G.; Malakian, K., Self-correcting approximate solutions by the iterative method for nonlinear stochastic differential equations, J. Math. Anal. Appl., 76, 309-327 (1980) · Zbl 0449.65093
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