zbMATH — the first resource for mathematics

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.

60H25 Random operators and equations (aspects of stochastic analysis)
Full Text: DOI
[1] Adomian, G, Stochastic systems, (1983), Academic Press New York · Zbl 0504.60066
[2] Adomian, G, The solution of general linear and nonlinear stochastic system, (), 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, (), 29-40
[5] Adomian, G, Stochastic systems analysis, (), 1-17
[6] Adomian, G, Solution of nonlinear stochastic physical problems, (), 1-22, Numero Speciale · Zbl 0825.35055
[7] \scG. Adomian, “Applications of Stochastic Systems Theory to Physics and Engineering,” Academic Press, New York, in press. · Zbl 0659.93003
[8] \scG. Adomian and R. E. Bellman, “New Methods in Partial Differential Equations,” Reidel, Dordrecht, in press. · Zbl 0557.35003
[9] Adomian, G; Rach, R, Inversion of nonlinear stochastic operators, J. math. anal. appl., 91, 39-46, (1983) · Zbl 0504.60066
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.