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Symbolic implementation of the algorithm for calculating Adomian polynomials. (English) Zbl 1033.65036

Summary: A symbolic implementation code is developed of a technique proposed by A.-M. Wazwaz [Appl. Math. Comput. 111, 53–69 (2000; Zbl 1023.65108)] for calculating Adomian polynomials for nonlinear operators. The algorithm proposed by him offers a promising approach for calculating Adomian polynomials for all forms of nonlinearity, but it is not easy to implement due to its huge size of algebraic calculations, complicated trigonometric terms, and unique summation rules.

It is well known that the algebraic manipulation language such as Mathematica is useful to facilitate such a hard computational work. Pattern-matching capabilities peculiar feature of Mathematica are used in index regrouping which is a key role in constructing Adomian polynomials. The computer algebra software Mathematica is used to collect terms to their order and to simplify the terms.

The symbolic implementation code developed by the author (appearing at appendix) has the flexibility that may easily cover any length of Adomian polynomial for many forms of nonlinear cases. A nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code.

65J15Equations with nonlinear operators (numerical methods)
68W30Symbolic computation and algebraic computation
76N15Gas dynamics, general
47J25Iterative procedures (nonlinear operator equations)
35L60Nonlinear first-order hyperbolic equations
65M55Multigrid methods; domain decomposition (IVP of PDE)
65M70Spectral, collocation and related methods (IVP of PDE)