An efficient algorithm for the multivariable Adomian polynomials

• Published in: Applied mathematics and computation 2010-11, Vol.217 (6), p.2456-2467
• Main Author: Duan, Jun-Sheng
• Format: Article
• Language: English
• Subjects: Acceleration of convergence; Adomian decomposition method; Adomian polynomials; Algorithms; Derivatives; Difference and functional equations, recurrence relations; Differential equation; Differential equations; Exact sciences and technology; Functions of a complex variable; Mathematical analysis; Mathematical models; Mathematics; Matrices; Matrix methods; Multivariable; Multivariable function; Nonlinear operator; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.07.046

In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u 1, u 2, … , u m about the initial solution components u 1,0, u 2,0, … , u m,0 ; of course the multivariable Adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations. The index matrices and their simplified forms of the multivariable Adomian polynomials are introduced. We obtain the recurrence relations for the simplified index matrices, which provide a convenient algorithm for rapid generation of the multivariable Adomian polynomials. Another alternative algorithm for term recurrence is established. In these algorithms recurrence processes do not require complicated operations such as parametrization, expanding and regrouping, derivatives, etc. as practiced in prior art. The MATHEMATICA program generating the Adomian polynomials based on the algorithm in this article is designed.