An efficient algorithm for the multivariable Adomian polynomials Jun-Sheng Duan College of Science, Shanghai Institute of Technology, Shanghai 201418, PR China article info Keywords: Adomian polynomials Adomian decomposition method Nonlinear operator Differential equation Multivariable function abstract 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. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The Adomian decomposition method (ADM) and its modifications [1–8] are techniques for solving nonlinear functional equations. The method, which requires no linearization or perturbation, efficiently works for a large class of initial value or boundary value problems, encompassing linear, nonlinear, and even stochastic systems. For example, it can be applied to ordinary-differential, partial-differential, delay-differential, algebraic, differential-algebraic, integral, integro-differential, non-integer-order differential equations, and so on, see e.g. [1–17]. Let us first recall the basic principles of the ADM. Consider an ordinary-differential equation in the form Lu þ Ru þ Nu ¼ gðtÞ; ð1Þ where L ¼ d p dt p is the highest order differential operator, R is the remaining linear part containing the lower order derivatives, N represents a nonlinear analytical operator and g(t) is a given function. For an initial value problem, for example, we assume that L 1 Lu = u  /, where L 1 is the p-fold definite integral operator from 0 to t, and thus / is determined by the initial values. Applying the operator L 1 on both sides of (1) gives u ¼ / þ L 1 g  L 1 Ru  L 1 Nu: ð2Þ The method supposes a series solution and decomposes the nonlinear term Nu into a series u ¼ X 1 n¼0 u n ; Nu ¼ X 1 n¼0 A n ; ð3Þ 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.046 E-mail addresses: duanjssdu@sina.com, duanjs@sit.edu.cn Applied Mathematics and Computation 217 (2010) 2456–2467 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc