A new definition of the Adomian polynomials

Abstract

PurposeTo provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.Design/methodology/approachDevelops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.FindingsThe four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.Originality/valueThis paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

An efficient algorithm for the multivariable Adomian polynomials

“…The convergence of the method was previously studied in [18][19][20][21][22]. Different algorithms for the Adomian polynomials have been considered for the sake of improving computational efficiency in [1][2][3][4][5][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation

“…Since it was first presented in the eighties, Adomian decomposition method has led to several modifications in an attempt to improve the accuracy or expand the application of the original method. One of such recent modification was presented in (Adomian & Rach, 1996), and an improved version of it that converges slightly faster was presented in (Rach, 2008). But inspite of the various types of modified Adomian polynomial available, the algorithm of the original Adomian polynomial is more simple and convenient in application.…”

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

“…The Adomian decomposition method (ADM) [2, 3, ( ) Nu x into the series of the Adomian polynomials [2,3,10,21,22].…”

“…Several convenient algorithms to readily generate the Adomian polynomials have since been developed by Adomian and Rach [3], Rach [21,23], Wazwaz [14,25], Abdelwahid [26], and others [27,30]. Recently, …”

“…The first Several convenient algorithms for the multivariable Adomian polynomials were proposed in [21,24,27,31]. Here we list the following convenient recursion formula for the m -variable Adomian polynomials n A [24] as…”