Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

Abstract

Purpose The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

“…The ADM (Adomian and Rach, 1993; Kumar and Umesh, 2022; Umesh and Kumar, 2021a; Bougoffa et al , 2015) is a powerful tool for the analytical and numerical treatment of NDEs that arise in the mathematical modelling of real-world problems. Consequently, the corresponding solution ξ decomposes into an infinite series as: And, the non-linear term N(ξ) transformed into the following series: where An=An(ξ0,ξ1,⋯,ξn) represents the Adomian polynomial and given by: while μ is a grouping constant.…”