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.

中文翻译：

---

步长不等分的Adomian分解方法求解Lane-Emden型方程

目的

本文的目的是使用改进的Adomian分解方法（MADM）获得不等步长分区的Lane-Emden型方程的高精度数值解。

设计/方法/方法

首先，作者描述了用于解决非线性微分方程的标准Adomian分解方案和Adomian多项式。之后，为了快速计算Adomian多项式，提出了一种基于Duan推论和Rach规则的算法。然后，对域的不等步长分区讨论MADM，以获得Lane-Emden型方程的数值解。此外，讨论了收敛性分析和近似解的误差界。

发现

与其他现有方法相比，该方法消除了问题的奇异行为，并在较大的有效收敛范围内提供了高精度的数值解，如测试示例所示。

创意/价值

与其他方法不同，所提出的方法不需要线性化或微扰来获得奇异微分方程的解析和数值解，并且所获得的结果在物理上更现实。