# Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane‐Emden equations of the first and second kinds

Abdul‐Majid Wazwaz (Department of Mathematics, Saint Xavier University, Chicago, Illinois, USA)
Randolph Rach (Hartford, Michigan, USA)

ISSN: 0368-492X

Article publication date: 18 October 2011

## Abstract

### Purpose

The purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden equations of the first and second kinds.

### Design/methodology/approach

The paper examines the theoretical framework of the Adomian decomposition method and compares it with the variational iteration method. The paper seeks to determine the relative merits and computational benefits of both the Adomian decomposition method and the variational iteration method in the context of the important physical models of the Lane‐Emden equations of the first and second kinds.

### Findings

The Adomian decomposition method is shown to readily solve the Lane‐Emden equations of both the first and second kinds for all positive real values of the system coefficient α and for all positive real values of the nonlinear exponent m. The decomposition series solution of these nonlinear differential equations requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The paper shows that the variational iteration method works effectively to solve the Lane‐Emden equation of the first kind for system coefficient values α=1, 2, 3, 4 but only for positive integer values of the nonlinear exponent m. The successive approximations of the solution of these nonlinear differential equations require the determination of the appropriate Lagrange multipliers, which are established in this paper. These two methodologies overcome the singular behavior at the origin x=0. The paper shows that the variational iteration method is impractical for solving either the Lane‐Emden equation of the first kind for non‐integer values of the system exponent m or the Lane‐Emden equations of the second kind. Indeed the Adomian decomposition method is shown to solve even the generalized Lane‐Emden equation for any analytic nonlinearity and all positive values of the system coefficient α in a practical and straightforward manner. The conclusions are supported by several numerical examples.

### Originality/value

This paper presents an accurate comparison of the Adomian decomposition method with the variational iteration method for solving the Lane‐Emden equations of the first and second kinds. The paper presents a new solution algorithm for the generalized Lane‐Emden equation for any analytic system nonlinearity and for any model geometry as characterized by all possible positive real values of the system coefficient α.

## Citation

Wazwaz, A. and Rach, R. (2011), "Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane‐Emden equations of the first and second kinds", Kybernetes, Vol. 40 No. 9/10, pp. 1305-1318. https://doi.org/10.1108/03684921111169404

## Publisher

:

Emerald Group Publishing Limited

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