Numerical analysis of Lane Emden–Fowler equations

Atta Ullah,Kamal Shah

doi:10.1080/16583655.2018.1451118

Atta Ullah, Kamal Shah

Open Access

https://doi.org/10.1080/16583655.2018.1451118

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Abstract

ABSTRACTThis paper is devoted to the numerical solutions of Lane Emden–Fowler partial differential equations. For the numerical analysis, we apply Laplace transform coupled with the Adomian decomposition method known as the Laplace Adomian decomposition method (LADM). We also compare our numerical results with some already existing methods such as homotopy perturbation method, which reveals that the LADM provides the same solutions without any need of perturbation or collocation. Some numerical test problems are also provided by using Maple software.

Highlights

Partial differential equations (PDEs) play significant roles in the modelling of many physical, biological and dynamical phenomena
This paper is devoted to the numerical solutions of Lane Emden–Fowler partial differential equations
We apply Laplace transform coupled with the Adomian decomposition method known as the Laplace Adomian decomposition method (LADM)

Summary

Introduction

Partial differential equations (PDEs) play significant roles in the modelling of many physical, biological and dynamical phenomena. A variety of problems in the dynamics and physics can be described by using the Lane–Emden-type differential equation. The Lane–Emden equation is a dimensionless form of Poisson’s equation describing the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid Solving such type of problems gives some particular solutions which lead to the Lane Emden-type equations. This is our aim to obtain the analytical solutions of the aforesaid problems for physical understanding Many methods such as the homotopy perturbation method (HPM) [9, 10] and semi-inverse method [11] were used to find the approximate analytical solutions of the Lane Emden–Fowler equation. We use the Laplace Adomian decomposition method (LADM) for the approximate analytical solutions of the Lane Emden–Fowler-type equation given by.

Preliminaries and procedure of LADM

Numerical examples

Conclusion