Solving the Systems of Equations of Lane-Emden Type by Differential Transform Method Coupled with Adomian Polynomials

Lie-jun Xie,Cai-lian Zhou,Song Xu

doi:10.3390/math7040377

Abstract

In this work, we applied the improved differential transform method to find the solutions of the systems of equations of Lane-Emden type arising in various physical models. With our proposed scheme, the desired solutions take the form of a convergent series with easily computable components. The results disclosing the relation between the differential transforms of multi-variables and the corresponding Adomian polynomials are proven. One can see that both the differential transforms and the Adomian polynomials of those nonlinearities have the same mathematical structure merely with constants instead of variable components. By using this relation, we computed the differential transforms of nonlinear functions given in the systems. The validity and applicability of the proposed method are illustrated through several homogeneous and nonhomogeneous nonlinear systems.

Highlights

The differential transform method (DTM) was firstly introduced by Pukhov [1,2,3]
We can obtain the coefficients of a Taylor series solution
Dx k x=0 from which one can see that the differential transform is derived from the classical Taylor series method

Summary

Introduction

The differential transform method (DTM) was firstly introduced by Pukhov [1,2,3]. his work passed unnoticed. We shall apply the DTM to solve the following systems of equations of Lane-Emden type:. We inevitably encounter the complicated differential transforms of those nonlinearities with multi-variables, if the traditional DTM is employed to obtain the solution of them. As far as we know, there is not any new work which engaged in calculating the differential transforms of nonlinear functions with multi-variables. The main difficulty of the systems of Lane-Emden type equations is the singular behavior at the origin Both the VIM and the ADM overcome this obstacle by finding a corresponding Volterra integral form for the given system. A brief conclusion is given in Section 5 to end this paper

Dtm and Adomian Polynomial

Adomian Polynomial

Differential Transform of Nonlinearities

Applications

Conclusions