A Reliable Treatment for Nonlinear Differential Equations

H R Marasi,M Sedighi,H Aydi,Y U Gaba,Ram Jiwari

doi:10.1155/2021/6659479

Abstract

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

Highlights

Initial value problems with singularity and of type Lane–Emden differential equation, d2y dx2 + 2 x dy dx yr (1)have been used to model a large category of phenomena in various science, such as mathematical physics and astrophysics. e first studies on these equations have been published by Lane in 1870 [1]
2 x dy dx have been used to model a large category of phenomena in various science, such as mathematical physics and astrophysics. e first studies on these equations have been published by Lane in 1870 [1]
In astrophysics and in the study of a self-gravitating spherically symmetric polytropic fluid, this equation appears as its gravitational Poisson’s equation. ere are several phenomena, such as astrophysics, aerodynamics, stellar structure, chemistry, biochemistry, and many others which can be modeled by the Lane–Emden equation [3,4,5]

Summary

Introduction

Initial value problems with singularity and of type Lane–Emden differential equation, d2y dx2 + 2 x dy dx yr (1)have been used to model a large category of phenomena in various science, such as mathematical physics and astrophysics. e first studies on these equations have been published by Lane in 1870 [1]. Initial value problems with singularity and of type Lane–Emden differential equation, d2y dx2 Wazwaz [8] used the Adomian decomposition method to Journal of Mathematics solve these equations; the appropriate choices of operator L were required to overcome the singularity behavior at the origin.

Results

Conclusion