Solution of Linear and Nonlinear Schrodinger Equations by Combine Elzaki Transform and Homotopy Perturbation Method

Mohannad H Eljaily

doi:10.11648/j.ajtas.20150406.24

Abstract

In this paper, the homotopy perturbation method (HPM) and ELzaki transform are employed to obtain the approximate analytical solution of the Linear and Nonlinear Schrodinger Equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. This method finds the solution without any discretization, linearization or restrictive assumptions and avoids the round-off errors,the results reveal that the ETHPM is very efficient, simple and can be applied to other nonlinear problems.

Highlights

The investigation of the exact solutions to nonlinear equations plays an important role in the study of nonlinear physical phenomena.The linear and nonlinear Schrodinger equations have been widely used in various application areas, e.g., quantum mechanics, optics, seismology and plasma physics
Since analytic approaches to the Schrodinger equation have limited applicability in science and engineering problems, there is a growing interest in exploring new methods to solve the equation more accurately and efficiently.In recent years, many research workers have paid attention to study the solutions of nonlinear partial differential equations by using various methods.Among thesethe Adomian decomposition method Hashim, Noorani, Ahmed.Bakar, Ismail and Zakaria, (2006), the tanh method, the homotopy perturbation method Sweilam, Khader (2009), Sharma and GirirajMethi (2011), Jafari, Aminataei (2010), (2011), the differential transform method (2008), homotopy perturbation transform method and the variational iteration method.He [12–25] developed the homotopy perturbation method (HPM) by merging the standard homotopy and perturbation for solving various physical problems
ELzaki transform is a useful technique for solving linear differential equations, but this transform is totally incapable of handling nonlinear equations [3] because of the difficulties that are caused by the nonlinear terms

Summary

Introduction

The investigation of the exact solutions to nonlinear equations plays an important role in the study of nonlinear physical phenomena.The linear and nonlinear Schrodinger equations have been widely used in various application areas, e.g., quantum mechanics, optics, seismology and plasma physics. This paper uses homotopy perturbation method to decompose the nonlinear term, so that the solution can be obtained by iteration procedure. The HPM and He’s polynomials and is mainly due to Ghorbani [8, 9].This method provides the solution in a rapid convergent series which may leads the solution in a closed form The advantage of this method is its capability of combining two powerful methods for obtaining exact solutions for nonlinear equations. Elzaki: Solution of Linear and Nonlinear Schrodinger Equations by Combine Elzaki Transform and Homotopy Perturbation Method research in this area, we apply the ETHPM in solving the linear and nonlinear Schrodinger equations to show the simplicity and straightforwardness of the method

Basic Idea

Applications

Conclusions