Analytical solutions of the time-fractional non-linear Schrodinger equation with zero and non zero trapping potential through the Sumudu decomposition method

T. Mathanaranjan,K. Himalini

doi:10.4038/josuk.v12i0.8015

Abstract

Sumudu decomposition method is used to construct the approximate analytical solutions of time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential. The Sumudu decomposition method is a combined form of the Sumudu transform and the Adomian decomposition method. The fractional derivatives are defined in the Caputo sense. The exact solutions of some nonlinear Schrodinger equations are given as a special case of our approximate analytical solutions. The computations show that the described method is easy to apply, and it needs smaller size of computation as compared to the other existing methods. Further, the solutions are derived in a convergent series form which shows the effectiveness of the method for solving a wide variety of nonlinear fractional differential equations.

Highlights

SUMUDU DECOMPOSITION METHOD we implement the Adomian decomposition method with Sumudu transformation to time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential given by Eq (1)
In order to illustrate the efficiency of this method, we compaire our solutions with the exact solutions for some specific choice of parameters
The Sumudu decomposition method is applied successfully for finding the approximate analytical solutions of the time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential that are considered in the Caputo sense

Summary

INTRODUCTION

The time-fractional nonlinear Schrodinger equations has the following form: iDt u(x, t). Adomian decomposition method is a semi-analytical method introduced by Adomian (1980), which provides an effective technique for finding explicit of a wider and general class of differential systems representing real physical problems (Ali and Al-Saif, 2008) This method and its modifications efficiently work for initial value or boundary value problems, for linear or nonlinear, ordinary or partial differential equations (Mahmoud and Borai, 1980). We use the Sumudu decomposition method (SDM) to construct the approximate analytical solutions of time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential. In the following two subsections, the analytical solution of the time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential are provided. SUMUDU DECOMPOSITION METHOD we implement the Adomian decomposition method with Sumudu transformation to time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential given by Eq (1). Convergence conditions of this series are examined by several authors, mainly by Cherruault (1993)

RESULTS AND DISCUSSION

Time-fractional nonlinear Schrodinger equations with trapping potential

CONCLUSION