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We present a new technique to obtain the solution of time-fractional coupled Schrödinger system. The fractional derivatives are considered in Caputo sense. The proposed scheme is based on Laplace transform and new homotopy perturbation method. To illustrate the power and reliability of the method some examples are provided. The results obtained by the proposed method show that the approach is very efficient and simple and can be applied to other partial differential equations.

The intuitive idea of fractional order calculus is as old as integer order calculus. It can be observed from a letter that was written by Leibniz to

Several schemes have been developed for the numerical solution of differential equations. The homotopy perturbation method was proposed by He [

We extend the homotopy perturbation and Laplace transform method to solve the time-fractional coupled Schrödinger system. The nonlinear time-fractional coupled Schrödinger partial differential system is as [

This paper is organized as follows. In Section

Some basic definitions and properties of the fractional calculus theory are used in this paper.

A real function

The left-sided Riemann-Liouville fractional integral operator of order

Some of the most important properties of operator

Amongst a variety of definitions for fractional order derivatives, Caputo fractional derivative has been used [

In this paper, we have considered time-fractional coupled Schrödinger system, where the unknown function

The Caputo time-fractional derivative operator of order

The Laplace transform of a function

If

The Mittag-Leffler function plays a very important role in the fractional differential equations and in fact it was introduced by Mittag-Leffler in 1903 [

If

For the convenience of the reader, we will first present a brief account of homotopy perturbation method. Let us consider the following differential equation:

The operator

To illustrate the basic ideas of this method, we consider the general form of a system of nonlinear fractional partial differential equations:

Now if we solve these equations in such a way that

To illustrate the power and reliability of the method for the time-fractional coupled Schrödinger system some examples are provided. The results reveal that the method is very effective and simple.

Consider the following linear time-fractional coupled Schrödinger system:

To solve (

Assume

Consider the following nonlinear time-fractional coupled Schrödinger system:

To solve (

Assume

Consider the following nonlinear time-fractional coupled Schrödinger system:

To solve (

Assume

In this paper, we have introduced a combination of Laplace transform and homotopy perturbation methods for solving fractional Schrödinger equations which we called homotopy perturbation and Laplace transform method. In this scheme, the solution considered to be a Taylor series which converges rapidly to the exact solution of the nonlinear equation. As shown in the three examples of this paper, a clear conclusion can be drawn from the results that the homotopy perturbation and Laplace transform method provide an efficient method to handle nonlinear partial differential equations of fractional order. The computations associated with the examples were performed using Maple 13.

The authors declare that there is no conflict of interests regarding the publication of this paper.