An Efficient Method for Time-Fractional Coupled Schr？dinger System

We present a new technique to obtain the solution of time-fractional coupled Schr？dinger 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. 1. Introduction 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？？H？pital. The fractional order calculus is a generalization of the integer order calculus to a real or complex number. Fractional differential equations are used in many branches of sciences, mathematics, physics, chemistry, and engineering. Applications of fractional calculus and fractional-order differential equations include dielectric relaxation phenomena in polymeric materials [1], transport of passive tracers carried by fluid flow in a porous medium in groundwater hydrology [2], transport dynamics in systems governed by anomalous diffusion [3, 4], and long-time memory in financial time series [5] and so on [6, 7]. In particular, recently, much attention has been paid to the distributed-order differential equations and their applications in engineering fields that both integer-order systems and fractional-order systems are special cases of distributed-order systems. The reader may refer to [8–10]. Several schemes have been developed for the numerical solution of differential equations. The homotopy perturbation method was proposed by He [11] in 1999. This method has been used by many mathematicians and engineers to solve various functional equations. Homotopy method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [12], nonlinear wave equations [13], and boundary value problems [14]. It can be said that He’s homotopy perturbation method is a universal one and is able to solve various kinds of nonlinear functional equations. For example, it was applied to nonlinear Schr？dinger equations [15], to nonlinear equations arising in heat transfer [16], and to other equations [17–20]. In this method, the solution is considered to be an infinite series which usually converges rapidly to exact solutions. In this paper we introduce a new form of homotopy perturbation and Laplace transform methods by extending the idea of [21]. We extend the homotopy perturbation and Laplace