Eur. Phys. J. Plus (2014) 129: 260
https://doi.org/10.1140/epjp/i2014-14260-6

Regular Article

A new Jacobi spectral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems

¹ Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
² Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
³ Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
⁴ Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt
⁵ Department of Mathematics, University of Central Florida, 32816-1364, Orlando, FL, USA

^* e-mail: alibhrawy@yahoo.co.uk

Received: 27 July 2014
Revised: 19 October 2014
Accepted: 6 November 2014
Published online: 5 December 2014

Abstract

The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrödinger equation (T-FSE) and the space-fractional Schrödinger equation (S-FSE). The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, the presented approach is also applied to solve the time-fractional coupled Schrödinger system (T-FCSS). In order to demonstrate the validity and accuracy of the numerical scheme proposed, several numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.