An improved collocation method for multi-dimensional space-time variable-order fractional Schrodinger equations

Current discretizations of variable-order fractional (V-OF) differential equations lead to numerical solutions of low order of accuracy. This paper explores a high order numerical scheme for multi-dimensional V-OF Schrbdinger equations. We derive new operational matrices for the V-OF derivatives of Caputo and Riemann-Liouville type of the shifted Jacobi polynomials (SJPs). These allow us to establish an efficient approximate formula for the Riesz fractional derivative. An operational approach of the Jacobi collocation approach for the approximate solution of the V-OF nonlinear Schrodinger equations. The main characteristic behind this approach is to investigate a space-time spectral approximation for spatial and temporal discretizations. The proposed spectral scheme, both in temporal and spatial discretizations, is successfully developed to handle the two-dimensional V-OF Schrodinger equation. Numerical results indicating the spectral accuracy and effectiveness of this algorithm are presented. (C) 2016 IMACS. Published by Elsevier B.V. All rights reserved.