Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi-Gauss-Lobatto collocation (J-GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge-Kutta (IRK) method of fourth order and the other based on Jacobi-Gauss-Radau collocation (J-GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations.