Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations

Abstract

In the present paper, we construct the numerical solution for time fractional (1 + 1)- and (1 + 2)-dimensional Schrödinger equations (TFSEs) subject to initial boundary. The solution is expanded in a series of shifted Jacobi polynomials in time and space. A collocation method in two steps is developed and applied. First step depends mainly on application of shifted Jacobi Gauss-Lobatto-collocation method for spatial discretization on the approximate solution and its spatial derivatives occurring in the TFSE and substitution in the boundary conditions or treatment of the non-local conservation conditions by the Jacobi Gauss-Lobatto quadrature rule. As a result, a system of fractional differential equation for the expansion coefficients is obtained. The second step is to use a shifted Jacobi Gauss-Radau- collocation scheme, for temporal discretization, to reduce such system into a system of nonlinear Newton iterative method. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithms demonstrating superiority over other methods.