Abstract
•Improved L1-Galerkin spectral methods for coupled nonlinear time-space fractional Schrödinger equations are proposed.•Sharp convergence rates reflecting the regularity of solution are obtained over uniform and nonuniform time-steps.•The well-posedness of the numerical solution is proved.•Two numerical examples with smooth and non-smooth solutions are presented to support our theoretical contributions.•The effects of fractional-order parameters on the pattern formations of coupled Schrödinger equations are studied.
Recently there has been a growing interest in designing efficient numerical methods for the solution of fractional differential equations. The solutions of such equations in general exhibit a weak singularity near the initial time. In this paper, we propose finite difference/spectral methods to solve the coupled nonlinear time-space fractional Schrödinger equations with non-smooth solutions in the time direction. The proposed methods combine the strength of the L1 scheme on both uniform and non-uniform grids and the Galerkin-Legendre method. The well-posedness of the numerical solution is proved. The convergence analysis shows clearly how the grading of the mesh and the regularity of the solution affect the order of convergence of the L1 scheme, so one can choose an optimal mesh grading which can be seen numerically by considering some numerical experiments.