Exponential time differencing method for the time space fractional Schrödinger equation

Fractional Schrödinger equations have played a major role in nonlocal models in quantum mechanics, optical physics and anomalous transport processes. The fundamental method of fractional Schrödinger equations, along with their long-range interactions, pose specific analytical and computational challenges compared with classical Schrödinger systems. In the present work, a numerical scheme is introduced for evaluating the time-space fractional Schrödinger equation (TSFSE) where time fractional derivatives are assumed in the sense of Caputo and in space Riesz derivative is used. To approximate the Riesz fractional derivative in space, the centered difference approach is employed by combining the exponential time differencing (ETD) method for Caputo derivative in time. The stability and convergence analyses are presented, establishing the unconditional stability and first-order temporal convergence and second-order spatial convergence under appropriate regularity assumptions. Numerical experiments confirm the convergence orders and verify the performance of recommended scheme.