On the stability preserving of L1 scheme to nonlinear time-fractional Schrödinger delay equations

Abstract

In this paper, we investigate the stability preserving of L1 scheme to nonlinear time fractional Schrödinger delay equations. This kind of Schrödinger equation contains both nonlocal effect and time memory reaction term. We derive sufficient conditions to ensure the asymptotic stability of the analytical equations. After that, we approximate the equations via the Galerkin finite element method in space. We show that the semidiscrete numerical solutions can inherit the long time behavior of the solutions. After that, a fully discrete approximation of the equations is obtained by the L1 scheme and a linear interpolation procedure. We provide detailed estimations on the discrete operators that are obtained by the Z transform and its inverse. Together with a discrete fractional comparison principle, we prove that the L1 scheme preserves the stability of the underlying equations. The main results obtained in this work do not depend on spatial and temporal step sizes. A numerical example confirms the effectiveness of our derived method and validates the theoretical findings.