Local error estimate of L1 scheme on graded mesh for time fractional Schrödinger equation

In this work, a time fractional Schr & ouml;dinger equation with Caputo fractional derivative of order alpha is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} is considered, whose solution exhibits a weak singularity at initial time. We divide the solution into two parts: imaginary part and real part, then the discrete system is constructed by the five-point difference scheme in space together with the L1 approximation of Caputo derivative on graded mesh in time. We provide stability and local error analysis for the discrete scheme, which shows that one can attain the optimal convergence order 2-alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2-\alpha $$\end{document} in positive time by selecting milder grading parameters. Numerical results are presented to verify the accuracy of the theoretical analysis.