ABSTRACT

In this paper, a finite difference-based numerical approach is developed for time-fractional Schrödinger equations with one or multidimensional space variables, with the use of fractional linear multistep method for time discretization and finite difference method for spatial discretization. The proposed method leads to achieve second order of accuracy for time variable. Stability and convergence theorems for the constructed difference scheme is achieved via z-transform method. Time-fractional Schrödinger equation is considered in abstract form to allow generalization of the theoretical results on problems which have distinct spatial operators with or without variable coefficients. Numerical results are presented on one and multidimensional experimental problems to verify the theoretical results.

摘要

在本文中，利用用于时间离散化的分数线性多步法和用于空间离散化的有限差分法，为具有一维或多维空间变量的时间分数薛定谔方程开发了一种基于有限差分的数值方法。所提出的方法导致时间变量达到二阶精度。构建的差分方案的稳定性和收敛定理是通过z变换方法实现的。时间分数薛定谔方程被认为是抽象形式，以允许对具有或不具有可变系数的不同空间算子的问题的理论结果进行概括。给出了一维和多维实验问题的数值结果以验证理论结果。