ABSTRACT In this paper, high order time discretization schemes for time fractional Schrödinger equations in one and two dimensions are proposed. Our schemes are based on high order weighted and shifted Grünwald-Letnikov (WSGL) difference operators for the time fractional derivatives, Sinc-Galerkin methods are used for the space variables. The stability of time semi-discrete schemes is analysed with the help of -transform. For the fully discretization schemes, the standard Sinc-Galerkin method and symmetric Sinc-Galerkin method are established by selecting proper weight functions. Finally, we apply our numerical schemes to solve one-dimensional and two-dimensional time fractional Schrödinger equations, verify the validity of present numerical schemes.

中文翻译：

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时间分数阶薛定谔方程的高阶WSGL差分算子结合Sinc-Galerkin方法

摘要 在本文中，提出了一维和二维时间分数阶薛定谔方程的高阶时间离散化方案。我们的方案基于时间分数阶导数的高阶加权和移位 Grünwald-Letnikov (WSGL) 差分算子，Sinc-Galerkin 方法用于空间变量。借助-变换分析了时间半离散方案的稳定性。对于完全离散化方案，通过选择合适的权函数建立标准的Sinc-Galerkin方法和对称Sinc-Galerkin方法。最后，我们应用我们的数值方案来求解一维和二维时间分数阶薛定谔方程，验证现有数值方案的有效性。