High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schrödinger equation

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 $\mathcal{Z}$-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.

Keywords:

• Time fractional Schrödinger equation
• WSGL difference operator
• $\mathcal{Z}$-transform
• stability
• Sinc-Galerkin method

2010 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 03F35
• 47B37
• 65M12

Acknowledgments

We would like to thank the anonymous referees for their carefully reading this manuscript and their valuable comments and constructive suggestions for improving this manuscript significantly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by the National Natural Science Foundation of China under Grant Nos. 11426174, 11702214, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2018JM1016, and the National Natural Science Foundation of China under Grant No. 11972286.