Finite difference schemes for time-fractional Schrödinger equations via fractional linear multistep method

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.

Keywords:

• Fractional linear multistep method
• time-fractional Schrödinger equation
• stability
• fractional derivative
• convergence

2010 Mathematics Subject Classifications:

• 65J10
• 35R11

Acknowledgments

The author is grateful to the referees for their insightful and constructive remarks which allowed to improve the paper in a remarkable way.

Disclosure statement

No potential conflict of interest was reported by the author(s).