Finite difference method for time-space-fractional Schrödinger equation

Abstract

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order of O(τ^2−α+h²), where τ and h are time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis.

Keywords:

• time-space-fractional Schrödinger equation
• Caputo derivative
• Riesz derivative
• finite difference scheme
• stability

2010 AMS Subject Classifications:

• 26A33
• 65M06
• 65M12
• 65M15
• 35R11

Acknowledgements

The work was partially supported by the National Natural Science Foundation of China (grant no. 11372170), the Key Program of Shanghai Municipal Education Commission (grant no. 12ZZ084), the grant of ‘The First-Cass Discipline of Universities in Shanghai’, and the grant of ‘085 Project of Shanghai’.