Finite difference schemes for two-dimensional time-space fractional differential equations

Abstract

In this paper, finite difference schemes for differential equations with both temporal and spatial fractional derivatives are studied. When the order of the time fractional derivative is in $(1,2)$, an alternating direction implicit (ADI) scheme with second-order accuracy in both space and time is constructed. For equations with time fractional derivatives of order lying in $(0,1)$, a scheme is derived and solved by the generalized minimal residual method. We also propose a preconditioner to improve the efficiency for the implementation of the scheme in this situation.

Keywords:

• two-dimensional fractional differential equation
• weighted and shifted Grünwald difference operator
• ADI scheme
• discrete energy method
• preconditioned GMRES method

2010 AMS Subject Classifications:

• 35R11
• 65M06
• 65M12
• 65M15

Acknowledgments

The authors would like to thank the editor and referees for their comments. With their valuable suggestions, the presentation of this manuscript has been greatly improved.

Disclosure Statement

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

Funding

This research is supported by the Macao Science and Technology Development Fund (FDCT) 001/2013/A, the grant MYRG086(Y1-L2)-FST12-VSW and MYRG071(Y2-L2)-FST13-LSL from University of Macau.