Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation

Abstract

In this study, one explicit and one implicit finite differencescheme is introduced for the numerical solution of time-fractionalRiesz space diffusion equation. The time derivative is approximatedby the standard Gr"{u}nwald Letnikov formula of order one, whilethe Riesz space derivative is discretized by Fourier transform-basedalgorithm of order four. The stability and convergence of theproposed methods are studied. It is proved that the implicit schemeis unconditionally stable, while the explicit scheme is stableconditionally. Some examples are solved to illustrate the efficiencyand accuracy of the proposed methods. Numerical results confirm thatthe accuracy of present schemes is of order one.