A fractional-order implicit difference approximation for the space-time fractional diffusion equation

Abstract

We consider a space-time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann--Liouville fractional derivative of order between one and two, and the first order time derivative by a Caputo fractional derivative of order between zero and one. A fractional order implicit finite difference approximation for the space-time fractional diffusion equation with initial and boundary values is investigated. Stability and convergence results for the method are discussed, and finally, some numerical results show the system exhibits diffusive behaviour.