Finite difference approximations for space-time fractional partial differential equation

An implicit difference scheme is presented for a space-time fractional convection-diffusion equation. The equation is obtained from the classical integer order convection-diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grunwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.