Finite difference approximations for the fractional advection–diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection–diffusion equation, the fractional Crank–Nicholson method (FCN) is given, which is based on shifted Grünwald–Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order (Δt)2(Δt)2. A numerical example is presented to confirm the conclusions.