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Finite difference approximations for the fractional advection-diffusion equation. (English) Zbl 1234.65034
Summary: 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 $(\Delta t)^2$. A numerical example is presented to confirm the conclusions.

65M06Finite difference methods (IVP of PDE)
60G22Fractional processes, including fractional Brownian motion
60J60Diffusion processes
35K57Reaction-diffusion equations
35R11Fractional partial differential equations
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