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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.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 60G22 Fractional processes, including fractional Brownian motion 60J60 Diffusion processes 35K57 Reaction-diffusion equations 35R11 Fractional partial differential equations
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