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Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. (English) Zbl 1242.65157

Summary: We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the ”fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank-Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank-Nicolson method for the fractional diffusion equation with using fractional centered difference approach.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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