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Compact finite difference method for the fractional diffusion equation. (English) Zbl 1179.65107

The author apply for solving the one-dimensional fractional diffusion equation

u t= 0 D t 1-γ [K γ 2 u x 2 ]+f(x,t),x(L 0 ,L 1 ),t(0,T)

a special finite difference method using the Grunwald discretization process for the fractional derivative. The approximate scheme has an error of fourth order for the spatial variable and of first order for the time variable. The stability of this scheme is proved using Fourier series to expand the error of the approximate system.

MSC:
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35R11Fractional partial differential equations
35K05Heat equation
65M15Error bounds (IVP of PDE)
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