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Finite difference/spectral approximations for the time-fractional diffusion equation. (English) Zbl 1126.65121
Consider the problem of finding a numerical solution for the time-fractional diffusion equation with the fractional derivative with respect to time being given in the sense of Caputo. For the solution of this problem, the authors propose a method based on a finite difference scheme with respect to time combined with a Legendre spectral method for the space variable(s). A proof of stability and convergence of the algorithm is provided. Error estimates and numerical results are given as well.

MSC:
65R20Integral equations (numerical methods)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
65M70Spectral, collocation and related methods (IVP of PDE)
45K05Integro-partial differential equations
26A33Fractional derivatives and integrals (real functions)
35K15Second order parabolic equations, initial value problems