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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:
65R20 Numerical methods for integral equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
45K05 Integro-partial differential equations
26A33 Fractional derivatives and integrals
35K15 Initial value problems for second-order parabolic equations
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