Convergence of the Grünwald–Letnikov scheme for time-fractional diffusion |
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Authors: | R. Gorenflo E.A. Abdel-Rehim |
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Affiliation: | 1. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 3, D-14195 Berlin, Germany;2. Department of Mathematics and Computer Science, Suez Canal University, Ismailia, Egypt |
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Abstract: | Using bivariate generating functions, we prove convergence of the Grünwald–Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier–Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier–Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation. |
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Keywords: | 26A33 33E12 45K05 60G50 60J60 65N06 |
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