Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion

Authors:
R. Gorenflo;E. A. Abdel-Rehim
Affiliations:
Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 3, D-14195 Berlin, Germany;Department of Mathematics and Computer Science, Suez Canal University, Ismailia, Egypt
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

Using bivariate generating functions, we prove convergence of the Grunwald-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.