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

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...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2007-08, Vol.205 (2), p.871-881
Main Authors: Gorenflo, R., Abdel-Rehim, E.A.
Format: Article
Language:English
Subjects:
Convergence in distribution
Difference schemes
Fractional derivative
Fractional diffusion
Potential well
Stochastic processes
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Summary: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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2005.12.043