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Zbl 1009.82016
Gorenflo, Rudolf; Mainardi, Francesco; Moretti, Daniele; Paradisi, Paolo
Time fractional diffusion: A discrete random walk approach.
(English)
[J] Nonlinear Dyn. 29, No.1-4, 129-143 (2002). ISSN 0924-090X; ISSN 1573-269X/e

Summary: The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta\in (0,1)$. From a physical view-point this generalized diffusion equation is obtained from a fractional Fick law which describes transport processes with long memory. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of slow anomalous diffusion. By adopting a suitable finite-difference scheme of solution, we generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.
MSC 2000:
*82C41 Dynamics of random walks, etc.
82C70 Transport processes
60G50 Sums of independent random variables

Keywords: anomalous diffusion; random walks; fractional derivatives

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