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Time fractional diffusion: A discrete random walk approach. (English) Zbl 1009.82016
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 $\beta \in \left(0,1\right)$. 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:
 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. 82C70 Transport processes (time-dependent statistical mechanics) 60G50 Sums of independent random variables; random walks
##### Keywords:
anomalous diffusion; random walks; fractional derivatives