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 \(\beta\in (0,1)\). 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.


82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
60G50 Sums of independent random variables; random walks
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