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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 β(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:
82C41Dynamics of random walks, random surfaces, lattice animals, etc.
82C70Transport processes (time-dependent statistical mechanics)
60G50Sums of independent random variables; random walks