Time fractional diffusion: A discrete random walk approach

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 is an element of (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.