Time fractional diffusion: a discrete random walk approach

The time fractional diffusion equation is obtained from the standard
diffusion e
quation 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 thi
s generalized diffusion equation is obtained from a fractional Fick law
which describes transport processes with long memory. The fundamental
solution f
or the Cauchy problem is interpreted as a probability density of a
self-similar non-Markovian stochastic process related to a phenomenon of
slow an
omalous diffusion. By adopting a suitable finite-difference
scheme of solution, we generate discrete models of random walk suitable for
simu
lating random variables whose spatial probability density evolves in
time according to this fractional diffusion equation.