  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 $\beta \in (0,1)$. 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 sub-diffusion (the
variance grows in time sub-linearly). A further generalization is obtained by
considering a continuous or discrete distribution of fractional time
derivatives of order less than one.
  Then the fundamental solution is still a probability density of a
non-Markovian process that, however, is no longer self-similar but exhibits a
corresponding distribution of time-scales.
