Some aspects of fractional diffusion equations of single and distributed order

Details Some aspects of fractional diffusion equations of single and distributed order Some aspects of fractional diffusion equations of single and distributed order

2007-11-27

arxiv.org/abs/0711.4261v1

Applied Mathematics and Computation, Vol. 187, No 1, pp. 295-305 (2007)

Physics

Mathematical Physics

14 pages. International Symposium on "Analytic Function Theory, Fractional Calculus and Their Applications", University of Vic

Scientific paper

10.1016/j.amc.2006.08.126

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.

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