Some aspects of fractional diffusion equations of single and distributed order

TL;DR

This paper explores fractional diffusion equations of single and distributed order, analyzing their fundamental solutions as probability densities of non-Markovian stochastic processes with sub-diffusive behavior and varying time-scales.

Contribution

It introduces a generalized framework for fractional diffusion equations with distributed orders, extending the understanding of associated stochastic processes.

Findings

Fundamental solutions are probability densities of non-Markovian processes.

Distributed order equations exhibit non-self-similar behavior.

Variance grows sub-linearly over time in these processes.

Abstract

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.