Fractional Cauchy problems on bounded domains: survey of recent results

TL;DR

This survey reviews recent advances in fractional Cauchy problems on bounded domains, highlighting solutions involving Mittag-Leffler functions, eigenvalue problems, and stochastic methods using inverse subordinators.

Contribution

It compiles recent results on fractional Cauchy problems on bounded domains and explores solutions with infinite sums of fractional derivatives and their eigenvalue connections.

Findings

Solutions expressed via Mittag-Leffler functions

Connection between fractional derivatives and eigenvalue problems

Stochastic solutions involve inverse subordinators

Abstract

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \citet{nigmatullin}, and \citet{zaslavsky} in $\mathbb R^d$ for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizations on bounded domains $D\subset \rd$ obtained in \citet{m-n-v-aop, mnv-2}. We also study the solutions of fractional Cauchy problem where the first time derivative is replaced with an infinite sum of fractional derivatives. We point out a connection to eigenvalue problems for the fractional time operators considered. The solutions to the eigenvalue problems are expressed by Mittag-Leffler functions and its generalized versions. The stochastic solution of the eigenvalue problems for the fractional derivatives are given by inverse subordinators.