Distributed-order fractional Cauchy problems on bounded domains

TL;DR

This paper studies distributed-order fractional Cauchy problems on bounded domains, providing explicit solutions and stochastic representations that model complex delay effects in diffusion processes.

Contribution

It introduces explicit strong solutions and stochastic analogues for distributed-order fractional Cauchy problems with Dirichlet boundary conditions.

Findings

Explicit strong solutions derived for distributed-order fractional problems.

Stochastic representations constructed via non-Markovian time changes.

Applicable to modeling complex delay effects in diffusion processes.

Abstract

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. The fractional derivative models time delays in a diffusion process. The order of the fractional derivative can be distributed over the unit interval, to model a mixture of delay sources. In this paper, we provide explicit strong solutions and stochastic analogues for distributed-order fractional Cauchy problems on bounded domains with Dirichlet boundary conditions. Stochastic solutions are constructed using a non-Markovian time change of a killed Markov process generated by a uniformly elliptic second order space derivative operator.