Space-time fractional Dirichlet problems

TL;DR

This paper provides explicit solutions and stochastic representations for space-time fractional Dirichlet problems, addressing an open problem in numerical analysis and demonstrating applications in multi-variable fractional diffusion models.

Contribution

It introduces explicit solutions and stochastic representations for space-time fractional diffusion problems on bounded domains, solving an open numerical analysis problem.

Findings

Explicit solutions for fractional diffusion problems

Stochastic solutions via Markov processes with inverse stable subordinator

Applications demonstrating well-posedness in multi-variable cases

Abstract

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.