A one-phase space -- fractional Stefan problem with no liquid initial domain

TL;DR

This paper studies a one-dimensional fractional Stefan problem with a non-local flux modeled by the Caputo derivative, proving existence and uniqueness of solutions for initial liquid and solid domains, including a limit case.

Contribution

It establishes the existence and uniqueness of solutions for a fractional phase-change problem with non-local flux and boundary conditions, extending to cases with initial solid domain.

Findings

Existence and uniqueness of solutions for the fractional Stefan problem.

Solution existence for both initial liquid and solid domains.

Limit solution existence when domain transformation is not possible.

Abstract

Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face $x=0$, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a limit solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.