A One-Phase Space-Fractional Stefan Problem With No Liquid Initial Domain

Abstract

We consider a phase-change problem for a one-dimensional material with a nonlocal 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 solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.