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

Abstract

Taking into account the recent works [14] and [17], 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.

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

“…Although in general η, ϑ may be nonzero outside Ω, except for the bilinear form ´Rd ×[0,T ] AD s ϑ • D s ξ, the other integral terms in the variational formulation (1.14) are only integrated over Ω in space, since the test function ξ is 0 in Ω c ×]0, T [. Different non-local versions of Stefan-type problems have previously been considered, including in [10] and [16] for nonsingular integral kernels, in [53], [8], [44] and [42] for the fractional Caputo derivatives, and in [22], [23], [24], [25] and [30] for the fractional Laplacian and its nonlocal integral generalization in [2]. Stefan-type problems that are fractional in the time derivative have also been considered (see, for instance, [43], [34] and [15].)…”