Explicit solution for a two--phase fractional Stefan problem with a heat flux condition at the fixed face

TL;DR

This paper derives an explicit generalized Neumann solution for a two-phase fractional Stefan problem with a heat flux boundary condition, extending classical results to fractional derivatives and establishing relationships with related problems.

Contribution

It provides a novel explicit solution for a fractional Stefan problem with a heat flux condition, linking it to classical solutions and inequalities for phase-change interface coefficients.

Findings

Explicit generalized Neumann solution for fractional Stefan problem.

Relationship established between fractional and classical problems.

Inequality for phase-change interface coefficient derived.

Abstract

A generalized Neumann solution for the two-phase fractional Lam\'e--Clapeyron--Stefan problem for a semi--infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order $\al \in (0,1)$ respect on the temporal variable is considered in two governing heat equations and in one of the conditions for the free boundary. Furthermore, we find a relationship between this fractional free boundary problem and another one with a constant temperature condition at the fixed face and based on that fact, we obtain an inequality for the coefficient which characterizes the fractional phase-change interface obtained in Roscani--Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249. We also recover the restriction on data and the classical Neumann solution, through the error function, for the classical two-phase Lam\'e-Clapeyron-Stefan problem for the case $\al=1$.