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

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 $\text{al}\in (0,1)$ 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 $\text{al}=1$.