A Generalization of the Hopf's Lemma for the 1-D Moving-Boundary Problem for the Fractional Diffusion Equation and its Application to a Fractional Free-Boundary Problem

This paper deals with a theoretical mathematical analysis of a one-dimensional-moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order $\al$ $\in (0,1)$ is taken in the Caputo's sense. A generalization of the Hopf's lemma is proved, and then this result is used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is considered.