Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem

We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α∈(0,1) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is investigated.