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

TL;DR

This paper extends Hopf's lemma to a 1-D fractional diffusion problem with moving boundaries, enabling analysis of fractional free-boundary Stefan problems and their monotonicity properties.

Contribution

It generalizes Hopf's lemma for fractional diffusion equations and applies this to establish monotonicity in fractional free-boundary Stefan problems.

Findings

Generalized Hopf's lemma for fractional diffusion

Proved monotonicity of free-boundary in fractional Stefan problem

Enhanced understanding of fractional moving-boundary problems

Abstract

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.