Two equivalent Stefan's problems for the Time Fractional Diffusion Equation

TL;DR

This paper investigates two related Stefan's problems involving the time fractional diffusion equation with Caputo derivatives, establishing their equivalence and analyzing their convergence to classical solutions as the fractional order approaches 1.

Contribution

The paper introduces and proves the equivalence of two fractional Stefan's problems and examines their convergence to classical heat equation solutions as fractional order approaches 1.

Findings

Proved the equivalence between two fractional Stefan's problems.

Analyzed the convergence to classical solutions as fractional order approaches 1.

Extended Stefan's problems to fractional diffusion equations with Caputo derivatives.

Abstract

Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order $ \al \in (0,1) $ is taken in the Caputo's sense. The first one has a constant condition on $ x = 0 $ and the second presents a flux condition $ T_x (0, t) = \frac {q} {t ^ {\al/2}} $. An equivalence between these problems is proved and the convergence to the classical solutions is analysed when $ \al \nearrow $ 1 recovering the heat equation with its respective Stefan's condition.