Determination of two unknown thermal coefficients through an inverse one-phase fractional Stefan problem

TL;DR

This paper develops a fractional Stefan problem model with memory effects to determine two unknown thermal coefficients in a phase-change material, providing explicit solutions and conditions for existence and uniqueness.

Contribution

It introduces a fractional time derivative approach to inverse Stefan problems, deriving explicit solutions and conditions for multiple unknown thermal coefficients.

Findings

Explicit solutions for temperature and thermal coefficients are obtained.

Necessary and sufficient conditions for solution existence and uniqueness are established.

Classical results are recovered as the fractional order approaches 1.

Abstract

We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed boundary and a known evolution for the moving boundary. We assume that the phase-change process presents latent-heat memory effects by considering a fractional time derivative of order $\alpha$ ($0<\alpha<1$) in the Caputo sense and a sharp front model for the interface. According to the choice of the unknown thermal coefficients, six inverse fractional Stefan problems arise. For each of them, we determine necessary and sufficient conditions on data to obtain the existence and uniqueness of a solution of similarity type. Moreover, we present explicit expressions for the temperature and the unknown thermal coefficients. Finally, we show that the results for the classical statement of this problem, associated with $\alpha=1$, are obtained through the fractional model when $\alpha\to 1^-$.