Determination of one unknown thermal coefficient through the one-phase fractional Lam\'e-Clapeyron-Stefan problem

TL;DR

This paper derives explicit formulas to determine an unknown thermal coefficient in a fractional Stefan problem using fractional calculus and special functions, extending classical methods to fractional derivatives.

Contribution

It introduces a method to identify unknown thermal coefficients in a fractional Stefan problem, generalizing classical approaches to fractional derivatives with explicit solutions.

Findings

Explicit expressions for unknown thermal coefficients are obtained.

Necessary and sufficient conditions for unique solutions are established.

The classical case is recovered as a limit when the fractional order approaches 1.

Abstract

We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lam\'e-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face $x=0$. The partial differential equation and one of the conditions on the free boundary include a time Caputo's fractional derivative of order $\alpha \in (0,1) $. Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in Roscani - Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802-815, Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249, and Voller, Int. J. Heat Mass Transfer, 74 (2014), 269-277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lam\'e-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74-82, which are recovered by taking the limit when the order $\alpha\nearrow 1$.