A note on models for anomalous phase-change processes

TL;DR

This paper reviews fractional free boundary problems for modeling anomalous phase-transitions, highlighting their differences, assumptions, and a heuristic approach that generalizes classical Stefan problems using fractional derivatives.

Contribution

It provides a comparative survey of fractional Stefan problems and introduces a heuristic fractional energy balance approach for generalizing classical models.

Findings

Problems are nonequivalent but reduce to classical Stefan problem when fractional order is one.

A fractional energy balance approach naturally generalizes classical Stefan problem.

Fractional derivatives in time model anomalous phase-change processes effectively.

Abstract

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier's law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones.