A New Mathematical Formulation for a Phase Change Problem with a Memory Flux

TL;DR

This paper introduces a novel mathematical model for a phase change problem incorporating memory effects via fractional derivatives, providing a new fractional Stefan condition and advancing the understanding of heat transfer with memory.

Contribution

It develops a new fractional derivative-based model for phase change problems with memory flux, including an integral form of the fractional Stefan condition.

Findings

Derived a fractional Stefan condition involving Caputo and Riemann--Liouville derivatives.

Established an integral relationship for the free boundary in the phase change problem.

Enhanced the mathematical framework for phase change problems with memory effects.

Abstract

A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense of Riemann--Liouville. An integral relationship for the free boundary which is equivalent to the `fractional Stefan condition' is also obtained.