A new mathematical formulation for a phase change problem with a memory flux

Abstract

A mathematical formulation for a one-phase change problem in a form of Stefan problem with a memory flux is obtained. The hypothesis that the integral of weighted backward fluxes is proportional to the gradient of the temperature is considered. The model that arises involves fractional derivatives with respect to time both in the sense of Caputo and of Riemann-Liouville. An integral relation for the free boundary, which is equivalent to the "fractional Stefan condition", is also obtained.

A Note on Models for Anomalous Phase-Change Processes

“…Finally, we notice that the assumptions in [28] (see the third problem discussed in section 2) do not imply those considered here. In fact: The (nonlocal) heat flux (2.3) in combination with a classical energy conservation law in the liquid phase yields…”

“…0 < x < s(t), t > 0, which in general differs from (3.2) since, as pointed out in [28], we cannot interchange the first integral operator with the classical spacial derivative in the right-hand side due to the loss of differentiability of the heat fluxq along the interface.…”

“…Recently, Roscani, Bollati, and Tarzia introduced a third approach in [28] that leads to a new fractional Stefan-like problem for one-dimensional phase-transitions.…”

“…For example, several (nonequivalent) time-fractional Stefan-like problems were proposed as possible models for one-dimensional phase-change processes that exhibit a characteristic time of diffusion ∼ t γ/2 with γ ∈ (0, 1), e.g. [21,28,43].…”

A self‐similar solution to time‐fractional Stefan problem

“…In papers, 6‐10 the authors discussed other possible formulations of time‐fractional Stefan problem and compare the formulas for special solutions. In paper, 11 there is shown that the time‐fractional sharp‐interphase model obtained in Falcini et al 1 is not a consequence of the assumption (). Moreover, the authors obtained a new model based on ().…”