A Class of Moving Boundary Problems with a Source Term. Application of a Reciprocal Transformation

TL;DR

This paper introduces a new Stefan-type moving boundary problem with a source term, demonstrating its equivalence to a nonlinear boundary value problem via reciprocal transformations, and provides explicit solutions for a special case.

Contribution

It establishes an equivalence between a Stefan problem with variable phase-change temperature and a nonlinear evolution equation using reciprocal transformations, including explicit solutions.

Findings

Proved the equivalence of Stefan and boundary value problems.

Derived explicit solutions for a specific case.

Applied reciprocal transformations to simplify complex boundary problems.

Abstract

We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems.