Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type

TL;DR

This paper introduces a new definition of Lie invariance for complex multi-dimensional boundary value problems, specifically applied to Stefan-type problems modeling melting and evaporation, enabling symmetry-based reductions and exact solutions.

Contribution

It proposes a generalized Lie invariance definition for nonlinear multi-dimensional BVPs and applies it to classify and reduce Stefan-type problems, providing a framework for exact solutions.

Findings

Solved the group classification problem for (1+3)-dimensional Stefan BVPs

Derived reductions to lower-dimensional BVPs with physical significance

Constructed exact solutions for specific coefficient cases

Abstract

A new definition of Lie invariance for nonlinear multi-dimensional boundary value problems (BVPs) is proposed by the generalization of known definitions to much wider classes of BVPs. The class of (1+3)-dimensional nonlinear BVPs of the Stefan type, modeling the process of melting and evaporation of metals, is studied in detail. Using the definition proposed, the group classification problem for this class of BVPs is solved and some reductions (with physical meaning) to BVPs of lower dimensionality are made. Examples of how to construct exact solutions of the (1+3)-dimensional nonlinear BVP with the correctly-specified coefficients are presented.