Exact solutions of nonlinear boundary value problems of the Stefan type

TL;DR

This paper employs Lie symmetry methods to find exact solutions for nonlinear Stefan-type boundary value problems modeling metal melting and evaporation, identifying conditions for explicit analytical solutions.

Contribution

It systematically determines all Lie symmetries that reduce the nonlinear heat equation to solvable ordinary differential equations, revealing specific heat conductivity forms for explicit solutions.

Findings

Identified all Lie operators enabling reduction to ODEs.

Established heat conductivity coefficients allowing explicit solutions.

Provided a classification of solvable nonlinear boundary value problems.

Abstract

The (1+1)-dimensional nonlinear boundary value problem, modeling the process of melting and evaporation of metals, is studied by means of the classical Lie symmetry method. All possible Lie operators of the nonlinear heat equation, which allow us to reduce the problem to the boundary value problem for the system of ordinary differential equations, are found. The forms of heat conductivity coefficients are established when the given problem can be analytically solved in an explicit form.