Lie symmetries of nonlinear boundary value problems

TL;DR

This paper extends Lie symmetry methods to nonlinear boundary value problems, providing a new definition, classifying symmetries for metal melting and evaporation models, and deriving exact solutions with physical relevance.

Contribution

It introduces a generalized definition of Lie invariance for BVPs and applies it to complex physical models, enabling symmetry-based reductions and exact solutions.

Findings

Classified all Lie symmetries for the studied BVPs

Derived reductions to ODEs with physical interpretation

Constructed exact solutions matching numerical results

Abstract

Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.