Group theoretic analysis of a class of boundary value problems for a nonlinear heat equation

TL;DR

This paper develops a group theoretic framework for analyzing boundary value problems of a nonlinear heat equation, enabling classification and invariant solutions, thus advancing mathematical modeling of heat conduction.

Contribution

It introduces a new invariance definition for BVPs, formulates a classification algorithm, and applies it to a nonlinear heat equation model.

Findings

Group classification of BVPs for nonlinear heat equations achieved

Invariant solutions for specific boundary value problems derived

A systematic algorithm for classifying BVPs based on symmetry is proposed

Abstract

A definition of invariance in Lie's sense for a boundary value problem (BVP) with the basic evolution differential equations is proposed. A problem of group classification at a wide class of BVPs parameterized by arbitrary elements is formulated and an algorithm for its solution is also proposed. The group classification of a class of BVPs for an (1+1)--dimensional nonlinear heat equation arising from mathematical modeling of heat conduction in semi-infinite body is carried out. An example of invariant solution of a BVP from the class under study is presented.