Symmetry Classification for the Nonlinear Heat Conductivity Equation

TL;DR

This paper analyzes the symmetry properties of nonlinear heat conductivity equations, classifies them based on their symmetries, and finds invariant solutions, enhancing understanding of their mathematical structure.

Contribution

It provides a comprehensive symmetry classification and invariant solutions for a broad class of nonlinear heat conductivity equations, extending previous analyses.

Findings

Symmetry properties of the equations are characterized.

A classification scheme based on Lie algebra extensions is developed.

Invariant solutions and equivalence transformations are identified.

Abstract

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie algebra of them. The invariant solutions of equivalence transformations and classification of the nonlinear heat conductivity equations among with additional operators are also given.