Projective analysis and preliminary group classification of the nonlinear fin equation $u_t=(E(u)u_x)_x + h(x)u$

TL;DR

This paper explores the symmetry properties of nonlinear fin equations, showing they cannot be linearized through projective symmetries and classifying equations based on their Lie algebra extensions.

Contribution

It provides a detailed symmetry analysis, including projective symmetries and an equivalence classification, for a broad class of nonlinear fin equations.

Findings

Equations cannot be reduced to linear form via projective symmetries.

Classification of equations based on extended Lie algebra structures.

Invariant solutions and additional symmetry operators are identified.

Abstract

In this paper we investigate for further symmetry properties of the nonlinear fin equations of the general form $u_t=(E(u)u_x)_x + h(x)u$ rather than recent works on these equations. At first, we study the projective (fiber-preserving) symmetry to show that equations of the above class can not be reduced to linear equations. Then we determine an equivalence classification which admits an extension by one dimension of the principal Lie algebra of the equation. The invariant solutions of equivalence transformations and classification of nonlinear fin equations among with additional operators are also given.