Group classification of heat conductivity equations with a nonlinear source

TL;DR

This paper develops a systematic classification method for heat conductivity equations with nonlinear sources based on Lie algebra symmetries, identifying all equations with nontrivial symmetries and their equivalence classes.

Contribution

It introduces a comprehensive classification framework combining Lie's method, equivalence transformations, and algebra theory, applied specifically to nonlinear heat equations.

Findings

Complete classification of equations with symmetry groups up to four dimensions.

Any equation with symmetry higher than four dimensions is equivalent to a linear equation.

Identification of new invariant equations with potential applications in heat transfer modeling.

Abstract

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.