Group classification of heat conductivity equations with a nonlinear source

Abstract

We suggest a systematic procedure for classifying partial differential equations (PDEs) invariant with respect to low-dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie method, the technique of equivalence transformations and the 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 3, 7, 28 and 12 inequivalent classes of PDEs of the type considered that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any PDE belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation. This classification is compared with 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 modelling of, say, nonlinear heat transfer processes.