Group classification of heat conductivity equations with a nonlinear source

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.

LaTeX, 51 pages

References in corpus (1)

• Symmetries and Exact Solutions of Nonlinear Dirac Equations

Cited by in corpus (30)

• Realizations of Real Low-Dimensional Lie Algebras
• Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations
• New results on group classification of nonlinear diffusion-convection equations
• Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
• Complete group classification of a class of nonlinear wave equations
• Symmetry classification of KdV-type nonlinear evolution equations
• Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
• Enhanced preliminary group classification of a class of generalized diffusion equations
• Symmetry preserving parameterization schemes
• Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations
• Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities
• Group classification of linear evolution equations
• Group analysis of general Burgers-Korteweg-de Vries equations
• Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes
• Group classification of nonlinear Schrödinger equations
• Variational Contact Symmetries of Constraint Lagrangians
• Group Classification of a Generalized Black--Scholes--Merton Equation
• Equivalence groupoid and group classification of a class of variable-coefficient Burgers equations
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras
• Lie Symmetries of (1+1)-Dimensional Cubic Schrödinger Equation with Potential
• Classification of classical and non-local symmetries of fourth-order nonlinear evolution equations
• Symmetry classification of third-order nonlinear evolution equations
• Group Classification of a generalization of the Heath Equation
• The group classification of one class of nonlinear wave equations
• Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid
• On contact symmetries of evolution equations
• Group classification via mapping between classes: an example of semilinear reaction-diffusion equations with exponential nonlinearity
• Group classification and symmetry reductions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy
• Invariance of the quasilinear equations of hiperbolic type with respect to the three-parametric Lie algebras