Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat Equations

TL;DR

This paper investigates symmetries of a nonlinear heat equation, using differential Gr"obner bases to find conditions for additional symmetries and to derive exact solutions, including new reductions for the linear case.

Contribution

It introduces a systematic method to determine symmetry conditions and provides a catalogue of new symmetry reductions and exact solutions for specific nonlinear heat equations.

Findings

Identified conditions for nontrivial symmetries of the nonlinear heat equation.

Developed a catalogue of symmetry reductions, including new ones for the linear case.

Derived exact solutions for cubic nonlinearities using roots of the equation.

Abstract

Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gr\"obner bases is used both to find the conditions on $f(u)$ under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic $f(u)$ in terms of the roots of $f(u)=0$.