Symmetry reductions and exact solutions of a class of nonlinear heat equations

Abstract

Classical and nonclassical symmetries of the nonlinear heat equation u t = u xx + f ( u) are considered. The method of differential Gröbner 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 the nonlinear heat equation for cubic f ( u) in terms of the roots of f ( u) = 0.