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

Abstract Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f (u) are considered. The method of differential Grobner 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.

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