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

Peter A. Clarkson·Elizabeth L. Mansfield
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Abstract

Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ 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 (1) for cubic $f(u)$ in terms of the roots of $f(u)=0$.

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Peter A. Clarkson, Elizabeth L. Mansfield. 1993-06-16. Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat Equations. https://arxiv.org/abs/solv-int/9306002

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