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

摘要

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