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Symmetry reductions and exact solutions of a class of nonlinear heat equations. (English) Zbl 0812.35017

Summary: 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 \(f(u)=0\).

MSC:

35C05 Solutions to PDEs in closed form
35K05 Heat equation
35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
58J70 Invariance and symmetry properties for PDEs on manifolds
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