Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1993-06-16
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Latex file 32 pages, 13 figures available from author
Scientific paper
Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gr\"obner 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 $f(u)=0$.
Clarkson Peter A.
Mansfield Elizabeth L.
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