Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat Equations
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 $f(u)=0$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 1993
- DOI:
- 10.48550/arXiv.solv-int/9306002
- arXiv:
- arXiv:solv-int/9306002
- Bibcode:
- 1993solv.int..6002C
- Keywords:
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- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- Latex file 32 pages, 13 figures available from author