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New exact solutions of the Boussinesq equation. (English) Zbl 0721.35074

Summary: In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation \(q_{tt}+qq_{xx}+q^ 2_ x+q_{xxxx}=0\). These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)

Software:

MACSYMA
PDFBibTeX XMLCite
Full Text: DOI

References:

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