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Some symmetries of the nonlinear heat and wave equations. (English) Zbl 1020.35501
From the text: In this note we are concerned with the construction of special solutions to the nonlinear heat-convection and wave equations, namely: (1) $u_t=[K(u)u_x]_x+[\varphi(u)]_x$ and (2) $u_{tt}=[K(u)u_x]_x$. First, the classical Lie group theory is used to construct the classical symmetries of (1) and (2). The results for the purely diffusive, diffusive-convective case and wave-equation (2) are summarized in tables 1a, 3a and 2a, respectively. Table 4 presents our results for the hyperbolic diffusion equation. We then present some special solutions whose underlying symmetry may not be attained via the classical Lie group theory.

35A30Geometric theory for PDE, characteristics, transformations
35K05Heat equation
35L05Wave equation (hyperbolic PDE)
Full Text: DOI
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