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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.

##### MSC:
 35A30 Geometric theory for PDE, characteristics, transformations 35K05 Heat equation 35L05 Wave equation (hyperbolic PDE)
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##### References:
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