Construction of exact solutions to nonlinear equations of the hyperbolic type

1Barannyk, AF, 2Barannyk, TA, 3Yuryk, II
1Institute of Mathematics, Pomeranian University, Slupsk, Poland
2V.G. Korolenko Poltava National Pedagogical University
3National University of Food Technologies, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 7:3-9
https://doi.org/10.15407/dopovidi2017.07.003
Section: Mathematics
Language: Ukrainian
Abstract: 
Substitutions that reduce the equation $u_{tt} = a(t)uu_{xx} + b(t)u_{x}^{2} + c(t)u$ to a system of ordinary differential equations are considered. An efficient method to integrate the corresponding reduced systems is proposed. It is shown that their integration can be reduced to the integration of a system of linear equations $w_{1}^{''} = \Phi_{1}(t)w_{1}$, $w_{2}^{''} = \Phi_{2}(t)w_{2}$, where $\Phi_{1}(t)$ and $\Phi_{2}(t)$ are arbitrary predefined functions.
Keywords: differential equations, generalized separation of variables, nonlinear hyperbolic equations
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