Barannyk Anatoly, Barannyk Tetyana, Yuryk Ivan (Institute of Mathematics, Pomeranian University, Slupsk, Poland; Poltava V.G. Korolenko National Pedagogical University, Ukraine; National University of Food Technology, Kyiv, Ukraine)

Construction of exact solutions to nonlinear equations of hyperbolic type

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 effective method to integrate the corresponding reduced systems is proposed. It is shown that their integration can be reduced to integration of 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.