2019
Том 71
№ 10

All Issues

Yuryk I. I.

Articles: 7
Article (Ukrainian)

Exact solutions of the nonliear equation $u_{tt} = = a(t) uu_{xx} + b(t) u_x^2 + c(t) u $

Barannik A. F., Barannik A. F., Yuryk I. I.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1180-1186

Ans¨atzes that reduce the equation$u_{tt} = = a(t) uu_{xx} + b(t) u_x^2 + c(t) u $ to a system of two ordinary differential equations are defined. Also it is shown that the problem of constructing exact solutions of the form $u = \mu 1(t)x_2 + \mu 2(t)x\alpha , \alpha \in \bfR$, to this equation, reduces to integrating of a system of linear equations $\mu \prime \prime 1 = \Phi 1(t)\mu 1, \mu \prime \prime 2 = \Phi 2(t)\mu 2$, where $\Phi 1(t)$ and \Phi 2(t) are arbitrary predefined functions.

Article (Ukrainian)

Generalized separation of variables and exact solutions of nonlinear equations

Barannik A. F., Barannik A. F., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1598 - 1609

We consider the generalized procedure of separation of variables of the nonlinear hyperbolic-type equations and the Korteweg - de Vries-type equations. We construct a wide class of exact solutions of these equations which cannot be obtained with the use of the S. Lie method and the method of conditional symmetries.

Article (Ukrainian)

Generalized procedure of separation of variables and reduction of nonlinear wave equations

Barannik A. F., Barannik A. F., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 892-905

We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries.

Article (Ukrainian)

On Exact Solutions of Nonlinear Diffusion Equations

Barannik A. F., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1011 – 1019

New classes of the exact solutions of nonlinear diffusion equations are constructed.

Article (Russian)

Nonlinear d'alembert equation in the pseudo-euclidean space $R_{2,n}$ and its solutions

Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 820-827

We investigate the nonlinear D'Alembert equation in the pseudo-Euclidean space $R_{2,n}$ and construct new exact solutions containing arbitrary functions.

Article (Ukrainian)

A new method for the construction of solutions of nonlinear wave equations

Barannik A. F., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 583-593

We propose a simple new method for the construction of solutions of multidimensional nonlinear wave equations.

Article (Ukrainian)

Classification of maximal subalgebras of rank n of the conformal algebra AC(1, n)

Barannik A. F., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 459–470

We obtain a complete classification of I-maximal subalgebras of rank n of the conformal algebra AC(1, n).