2019
Том 71
№ 11

All Issues

Barannyk A. F.

Articles: 16
Article (Ukrainian)

A method for the construction of exact solutions to the nonlinear heat equation $u_t = \left(F(u)u_x \right)_x +G(u)u_x +H(u)$

Barannyk A. F., Barannyk T. A., Yuryk I. I.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1443 -1454

UDC 517.9
We propose a method for the construction of exact solutions to the nonlinear heat equation based on the classical method of separation of variables and its generalization. We consider substitutions used to reduce the nonlinear heat equation to a system of two ordinary differential equations and construct the classes of exact solutions by the method of generalized separation of variables.

Article (Ukrainian)

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

Barannyk A. F., Barannyk T. A., Yuryk I. I.

↓ Abstract   |   Full text (.pdf)

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

Barannyk A. F., Barannyk T. A., 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

Barannyk A. F., Barannyk T. A., 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

Barannyk 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 (Ukrainian)

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

Barannyk 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)

Barannyk 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).

Article (Russian)

Reduction of the multidimensional d’Alembert equation to two-dimensional equations

Barannik L. F., Barannyk A. F., Fushchich V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 651–662

We give a classification of the maximal subalgebras of rank $n - 1$ for the extended Poincare algebra $A\bar P (1.n)$, which is realized on the set of solutions of the d'Alembeit equation $\square u + \lambda u^k = 0$. These subalgebras are used for constructing the anzatses reducing this equation to differential equations with two invariant variables.

Article (Ukrainian)

Reduction of a multidimensional poincare-invariant nonlinear equation to two-dimensional equations

Barannik L. F., Barannyk A. F., Fushchich V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1311–1323

Article (Ukrainian)

Connected subgroups of the conformal group C(1,4)

Barannik L. F., Barannyk A. F., Fushchich V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 870–884

A method is given for describing maximal subalgebras of rank r, 1 ≤r ≤ 4, of the conformal algebra AC(1,4), which is the maximal invariance algebra of the eikonal equation. With the help of this method a classification is made up to C(1,4)-equivalence of all maximal subalgebras L of rank 1, 2, 3, and 4 of the algebra AC(1,4) satisfying the condition L ∩ V ⊂〈p1, P2, P3, p4〉, where V is the space of translations.

Article (Ukrainian)

Reduction and exact solutions of the eikonal equation

Barannik L. F., Barannyk A. F., Fushchich V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 461-474

Article (Ukrainian)

Maximal subalgebras of rank n?1 of the algebra AP(1, n) and the reduction of nonlinear wave equations. II

Barannyk A. F., Fushchich V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 12. - pp. 1693–1700

Article (Ukrainian)

Continuous subgroups of a generalized Euclidean group

Barannik L. F., Barannyk A. F., Fushchich V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 67–72

Article (Ukrainian)

Crossed group rings in which solutions of the equation x n - μ = 0 are trivial

Barannik L. F., Barannyk A. F.

Full text (.pdf)

Ukr. Mat. Zh. - 1983. - 35, № 2. - pp. 137—143

Article (Ukrainian)

Nearly Hamiltonian groups

Barannyk A. F.

Full text (.pdf)

Ukr. Mat. Zh. - 1978. - 30, № 5. - pp. 579–585

Article (Ukrainian)

A generalization of metahamiltonian groups

Barannyk A. F.

Full text (.pdf)

Ukr. Mat. Zh. - 1973. - 25, № 6. - pp. 792—795