# Barannyk T. A.

### 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.

### 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.

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.

### Conditional Symmetry of a System of Nonlinear Reaction-Diffusion Equations

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1443-1449

The conditional symmetry of a system of nonlinear reaction-diffusion equations is investigated. It is shown that the operators of conditional symmetry exist for the systems of nonlinear reaction-diffusion equations with an arbitrary number of independent variables. Moreover, these operators are found in the explicit form.

### Generalized separation of variables and exact solutions of nonlinear equations

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

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.

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

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

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.

### Conditional Symmetry and Exact Solutions of a Multidimensional Diffusion Equation

Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1416-1420

We investigate the conditional symmetry of a multidimensional nonlinear reaction–diffusion equation by its reduction to a radial equation. We construct exact solutions of this equation and infinite families of exact solutions for the corresponding one-dimensional diffusion equation.