# Berezovsky A. A.

### Finite-Time Stabilization in Problems with Free Boundary for Nonlinear Equations in Media with Fractal Geometry

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 997–1001

By using the method of a priori estimates, we establish differential inequalities for energetic norms in $W^l_{2,r}$ of solutions of problems with a free bound in media with the fractal geometry for one-dimensional evolutionary equation. On the basis of these inequalities, we obtain estimates for the stabilization time $T$.

### On One Nonlocal Problem with Free Boundary

Berezovsky A. A., Mitropolskiy Yu. A., Netesova T. M.

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 908-918

We investigate group-theoretic properties of a nonlocal problem with free boundary for a degenerating quasilinear parabolic equation. We establish conditions for the invariant solvability of this problem, perform its reduction, and obtain an exact self-similar solution.

### Problems with free boundaries for nonlinear parabolic equations

Berezovs'kyi M. A., Berezovsky A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1360–1372

We establish necessary conditions for the existence of effects of space localization and stabilization in time that are qualitatively new for evolutionary equations. We suggest constructive methods for the solution of the corresponding one-dimensional problems with free boundaries that appear in ecology and medicine.

### Seminar-School “Mathematical Simulation”

Berezovsky A. A., Khomchenko A. N., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1152

### Ukrainian school-seminar "Nonlinear boundary value problems of mathematical physics and applications"

Berezovsky A. A., Konet I. M., Lenyuk M. P., Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 613–617

### Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 244–254

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an *a priori* estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small.

### Problems with free boundaries and nonlocal problems for nonlinear parabolic equations

Berezovsky A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 84–97

We present statements of problems with free boundaries and nonlocal problems for nonlinear parabolic equations arising in metallurgy, medicine, and ecology. We consider some constructive methods for their solution.

### School-Seminar “Nonlinear boundary-value problems in Mathematical Physics and their applications”

Berezovsky A. A., Lenyuk M. P., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 863-865

### Space-time localization in problems with free boundaries for a nonlinear second-order equation

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 202-211

For thermal and diffusion processes in active media described by nonlinear evolution equations, we study the phenomena of space localization and stabilization for finite time.

### Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Berezovsky A. A., Kerefov A. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1289–1398

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.

### Equivalent linearization of systems with distributed parameters

Berezovsky A. A., Konovalova N. R., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 464-471

### Qualitative investigation of a mathematical model of the thermocline

Berezovsky A. A., Boguslavsky A. S., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1982. - 34, № 5. - pp. 631—633

### Curved stationary waves in rods under a nonlinear law of elasticity

Berezovsky A. A., Zhernovoi Yu. V.

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 493–498

### Invariant solutions of one quasilinear equation

Berezovsky A. A., Netesova T. M.

Ukr. Mat. Zh. - 1977. - 29, № 4. - pp. 509–513

### Longitudinal-transversal vibrations of viscoelastic rods with consideration of physical and geometric nonlinearities

Berezovsky A. A., Kurbanov I., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1976. - 28, № 5. - pp. 629–638

### Parametrically excited oscillations of a rod subject to a nonlinear law of elasticity

Berezovsky A. A., Mitropolskiy Yu. A., Turgunov N.

Ukr. Mat. Zh. - 1975. - 27, № 3. - pp. 395–400

### The increase in the stability of a flexible circular flat plate by the use of high frequency compressive forces

Berezovsky A. A., Mitropolskiy Yu. A., Turgunov N.

Ukr. Mat. Zh. - 1974. - 26, № 3. - pp. 402–408

### Integro-differential Equations of the Nonlinear Theory of Depressed Thin Shells

Ukr. Mat. Zh. - 1960. - 12, № 4. - pp. 373 - 380

A method is presented for obtaining nonlinear integro-differential equations equivalent to the differential equations of the nonlinear theory of depressed shells. It is shown fhat in a number of cases the integral equations obtained lead to the same results äs^be nonlinear differential equations.

### On the Motion of a Load on a Depressed Shell

Ukr. Mat. Zh. - 1960. - 12, № 1. - pp. 79-87