# Mel'nik T. A.

### Homogenization of a quasilinear parabolic problem with different alternating nonlinear Fourier boundary conditions in a two-level thick junction of the type 3:2:2

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1632-1656

We investigate the asymptotic behavior of a solution of a quasilinear parabolic boundary-value problem in a two-level thick junction of the type 3:2:2. This junction consists of a cylinder on which thin disks of variable thickness are $\varepsilon$-periodically threaded. The thin disks are divided into two levels, depending on their geometric structure and the conditions imposed on their boundaries. In this problem, we consider different alternating inhomogeneous nonlinear Fourier conditions. Moreover, the Fourier conditions depend on additional perturbation parameters. We prove theorems on the convergence of a solution of this problem as $\varepsilon \rightarrow 0$ for different values of these parameters.

### Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 195–216

A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are $\varepsilon$-periodically alternated. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as $\varepsilon \rightarrow 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. The Hausdorff convergence of the spectrum is proved as $\varepsilon \rightarrow 0$, the leading terms of asymptotics are constructed and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

### Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1524-1533

We prove a convergence theorem and obtain asymptotic (as ε → 0) estimates for a solution of a parabolic initial boundary-value problem in a junction Ω_{ε} that consists of a domain Ω_{0} and a large number *N* ^{2} of ε-periodically located thin cylinders whose thickness is of order ε = *O*(*N* ^{−1}).

### Asymptotics of solutions of discontinuous singularly perturbed boundary-value problems

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 861–864

We construct an asymptotic expansion of a boundary-value problem for a singularly perturbed system of differential equations with the right-hand side discontinuous at certain surface.

### Decomposition of systems of quasidifferential equations with rapid and slow variables

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 413–417

We obtain the decomposition of systems of quasidifferential equations with rapid and slow variables.

### Linear singularly perturbed problems with pulse influence

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 133–139

We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically large pulse influence and the corresponding degenerate problem.

### An iteration method for the problem of averaging in the standard form

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 426–429

An iteration method for the enhancement of the precision of an approximate solution of the problem of averaging in the standard form is considered.

### Aniterative method for the solution of some singularly perturbed cauchy problems

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1055–1060

We construct an iterative method for the solution of Cauchy problems for systems of singularly perturbed equations with fast time.

### Asymptotics of a solution of a discontinuous singularly perturbed Cauchy problem

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1502–1508

We construct the asymptotic expansion of a solution of the Cauchy problem for a singularly perturbed system of differential equations whose right-hand side is discontinuous on a certain surface. We consider the case where the surface of discontinuity is crossed and estimate the remainder of the constructed asymptotic expansion.