T. A. Mel’nyk

Homogenization of 3D thick cascade junction with a random transmission zone periodic in one direction

In the paper, we deal with the homogenization problem for the Poisson equation in a singularly perturbed three-dimensional junction of a new type. This junction consists of a body and a large number of thin curvilinear cylinders, joining to body through a random transmission zone with rapidly oscillating boundary, periodic in one direction. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and with random perturbed coefficients on the boundary of the transmission zone. We prove the homogenization theorems and the convergence of the energy integrals.

Asymptotic Analysis of the Neumann Problem for the Ukawa Equation in a Thick Multi-structure of Type 3:2:2

We propose two different approaches for asymptotic analysis of the Neumann boundary-value problem for the Ukawa equation in a thick multistructure Ω e , which is the union of a domain Ω0 and a large number N of e—periodically situated thin annular disks with variable thickness of order \(\varepsilon = \mathcal{O}\left( {N^{ - 1} } \right)\), as e → 0. In the first approach, using some special extension operator, the convergence theorem is proved as e → 0. In the second one, the leading terms of the asymptotic expansion for the solution are constructed and the corresponding estimates in the Sobolev space H1(Ωe) are proved.

Asymptotic Analysis of Spectral Problems in Thick Multi-Level Junctions

Spectral boundary-value problems are considered in a new kind of perturbed domain, namely, thick multi-level junctions. Boundary-value problems in thick one-level junctions (thick junctions) have been intensively investigated recently (see, for instance, [BlGaGr07], [BlGaMe08], [Me08] and, the references there). In [MeNa97]–[Me(3)01], classification of thick one-level junctions was given and basic results were obtained both for boundary-value and spectral problems in thick junctions of different types. It was shown that qualitative properties of solutions essentially depend on the junction type and on the conditions given on the boundaries of the attached thin domains. It is known that the asymptotic behavior of the spectrum of a perturbed spectral problem is highly sensitive to perturbation, and it is unexpected. This was also observed for spectral problems in thick junctions with Neumann conditions ([MeNa97] and [Me00]), with Dirichlet conditions ([Me99] and [Me(3)01]), with Fourier conditions ([Me(2)01]) and with Steklov ones ([Me(1)01]).

Asymptotic Approximations for Chemical Reactive Flows in Thick Fractal Junctions

The asymptotic analysis of a reaction-diffusion system with nonlinear boundary conditions in a thick fractal junctions is presented. In particular, the corresponding homogenized problem is found, the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved, and the approximation for the solution is constructed and justified.