ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

# асимптотика

## Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

The object of the paper is Dirac system with antiperiodic boundary conditions and complex-valued conditions potential. A new method is suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transform operators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it is proved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensional vector-functions since eigenvalues may be multiple.

## The asymptotic separation of variables in thermoelastic problem for anisotropic layer with inhomogeneous boundary conditions

A method for resolving a thermoelasticity problem with inhomogeneous boundary conditions is presented. Boundary conditions represent uneven surface heating of the layer. An asymptotic procedure for separation of variables based on introduction of additional dimensional scales is used. With an additional assumption that the unevenness of the heating is small enough this procedure makes it possible to obtain the solution. The method is shown for periodic heating case. After the separation of variables the solution is obtained using Fourier series.

## Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight

Current work is devoted to investigation of properties of polynomials, orthogonal with Jacobi weight on nonuniform grid where. In case of integer for such discrete orthonormal polynomials  asymptotic formula  with  was obtained, where classical Jacobi polynomial, remainder term. As corollary of asymptotic formula it was deduced weighted estimation  polynomials on segment [−1,1].

## About Asymptotics of Chebyshev Polynomials Orthogonal on an Uniform Net

In this article asymptotic properties of the Chebyshev polynomials Tn(x,N) (0 ≤ n ≤ N − 1) orthogonal on an uniform net ΩN = {0,1,...,N − 1} with the constant weight µ(x) = 2 N (discrete analog of the Legendre polynomials) by n = O(N 1 2 ), N → ∞ were researched. The asymptotic formula that is relating polynomials Tn(x,N) with Legendre polynomials Pn(t) for x = N 2 (1 + t) − 1 2 was determined.

## Development of Asymptotic Methods for the Analysis of Dispersion Relations for a Viscoelastic Solid Cylinder

Propagation of time-harmonic waves in a viscoelastic solid cylinder is considered. Vibrations of the cylinder are described by three-dimensional viscoelasticity equations in  cylindrical coordinates. The stress-free surface boundary conditions are imposed. Viscoelastic properties are described by integral operators with a fractional-exponential kernel. For the case of a rational singularity parameter the method of asymptotic analysis of dispersion relations is proposed, which is based on the generalized power series expansion.

## Mode-Series Expansion of Solutions of Elasticity Problems for a Strip

Oscillations of a strip are considered as a plane problem of elasticity theory. Description of oscillation modes is provided. Properties of eigenvalues and eigenfunctions are studied for a boundary value problem for their amplitudes. Green’s function is constructed as a kernel of the inverse operator. Completeness and expansion theorems are proved which allow one to solve problems for finite and infinite membranes under arbitrary boundary conditions.

## Inverse Problem for Sturm – Liouville Operators in the Complex Plane

The inverse problem for the standard Sturm – Liouville equation with a spectral parameter ρ and a potential function, piecewise-entire on a rectifiable curve γ ⊂ C, on which only the starting point is given, is studied for the first time. A function Q that is bounded on a curve γ is piecewise-entire on it if γ can be splitted by a finite number of points into parts on which Q coincides with entire functions, different in neighboring parts. The split points, the initial and final points of the curve are called critical points.

## Low-Frequency Vibration Modes of Strongly Inhomogeneous Elastic Laminates

The dynamic behaviour of thin multi-layered structures, composed of contrasting “strong” and “weak” layers, is considered. An asymptotic procedure for analysing the lowest cutoffs is developed. A polynomial frequency equation is derived, along with the linear equations for the associated eigenforms corresponding to displacement variation across the thickness.