асимптотика

Low-Frequency Vibration Modes of Strongly Inhomogeneous Elastic Laminates

Authors: Kaplunov J. D., Prikazchikova L. A.

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. For a five-layered laminate with clamped faces two term expansions for eigenfrequencies and eigenforms are compared with those obtained from the exact solution of the original problem for thickness resonances.

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

Authors: Kossovich L. Y., Yurko V. A., Kirillova I. V.

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.

About Asymptotics of Chebyshev Polynomials Orthogonal on an Uniform Net

Authors: Sultanov E. S.

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.

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

Authors: Vestyak V. A., Zemskov A. V., Fedorov I. A.

 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.

Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

Authors: Kornev V. V., Khromov A. P.

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