# eigenfunctions

## The equilibrium equations of shells in the coordinates of the general form

A mathematical model of homogeneous elastic shells is consider under kinematics Reissner–Mindlin type. Through direct (coordinateless) methods of the tensor calculus equations of equilibrium are obtained in terms of displacements in an arbitrary (not necessarily orthogonal) coordinate system, taking into account the asymmetry of the location of the front surface.

## Non-stationary vibration of growth circular cylindrical shell

Small forced vibrations of growing cylindrical shell fixed on circular boundaries is studied in the framework of Kirchhoff–Love shell theory. The process of the accretion are characterized by the continuous adherence of material particles to its facial surface. Since the shell bends during the accretion, its stressed-strained state depends not only on loading, but also on the history of the process of accretion, i.e. the schedule of accretion.

## On Propertiesofthe Eigenfunctionsofa Quadratic Pencilofthe Second Order Differential Operators

The degenerated second order ordinary differential quadratic pencil with constant coefﬁcients is considered. The case is studied, when the roots of characteristic equation lie on a straight line coming through the origin and on the both side of the origin. Properties of the system of its eigenfunctions in the spaces L2[0,σ], σ > 0 is investigated. Criteria of one-fold completeness and minimality of this systemareprovedandsufﬁcientconditionsofone-foldcompleteness and minimality are found.

## On Riesz Basises of Eigenfunctions of Integral Operators with Kernels Discontinuous on Broken Lines

For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.

## Dynamics of Multilayered Thermoviscoelastic Plates

This paper deal with laminated thin-walled structures. The laminated structures considered herein consist of three layers. The following assumptions are assumed. The thickness of inner layer is considerably greater the others.The kine matic relationsf ortheinnerlayerare examined in the form of Mindlin – Reissner shell theory, for the outer layers are in the form of membrane theory. The deformations of the whole layered structure are deﬁned by the polyline hypothesis.

## 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.

## Equiconvergence Theorem for Integral Operator with Involution

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

## The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange – Sturm – Liouville

Let us say that the principle of localization holds at the class of functions *F* at point *x _{0}* ∈ [0,

*π*] for the Lagrange –Sturm – Liouville interpolation process

*L*(

^{SL}_{n}*f, x*) if lim

_{n→∞}

*L*(

^{SL}_{n}*f*,

*x*) −

_{0}*L*(

^{SL}_{n }*g*,

*x*) = 0 follows from the fact that the condition

_{0}*f*(

*x*) =

*g<*