Vol 59, No 5 (2023)
Articles
Asymptotic Properties of a Class of Systems with Linear Delay
Abstract
Sufficient conditions for the asymptotic stability of linear systems of differential equations with linear delay are obtained. On the basis of these conditions, some systems of linear differential equations are studied, and one of them is stabilized on an infinite time interval.
569-581
Spectral Properties of a Singular Differential Operator on an Interval with Transmission Conditions
Abstract
We study the first boundary value problem for a second-order differential operator with a singular coefficient on an interval with transmission conditions at an interior point. Asymptotic formulas are obtained for the eigenfunctions and eigenvalues of both the original and the adjoint operator. The completeness and unconditional basis property of the eigenfunction systems of these operators in the space of square integrable functions on the interval are established. Il’in’s method and Il’in’s conditions are applied to establish the Bessel inequality.
582-587
On the Control of the Spectra of Upper Strong Oscillation Exponents of Signs, Zeros, and Roots of Third-Order Differential Equations
Abstract
We construct an example of a third-order linear homogeneous differential equation with continuous coefficients on the half-line whose spectra of upper strong oscillation exponents of the signs, zeros, and roots coincide with a given Souslin set containing zero in the nonnegative half-line of the extended real line.
588-595
On the Pinsky Phenomenon for B-Elliptic Operators
Abstract
Necessary conditions for the summability of spectral expansions in eigenfunctions of an elliptic operator with a Bessel operator in one of the variables in an arbitrary
-dimensional domain adjacent to the singularity hypersurface are obtained. It is proved that if the spectral expansion of an arbitrary function at some point of this hypersurface is summable by Riesz means, then its average value over the half-ball centered at the specified point has generalized smoothness.
596-607
On the Unique Solvability of Initial–Boundary Value Problems for Parabolic Systems in Bounded Plane Domains with Nonsmooth Lateral Boundaries
Abstract
We consider the first and second initial–boundary value problems for inhomogeneous second-order parabolic systems with Dini continuous coefficients under nonzero initial conditions in bounded domains on the plane with nonsmooth lateral boundaries that, in particular, admit cusps. Theorems are proved on the unique classical solvability of these problems in the space of functions that are continuous together with their first spatial derivatives in the closure of these domains.
608-618
Traveling Wave Method
Abstract
A survey of the development of the traveling wave method for one-dimensional media is presented. The main results and changes in the statement of the problem of representing solutions of linear systems of partial differential equations in terms of “traveling waves” (more precisely, in terms of a system of wave transport equations) are presented. It is shown that as the study of systems becomes more complicated, the problem of representing the solution by the traveling wave method turns out to be applicable not only for hyperbolic systems but also for systems containing (even implicitly) both parabolic and elliptic components and thereby approaches the general problem of decomposition of an arbitrary system of linear equations into a system of first-order equations with a main part of the canonical type and with a subordinate linear part.
619-634
On the Fundamental Solution Matrix of the Plane Anisotropic Elasticity Theory
Abstract
An explicit expression (in polar coordinates) for the fundamental solution matrix of the Lamé system of the plane anisotropic theory of elasticity is given. It is shown that the operator of convolution with this matrix in a finite domain with Lyapunov boundary is bounded in the Hölder spaces. A similar result is also established for an infinite domain in the corresponding weighted Hölder spaces (with a power-law behavior at infinity).
635-641
On the Darboux Problem for Hyperbolic Systems
Abstract
For a hyperbolic system with simple characteristics in the
-dimensional space of independent variables, the existence and uniqueness of a solution of the Darboux problem is proved. The Riemann–Hadamard matrix is determined, and the solution of the Darboux problem is constructed in terms of this matrix. As an example of application of the results, the solution of the Darboux problem for a system with four independent variables is constructed in detail.
642-651
On the Effect of Irregularity of the Domain Boundary on the Solution of a Boundary Value Problem for the Laplace Equation
Abstract
We consider an inhomogeneous boundary value problem with mixed boundary conditions for the Laplace equation in a domain representing a perturbation
of a rectangle where one of its sides is replaced by some curve of minimal smoothness. An estimate is obtained for the difference between the solutions of the perturbed and unperturbed problems in the norm of the Sobolev space on their common domain.
652-657
On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates
Abstract
We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.
658-674
On the Existence of Solutions of Degenerate Discrete-Time Systems
Abstract
We consider a nonstationary linear discrete-time descriptor system with rectangular matrix coefficients defined on a finite horizon. An answer is obtained to the question as to what the largest number of unknown vectors that can be found from a given finite number of equations is. In a similar way, the solvability of nonstationary linear continuous- or discrete-time systems, as well as (in the local sense) nonlinear discrete-time systems, is studied. It is shown that in cases where the considered linear (or nonlinear) system retains its internal structure, it is possible to find its solutions on an infinite horizon. The proposed approach has sufficient generality and automatically solves the problem of consistency of the initial data.
675-692
Singularly Perturbed Integro-Differential Systems with Kernels Depending on Solutions of Differential Equations
Abstract
We consider integro-differential equations (IDEs) with a rapidly oscillating inhomogeneity and with a Volterra-type integral operator whose kernels can contain both a classical rapidly decreasing exponential (the simplest case) and fundamental solutions of differential systems (the general case). Difficulty in constructing a regularized (according to S.A. Lomov) asymptotics in the general case is due to the complex asymptotic structure of the fundamental solution matrix (Cauchy matrix) of the homogeneous differential system. In the present paper, we first construct a regularized asymptotics of the Cauchy matrix, which is then used to construct a regularized asymptotics of the solution of the IDE.
693-704