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Том 54, № 1 (2018)
- Год: 2018
- Статей: 11
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9340
Ordinary Differential Equations
1-6
Weinstein Criteria and Regularized Traces in the Case of Transverse Vibrations of an Elastic String with Springs
Аннотация
The transverse vibrations of a string with additional restrictions in the form of elastic point constraints are studied. In contrast toWeinstein’s original approach, the constraints are not represented as orthogonality-type conditions in the case under study. Nevertheless, it is shown that the main results of Weinstein’s theory remain valid. It is also shown that the string rigidity coefficients can uniquely be reconstructed from the first-order regularized traces of the corresponding operators. This permits one to give a physical interpretation of regularized traces.
7-12
Dynamics of Delay Systems with Rapidly Oscillating Coefficients
Аннотация
The problems of generalization of the averaging principle to delay systems are considered. New effects are revealed in the study of bifurcation problems, as are new phenomena that arise in the case of rapid oscillations of the delay. As an application of the results, the dynamics of a logistic equation with rapidly oscillating coefficients is studied.
13-27
Construction of Asymptotics of Solutions of Differential Equations with Cusp-Type Degeneration in the Coefficients in the Case of Multiple Roots of the Highest-Order Symbol
Аннотация
The asymptotics of linear differential equations with cusp-type degeneration are studied. The problem of constructing asymptotics at infinity for equations with holomorphic coefficients can be reduced to that problem. The main result is the construction of asymptotics of solutions of such equations in the case of multiple roots of the highest-order symbol under certain additional conditions on the lower-order symbol of the differential operator.
28-37
Uniform Boundedness in the Sense of Poisson of Solutions of Systems of Differential Equations and Lyapunov Vector Functions
Аннотация
We introduce several generalizations of the properties of equiboundedness and uniform boundedness of solutions of ordinary differential systems, which are united by the common names of equiboundedness in the sense of Poisson and uniform boundedness in the sense of Poisson. For each of the above-introduced properties, we use the method of Lyapunov vector functions to obtain sufficient criteria for the system to have a certain property. In terms of the upper Dini derivative of the Lyapunov function given by a system, several criteria are established for the solutions of this system to have the relevant type of uniform boundedness in the sense of Poisson.
38-48
On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials
Аннотация
We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials lr,kα(x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product
49-66
Partial Differential Equations
Solvability of a Boundary Value Problem for Second-Order Elliptic Differential Operator Equations with a Spectral Parameter in the Equation and in the Boundary Conditions
Аннотация
In a Hilbert space H, we study noncoercive solvability of a boundary value problem for second-order elliptic differential-operator equations with a spectral parameter in the equation and in the boundary conditions in the case where the leading part of one of the boundary conditions contains a bounded linear operator in addition to the spectral parameter. We also illustrate applications of the general results obtained to elliptic boundary value problems.
67-85
On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions
Аннотация
We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.
86-105
On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations
Аннотация
We study a nonlinear reaction–diffusion system modeled by a system of two parabolic-type equations with power-law nonlinearities. Such systems describe the processes of nonlinear diffusion in reacting two-component media. We construct multiparameter families of exact solutions and distinguish the cases of blow-up solutions and exact solutions periodic in time and anisotropic in spatial variables that can be represented in elementary functions.
106-120
Initial Value Problem for B-Hyperbolic Equation with Integral Condition of the Second Kind
Аннотация
For the hyperbolic equation with Bessel operator, we study the initial boundaryvalue problem with integral nonlocal condition of the second kind in a rectangular domain. The integral identity method is used to prove the uniqueness of the solution to the posed problem. The solution is constructed as a Fourier–Bessel series. To justify the existence of the solution to the nonlocal problem, we obtain sufficient conditions to be imposed on the initial conditions to ensure the convergence of the constructed series in the class of regular solutions.
121-133
Inverse Problem for an Integro-Differential Equation of Acoustics
Аннотация
We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t > 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.
134-142