Vol 233, No 6 (2018)

We consider a nonstationary initial boundary-value problem describing complex (radiative-conductive) heat transfer in a system of semitransparent bodies. To describe radiation propagation, we use the radiation transfer equation with boundary-value conditions of diffuse reflection and refraction of radiation. We take into account that the radiation intensity and optical properties of bodies depend on the radiation frequency. The unique solvability of a weak solution is established. The comparison theorem is proved. A priori estimates of a weak solution as well as its regularity are obtained.

We consider boundary-value problems for differential-difference operators with perturbations in translations of the independent variable. We prove that the family of differential-difference operators is positive definite uniformly with respect to translations of the independent variable. Solutions of such problems depend continuously on these translations. We consider the coercivity problem for differential-difference operators with incommensurable translations of the independent variable and study the approximation of such operators by rational operators.

We develop the theory of generalized Jordan chains of multiparameter operator functions A(λ) : E₁→ E₂, λ ∈ Λ, dimΛ = k, dimE₁ = dimE₂ = n, where A₀ = A(0) is an irreversible operator. For simplicity, in Secs. 1–3, the geometric multiplicity of λ₀ is equal to one, i.e., dimN(A₀) = 1, N(A₀) = span{φ}, dimN^*(\( {A}_0^{\ast } \)) = 1, N^*(\( {A}_0^{\ast } \)) = span{ψ}, and it is assumed that the operator function A(λ) is linear with respect to λ. In Sec. 4, the polynomial dependence of A(λ) is linearized. However, the results of existence theorems for bifurcations are obtained for the case where there are several Jordan chains. Applications to degenerate differential equations of the form [A₀ + R(·, x)]x′= Bx are provided.

In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group G. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained.

In designing structures and devices interacting with the flow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The definition of stability of an elastic body taken in the article corresponds to Lyapunov’s concept of stability of a dynamical system. On the base of the proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic flow of gas or liquid (in an ideal model of an incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing, is considered. The action of these devices is based on the oscillations of elastic elements during the gas or liquid flow. The dynamic stability of the elastic element located on one wall of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. Both models are described by a coupled nonlinear system of differential equations for the unknown functions — the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, we obtain the sufficient conditions of the stability, the imposed restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.