Boundary-value problems of the asymmetric theory of elasticity for thin plates

Boundary-value problems of the three-dimensional asymmetric micropolar, moment theory of elasticity with free rotation are considered for thin plates. It is assumed that the total stress-strain state is the sum of the internal stress-strain state and the boundary layers, which are determined in an approximation using asymptotic analysis. Three different asymptotic forms are constructed for the three-dimensional boundary-value problem posed, depending on the values of dimensionless physical constants of the plate material. The initial approximation for the first asymptotic form leads to a theory of micropolar plates with free rotation, the initial approximation for the second asymptotic form leads to a theory of micropolar plates with constrained rotation, and the initial approximation for the third asymptotic form leads to a theory of micropolar plates with “small shear stiffness.” The corresponding micropolar boundary layers are constructed and studied. The regions of applicability of each of the theories of micropolar plates constructed are indicated.     

Boundary-value problems with a triple periodic solution for the Laplace equation in R3

Starting from the triple periodic analogues of simple and double layer potentials, existence and uniqueness theorems for the triple periodic solutions of the fundamental boundary-value problems for the Laplace equation are proved.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 132–139, 1986.     

Boundary-value problems for systems of difference equations

Boundary-value problems for systems of difference equations with discrete argument whose linear part is the Noetherian operator are considered. The necessary and sufficient conditions of the solvability of difference boundary-value problems of this sort are obtained.     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

Asymptotic solution of heat-conduction problems for shells of variable thickness

The temperature field in thin cylindrical shells of variable thickness on heating by a plane-parallel pulsed heat flux in the presence of convective heat transfer with the surrounding medium are approximately determined. By introducting well-founded practical constraints on the thermophysical parameters, the nonlinear problem is reduced to linear form.Translated frm Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 94–97, 1987.     

Boundary-value problems for stationary Hamilton-Jacobi and Bellman equations

We introduce solutions of boundary-value problems for the stationary Hamilton-Jacobi and Bellman equations in functional spaces (semimodules) with a special algebraic structure adapted to these problems. In these spaces, we obtain representations of solutions in terms of “basic” ones and prove a theorem on approximation of these solutions in the case where nonsmooth Hamiltonians are approximated by smooth Hamiltonians. This approach is an alternative to the maximum principle.     

Asymptotic solution of strongly oscillating problems for the Klein-Gordon equation

An asymptotic integration method is proposed for a class of nonlinear one-dimensional wave problems with strongly oscillating initial values.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 167–175, 1985.     

Boundary-value problems for analytic and generalized analytic functions

This paper deals with boundary value problems of linear conjugation with shift for analytic functions in the case of piecewise continuous coefficients. Int main goal is the construction of a canonical matrix for these problems. Boundary value problems with shift for generalized analytic functions and vectors as well as differential boundary value problems are studied. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Boundary-value problems for Helmholtz equations in an angular domain. I

The boundary-value problems are investigated that arise when studying the diffraction of acoustic waves on an infinite cylinder with cross-section of an arbitrary shape situated inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory is worked out which enables one to reduce these boundary-value problems to integral equations on a one-dimensional contour — the boundary of the cross-section of this cylinder. The theorems on existence and uniqueness of solutions to the boundary-value problems and the corresponding integral equations are proved. For this case, a principle of limit absorption is established. Effective algorithms for calculating the kernels of the integral operators are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 403–418, March, 1993.     

Boundary-value problems for systems of ordinary differential equations

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3–103, 1987.     

Asymptotic behaviour of the solution of bending of a thin infinite plate

The single-valuedness is investigated of the regular solutions of bending of elastic plates with transverse shear deformation in finite and infinite domains. The results are used to justify the introduction, and interpret the physical meaning, of a special solvability class defined by the author in previous work.     

Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1289–1398, September, 1993.     

ASYMPTOTIC ESTIMATION FOR SOLUTION OF A CLASS OF SEMI-LINEAR ROBIN PROBLEMS

A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.