Asymptotics of solutions to the periodic problem for a Burgers type equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear Burgers type equation

Asymptotics of solutions of a cauchy problem

We consider the Cauchy problem (x)ⁿ=at^r + bx + f(t, x, x), x(0)=0. We prove the existence of continuously differentiable solutions and study their asymptotic behavior for t + 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 187–193, February, 1991.     

Asymptotics of the solution to the Neumann problem in a thin domain with sharp edge

An asymptotic expansion of the solution to the Neumann problem for a second-order equation in a thin domain with peak-like edge is constructed and justified. Owing to the sharpness of the edge, the procedure of dimension reduction leads to a degenerate limit equation on the longitudinal cross-section of the domain and a solution has irregular behavior near the boundary. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 193–219.     

The spectrum of the elasticity problem for a spiked body

We establish the existence of continuous spectrum for the operator of the linear elasticity problem in a three-dimensional domain with a sufficiently sharp spiked singularity of the boundary. We obtain some information about the structure of the spectrum and verify the weighted Korn inequality, which enables us to prove that the spectrum is discrete for insufficiently sharp spikes. We state some open questions.     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Increasing the accuracy of solutions to a discrete spectral problem

Two methods for increasing the accuracy of discrete Sturm — Liouville problems are examined. In one of these the principal term of the expansion for the error of the eigenvalues by small parameter steps is used. The other method is based on a minimized functional, which corresponds to a discrete scheme of fourth-order accuracy relative to the discretization parameter.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 56–63, 1989.     

Asymptotics of the solution of the Steklov spectral problem in a domain with a blunted peak

Mathematical Notes - We construct and justify the asymptotics of the eigenvalues and eigenfunctions of the Laplace equation with Steklov boundary conditions in a domain with an acute peak whose end...     

On the structure of the spectrum for the elasticity problem in a body with a supersharp spike

We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity O(r ^1+γ), where r is the distance to the vertex of the spike and γ > 1 is the sharpness exponent.     

Complex potentials for the plane problem of elasticity theory for a multiconnected anisotropic body with cracks

Results are given for a study and modernization of basic relationships obtained previously by the author for generalized complex potentials of crack theory generated by solving contact problems in the case of multiconnected anisotropic plates. In contrast to the author's previous work devoted to this question, general presentations of complex potentials, and boundary and additional conditions for finding them, are simplified. Use of them is shown in the case of a crack and concentrated forces, and when a plate has a finite number of cracks along a single line. Independence of stress intensity factor on anisotropy parameters is demonstrated in the last case.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 24–34, 1990.     

Numerical solution of a boundary-value problem of elasticity theory for a body with an inclusion

We consider the problem of steady-state oscillations of a plane body with an inclusion. A solution algorithm is constructed. Computational experiments are described, which establish the applicability of the proposed procedure for analyzing the effect of defects inside the body on wave fields excited by ultrasonic sources.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 83–87, 1991.     

Regular solutions of a multiparameter inverse spectral problem

Mathematical Notes -     

Fundamental system of solutions of the axially symmetric problem of the theory of elasticity for a body with a plane sheet of volume moment dipoles and forces

We have constructed the fundamental system of solutions of the axially symmetric problem of the theory of elasticity for an unbounded body with a sheet of volume forces, normal to a chosen plane, and moment dipoles, which is a mathematical model of the internal boundary layer of a certain type. With the help of such layers, one succeeds in formulating some inverse problems of elasticity and, hence, the related problems of the control of stress-strain state on the corresponding surfaces. We have also formulated and solved the generalized Kelvin problem and, according to it, for the plane of distributed normal load, and established the law of distribution of the moment dipoles (sheet parameters), which provides the vertical displacements assigned for points of the plane, in particular, zero, by the corresponding tension.     

Asymptotics for solutions of a defective renewal equation with applications

In this paper, we derive non-exponential asymptotic forms for solutions of defective renewal equations. These include as special cases asymptotics for compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the Gerber-Shiu discounted penalty function in the classical risk model and the reliability of a two-unit cold standby system in reliability theory.