An axially symmetric contact problem for a half-space with an elastically reinforced cylindrical cavity

We consider the pressure of a plate on a half-space with a round cylindrical cavity. The surface of the cavity is reinforced by elastic elements that are modeled by very general operators. The problem is reduced to a Fredholm integral equation of second kind. A detailed study is made of the case of reinforcement described by the Winkler law. An approximate solution is obtained in the form of the asymptotics with respect to the radii of the plate and the cavity.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 80–85.     

Pseudodifferential operator method in a problem on the diffraction of an electromagnetic wave on a dielectric body

We study the problem on the diffraction of electromagnetic waves on a solid body in free space. To analyze the integro-differential equations describing this phenomenon, we use the theory of pseudodifferential operators. We evaluate the asymptotic expansion of the symbol and prove the ellipticity and Fredholmness with index zero of the problem operator.     

An identification problem for a time periodic nonlinear wave equation in non cylindrical domains

The existence of a time-periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established. The problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion.     

Critical load of an axially compressed sandwich cylindrical panel

The problem of critical load of an axially compressed open sandwich cylindrical shell with two straightlinear edges free is presented. Large angle of the shell sector, geometrical and physical symmetry of the facings are assumed. The simply formula for calculation critical stress in the shell is presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An integral equation approach to electromagnetic diffraction by a cylinder

A two-dimensional perturbation theory of electromagnetic diffractionby a cylindrical conductor is developed, based upon Robin’sintegral equation of electrostatics. An innovative featureof the theory is of not requiring a boundary condition of zerotangential electric field at the conducting surface, but requiringexplicitly that the bulk conductivity be taken into account,and solutions of Maxwell’s equations are obtained for both the interior and exterior regions. The theory is a near-field, low-frequency one, being perhaps valid for the highest radio frequencies, but not for optical frequencies or for great distances from the cylinder. Received 3 June, 1999. ⁺ antoine@mip.ups-tlse.fr     

Plane problem of interaction between an elastic longitudinal wave and a cylindrical shell with axial cut

We have proposed a procedure for investigating the stressed state of a thin-walled round cylindrical shell with a cut along its generatrix, which is placed into an elastic space. A longitudinal elastic wave is incident on the shell. The procedure is based on the use of the Rayleigh expansion in partial waves. We have obtained relations for determining the displacement jumps and angle of rotation at the lips of the shell cut, the radial component of the displacement vector and the normal force in the cylindrical shell, as well as the stresses and displacements in the elastic medium. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 131–137, January–March, 2008.     

Elastic buckling of an axially compressed open circular cylindrical shell

The paper is devoted to stability problem of an axially compressed open elastic circular cylindrical shell. Two curvilinear edges of the open cylindrical shell are simply supported while two straight edges are free. Two load cases of the shell are assumed. The first load case ‐ was the invariable axially normal force intensity, and the second load case ‐ the linearly varying axially normal force intensity. Critical loads of the shell for both load cases are determined. These critical loads are compared with a classical load for closed circular cylindrical shell. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The edge wave in the problem of diffraction on a boundary with jump of curvature

The problem of the diffraction of creeping waves on a point of transition of the convex boundary to the straight boundary of a domain is investigated. It is assumed that at the point of jump of curvature, the tangent to the boundary is continuous and its derivative has a jump. An expression for the edge wave is obtained and investigated. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 274–287. Translated by N. Ya. Kirpichnikova.     

Closed form solution of electromagnetic wave diffraction problem in a homogeneous bi-isotropic medium

In this study, the problem of wave scattering of an electromagnetic field in a homogeneous bi-isotropic medium by a perfectly conducting strip is theoretically analyzed. The crux of the study is a rigorous construction of a closed form solution in the complex domain. A series solution of electromagnetic plane wave diffraction problem in terms of the eigenfunctions that happen to be the generalized Gamma functions is found. In the transformed domain, the scattered field is physically interpreted by computing the convergence history, and thereby, higher order accurate solution has been obtained in complex domain in closed form. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal control problem for steady-state equations of acoustic wave diffraction

An optimal control problem is considered for the steady-state equations of acoustic wave diffraction caused by a three-dimensional inclusion in an unbounded homogeneous medium. The task is to minimize the L ²-deviation of the pressure field inside the inclusion from a certain prescribed value due to changing the field sources in the external medium. The solvability of the problem is proved. A solution algorithm is proposed, and its convergence is proved.     

An asymmetric Neumann problem with weights

We prove the existence of a first nonprincipal eigenvalue for an asymmetric Neumann problem with weights involving the p-Laplacian (cf. (1.2) below). As an application we obtain a first nontrivial curve in the corresponding Fu?ik spectrum (cf. (1.4) below). The case where one of the weights has meanvalue zero requires some special attention in connexion with the (PS) condition and with the mountain pass geometry.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

An initial boundary value problem in ideal Magneto-Hydrodynamics

We consider the initial-boundary value problem for the system of equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall boundary condition. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the natural functional setting is provided by anisotropic Sobolev spaces, which take account of such singular behavior. We show the existence of the regular solution in the anisotropic Sobolev space for . The result improves previous results of the author and of Yanagisawa - Matsumura. RID="h1" ID="h1"Research of the project Cofin. MURST 1998 "Problems and Methods in the Theory of Hyperbolic Equations".     

Bifurcation analysis of the motion of a cylinder and a point vortex in an ideal fluid

We consider an integrable Hamiltonian system describing the motion of a circular cylinder and a vortex filament in an ideal fluid. We construct bifurcation diagrams and bifurcation complexes for the case in which the integral manifold is compact and for various topological structures of the symplectic leaf. The types of motions corresponding to the bifurcation curves and their stability are discussed.     

Flow problem for the equations of an ideal fluid

One considers the problem of the flow of an ideal fluid through a bounded domain with prescribed velocity vector at the entry and prescribed pressure on the part of the outflow. One proves the solvability of the problem, locally with respect to time, in the case of a plane-parallel or axisymmetric motion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 84–96, 1980.The authors express their sincere gratitude to O. A. Ladyzhenskaya for the helpful discussion of the results of the paper.     

Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies

The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space ?³ but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov-Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.