Volume and surface integral equations for electromagnetic scattering by a dielectric body

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and it is used to derive properties of the volume integral equation.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.     

The equivalence of team theory's integral equations and a Cauchy system: sensitivity analysis of a variational problem

Team decision theory studies the problem of how a group of decision makers should use information to coordinate their actions. Mathematically, the task is to find functions that maximize an objective functional. The Euler equations take the form of a system of integral equations. In this paper, it will be shown that a class of such integral equations has solutions that are identical to the solutions of a system of initial-valued integrodifferential equations. This Cauchy system describes the sensitivity of the solutions to underlying parameters and provides an efficient technique for solving difficult team decision problems. An analysis of a profit maximizing firm demonstrates the usefulness of the Cauchy system.     

A Mixed Symmetric Problem of Electroelasticity Theory for a Layer Weakened by Through-the-Thickness Tunnel Holes

A new solution procedure for spatial mixed symmetric problems of electroelasticity theory for a layer weakened by through-the-thickness holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 4k (k = 1, 2, ...) equations. Calculation results for characteristic stresses are presented.     

Conical diffraction by multilayer gratings: a recursive integral equation approach

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ?² coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with 2×2 operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

Solving some problems of elasticity theory by the integral equation method for a holomorphic vector

Under study are some problems of elasticity theory with nonclassical boundary value conditions. We assume that the load and displacement vectors are given on a part of the boundary, while on the other parts of the boundary, the load vector or the displacement vector may be given separately, and no conditions are imposed on the remaining part of the surface (of some nonzero measure).We consider the questions of uniqueness for the solutions to these problems. Solving the nonclassical problems is reduced to a system of singular integral equations for a holomorphic vector.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

Exact solutions of some mixed problems of uncoupled thermoelasticity for a truncated hollow circular cone with a groove along the generatrix

An elastic body of finite dimensions in the form of a truncated hollow circular cone with a groove along the generatrix is considered. The uncoupled problem of thermoelasticity is formulated for this body for different types of boundary conditions on all the surfaces. These are the conditions for specifying the displacements or sliding clamping on surfaces with fixed angular coordinates and the conditions for specifying the stresses on surfaces with a fixed radial coordinate (shear stresses are assumed to be zero). It is assumed that the temperature is a specified function of all the spherical coordinates. Some auxiliary functions, related to the displacements, are introduced first, and equations for these functions are then derived using Lamé's equations. A finite integral Fourier transformation with respect to one of the angular variables is then employed. After this, by solving certain Sturm-Liouville problems, a new integral transformation is constructed and is applied to the equations with respect to the other angular variable. As a result a one-dimensional system of differential equations is obtained, to solve which an integral Mellin transformation is employed in a special way. Finally, exact solutions of some problems of thermoelasticity are constructed in series for this body.     

The excitation of symmetric and antisymmetric lamb waves in a piezoelectric strip by surface electrodes

Solutions of the problems of the excitation by surface electrodes of symmetric and antisymmetric Lamb waves in a strip of piezoelectric material are obtained. The solutions of the problems are reduced to systems of singular integral equations in auxiliary functions, proportional to the jump in charge density at the electrodes. Approximate solutions of the systems of singular integral equations are obtained by the Bubnov-Galerkin method using Chebyshev polynomials of the first kind. The results of numerical calculations of the electro-elastic fields for a strip of cadmium sulphide are given.     

Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

ABSTRACT

A mixed boundary value problem (BVP) for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based boundary-domain integral equations (BDIEs). We use a parametrix different from the one employed by Mikhailov [Localized boundary-domain integral formulations for problems with variable coefficients. Eng Anal Bound Elem. 2002;26:681–690], Mikhailov and Portillo [A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient. In: Paul Harris, editor. Proceedings of the 10th UK conference on boundary integral methods. Brighton: Brighton University Press; 2015. p. 76–84] and Chkadua, Mikhailov, Natroshvili [Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: equivalence and invertibility. J Integral Eqs Appl. 2009;21:499–543]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.     

3D Fredholm integral equations for scattering by dielectric structures

We consider 3D singular integral equations that describe problems of interaction of an electromagnetic wave with 3D dielectric structures. By using the theory of singular integral equations, we reduce these equations to Fredholm integral equations of the second kind.     

Numerical Solution of a System of Boundary Integral Equations with a Logarithmic Singularity for Problems in the Theory of Shells with Holes

A quadrature formula for integrals having a logarithmic singularity is investigated. The formula permits the method of mechanical quadratures to be used in solving a system of boundary integral equations with an analogous singularity in problems associated with the theory of perforated shells.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Temperature Stresses in a Piecewise-Homogeneous Cylindrical Shell Caused by the Presence of a Longitudinal Crack

By using the method of generalized conjugation problems, we propose a numerical scheme for investigation of the redistribution of temperature stresses in a piecewise-homogeneous cylindrical shell caused by a longitudinal crack. This scheme is based on systems of integral equations (some of them are singular) to determine the unknown jumps of integral characteristics of the disturbed temperature field and displacements and their derivatives on the line of a crack and on the interface as well as the derivatives of these integral characteristics with respect to the longitudinal coordinate at the interface.     

Rank criteria for the controllability of a boundary-value problem for a linear system of integro-differential equations with pulse influence

We determine necessary and sufficient conditions for the solvability of boundary-value problems for a linear system of integro-differential equations with pulse influence. We prove theorems on the existence and integral representation of solutions of linear second order integral-sum Volterra equations and linear systems of integro-differential equations with pulse influence at fixed times. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 723–730, June, 2000.     

Reconstructing parameters of a medium from incomplete spectral information

The problem of the synthesis of a stratified medium with specified amplitude and phase properties is investigated. The wave propagation in the medium is described by a system of differential equations. The synthesis problem considered in the paper relates to inverse problems of spectral analysis with incomplete spectral information. Using the contour integral method we study properties of spectral characteristics and obtain algorithms for the solution of the synthesis problem for differential equations with singularities.     

Concentration of Mechanical Stresses near a Hole in a Piezoceramic Layer

A new procedure for solving three-dimensional mixed antisymmetric problems of elasticity theory for a layer weakened by through tunnel holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 3k (k = 1, 2, ...) equations. The calculation results for characteristic stresses are presented.