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.     

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

Method of integral equations in the scalar problem of diffraction on a system consisting of a “soft” and a “hard” screen and an inhomogeneous body

We consider the scalar problem on the diffraction of a plane wave on a system of two screens with boundary conditions of the first and the second kind and a solid inhomogeneous body in the semiclassical setting. The original boundary value problem for the Helmholtz equation is reduced to a system of singular integral equations over the body domain and the screen surfaces. We prove the equivalence of the integral and differential statements of the problem, the solvability of the system of integral equations in Sobolev spaces, and the smoothness of its solutions. To solve the integral equations approximately, we use the Bubnov-Galerkin method; we introduce basis functions on the body and the screens and prove the consistency and convergence of the numerical method.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary integral equations for screen problems in IR3

Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR³ via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.     

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.     

Integral equations for the Fourier-transformed boundary values for the transmission problems for right-angled wedges and octants

A 4×4-system of integral equations for the Fourier transformed boundary values of the normal derivatives of the wave functions defined in the four quadrants of R ²-space is derived. This system results from the scalar transmission problem with continuous passage of the boundary values of the total wave-fields and of the weighted normal derivatives corresponding to the case of magnetically polarized fields. Several equivalent systems of integral equations are deduced then which show that Banach's fixed point principle may be applied at least for slightly differing media in the four quadrants. The method, which is equivalent to a compatibility condition for holomorphic functions, may be generalized to the case of the scalar transmission problem for octants in R ³-space. There a 12 × 12-system of integral equations for the Fourier transformed normal derivatives on the quarter-plane faces is established.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

Method of Integral Equations for the Three-Dimensional Problem of Wave Reflection from an Irregular Surface

The problem on the reflection of the field of a plane H-polarized three-dimensional electromagnetic wave from a perfectly conducting interface between media which contains a local perfectly conducting inhomogeneity is considered. To construct a numerical algorithm, the boundary value problem for the system of Maxwell equations in an infinite domain with irregular boundary is reduced to a system of singular integral equations, which is solved by the approximation–collocation method. The elements of the resulting complex matrix are calculated by a specially developed algorithm. The solution of the system of singular integral equations is used to obtain an integral representation for the reflected electromagnetic field and computational formulas for the directional diagram of the reflected electromagnetic field in the far region.     

A generalization of Vinograd's theorem for dynamical systems

The theory of dynamical systems has been expanded by the introduction of local dynamical systems [10, 4, 9] and local semidynamical systems [1]. Using integral curves of autonomous ordinary differential equations to illustrate these generalizations, we find that, roughly, the integral curves form a local dynamical system if solutions exist and are unique without requiring existence for all time, and the integral curves form a local semidynamical system if solutions exist and are unique in the positive sense but need not exist for all positive time. In addition to autonomous ordinary differential equations, the enlarged theory of dynamical systems has applications to nonautonomous ordinary differential equations, certain partial differential equations, functional differential equations, and Volterra Integral equations [9, 1, 2, 8], respectively. All of these have metric phase spaces. Since many dynamic considerations are invariant to reparameterizations, it is of interest to known when a local dynamical (or semidynamical) system can be reparameterized to yield a “global” dynamical (or semidynamical) system. For autonomous ordinary differential equations, Vinograd [7] has shown that the local dynamical system on an open subset ofRⁿ formed by integral curves is isomorphic (in the sense of Nemytskii and Stepanov) to a global dynamical system. In an extensive study of isomorphisms, Ura [12] has expanded the Gottschalk-Hedlund notion of an isomorphism and restated Vinograd's result in terms of a reparameterization. In this paper we study the problem of finding a global dynamical (or semidynamical) system which is isomorphic to a given local system. A necessary and sufficient condition is found which is then used to show that the Vinograd result holds on metric spaces.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.     

Acoustic scattering by a thin cylindrical screen with Dirichlet and impedance boundary conditions on opposite sides of the screen

A problem on scattering acoustic waves by a thin cylindrical screen is studied. In doing so, the Dirichlet condition is specified on one side of the screen, while the impedance boundary condition is specified on the other side of the screen. The solution of the problem is subject to the radiating condition at infinity and to the propagative Helmholtz equation. By using the potential theory the scattering problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original scattering problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

The method of potential functions in problems of the theory of elasticity for bodies with a defect

The method of potential functions using a Fourier transformation in the class of slowly increasing distributions, corresponding to the classical method of complex potentials, is proposed for solving well-known problems of the theory of elasticity for bodies with a defect. It is shown that when a Fourier transformation with respect to all the spatial variables is used, the solution of the dynamic problem of the theory of elasticity can also be represented in terms of a jump in the stresses and displacements at the defect. The correctness of the transformed problem is considered (in terms of an analogue of the Lopatinskii condition). The solution of the system of Helmholtz equations, to which the system of Lamé equations is reduced in the case of the two-dimensional dynamic problem, is expressed in terms of the jump in the stresses and displacements at the defect as a result of solving the corresponding singular integral equations.     

The stressed state of piecewise-homogeneous bodies subject to nonsteady thermal fields

The study of the distribution of thermal fields and the thermal stresses they generate, and the influence of these factors on the strength of piecewise-homogeneous bodies was begun and developed in the work of Ya. S. Pidstrigach, Yu. M. Kolyano, and their students [5, 12, 13]. In the present paper we use the theory of generalized functions to reduce the two-dimensional quasistatic problem of thermal stresses for piecewisehomogeneous bodies to solving systems of integral equations. The kernels of these equations are chosen as the corresponding Green's functions. We obtain a single system of integral equations for the entire region under consideration. To solve the system we apply the method of boundary elements [2]. On the basis of the results obtained we carry out a study of the thermostressed state of a half-plane with rectangular inclusions. Translated fromMatematichni Metodi i Fiziko-mekhnichni Polya, Vol. 40, No. 1, 1997, pp. 53–58.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

Lp-Lq time decay estimate to the solution of the Cauchy problem of the system of equations for nonlocal model of hyperbolic thermoelasticity theory

The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e(x), and heat flux q(x), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. Using the modified Cagniard–de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three–dimensional space. Basing on the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the L^p−L^q time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)