Boundary-value problems for the helmholtz equation in an angular domain. II

We investigate boundary-value problems that appear in the study of the diffraction of acoustic waves on an infinite cylinder (with a cross section of an arbitrary shape) placed inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory which enables one to reduce these boundary-value problems to integral equations is elaborated.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 4, pp. 500–519, April, 1993.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

Potential theory for problems of diffraction on a layer between two parallel planes

We investigate the boundary-value problems that appear when studying the diffraction of acoustic waves on obstacles in a layer between two parallel planes. By using potential theory, these boundary-value problems are reduced to the Fredholm integral equations given on the boundary of the obstacles. The theorems on existence and uniqueness are proved for the Fredholm equations obtained and, hence, for the boundary-value problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 647–662, May, 1993.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

Integral equations in the linear theory of elasticity in semiinfinite domains

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 50, No. 5, pp. 613–622, May, 1998.     

A solution method for diffraction problems of electromagnetic waves on bodies of revolution in a stratified medium

We consider the diffraction of electromagnetic waves on bodies of revolution located in stratified media with arbitrary excitation. The problem is reduced to boundary-value problems on the azimuthal halfplane for systems of elliptical differential equations. A system of one-dimensional integral equations of the first kind is obtained.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 182–186, 1985.     

Solution of boundary-value problems of p-analytical functions with the characteristic p=x/(x2 + y2) on a halfplane with cuts

A method is developed for solution of boundary-value problems of p-analytical functions with the characteristic p=x/(x₂ + y₂) on a halfplane with parallel straight cuts. Two problems are considered, with the boundary values of the real or imaginary part of the p-analytical function specified on the lips of the cuts. Solution of these problems is reduced to solution of a system of Fredholm integral equations of the second kind. As an application of these boundary-value problems, we consider plane seepage in a nonhomogeneous medium under a hydrotechnical structure with two cutoff walls forming a subsurface circle.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 17–26, 1988.     

Some integral transforms with the hypergeometric function F4(α, β, γ, δ; x,y)

We investigate one of the most efficient methods for solving differential equations and boundary-value problems —the integral transform method. The properties of the Jacobi polynomial are used to construct a new integral transform with the hypergeometric function F ₄ in the kernel.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 26–30, 1986.     

Concentration of electroelastic fields in a composite piezoceramic plate with defects crossing the interface

The characteristics of fracture and strength of a composite piezoceramic plate with defects in the form of cracks and holes situated in both of the plate components are investigated. The corresponding boundary-value problems of electroelasticity are reduced to systems of singular integral equations by constructing integral representations of the complex potentials. The results of numerical realization of the constructed algorithms are reported.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 777–786, November–December, 1998.     

Electromagnetic fields in the presence of an infinite dielectric wedge: the phased line source excitation case

Electromagnetic fields, excited by an electric phased line sourcein the presence of an infinite dielectric wedge, are determinedby application of the Kontorovich–Lebedev transform. TheMaxwell's equations together with the conditions of continuityof the tangential field components at the material interfacesare formulated as a vector boundary-value problem. By representingthe field components as Kontorovich–Lebedev integrals,the problem is reduced to a system of singular integral equationsfor the unknown spectral functions. We construct numerical solutionsto those equations that permit fields evaluation for valuesof the wedge refractive index, not necessarily close to unity,and for arbitrary positioned source and observer. Numericalresults showing the influence of a wedge presence on the directivityof a phased line source are presented and verified through finite-differencefrequency-domain simulations.     

The Green's function method in stationary and nonstationary nonlinear heat-conduction problems

By constructing the Green's function we reduce quasilinear boundary-value problems for second-order parabolic and elliptic equations to nonlinear integral equations of second kind. The method is illustrated using the examples of the stationary and nonstationary heat-conduction problems in the case of radiant heat transfer.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 42–47.     

Solution of a mixed boundary-value problem of thermodiffusion for layered bodies of canonical shape

We propose a method of solving coupled thermodiffusion problems for layered bodies of canonical shape. The method is based on separating the coupled system of equations and boundary conditions into independent boundary-value problems. The solution contains arbitrary functions of time determined from a system of second-order integral equations of convolution type obtained as a result of satisfying the boundary conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 87–91.     

The antiplane problem of electroelasticity for a cylinder with a tunnel crack excited by an arbitrary system of surface electrodes

The antiplane mixed boundary-value problem of electroelasticity of the oscillations of an infinite piezoceramic cylinder, weakened by a curvilinear tunnel crack, is considered. Using special integral representations of the solution, the boundary-value problem is reduced to a system of singular integro-differential equations of the second kind with discontinuous kernels. The results of a numerical realization of the algorithm, characterizing the amplitude-frequency characteristics of a piecewise-uniform cylinder and the behaviour of the components of the electroelastic field in the region and on the boundary of the cylinder under conditions of the inverse piezoelectric effect, are presented.     

On the diffraction of Poincaré waves

The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two-dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary-value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary-value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non-empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Necessary conditions of optimality in constrained planar domain optimization

The article considers optimization of the shape of a planar domain with an integral functional subject to integral inequality constraints and constraints on domain location. The boundary-value problem is a Dirichlet problem for a quasilinear elliptical equation. A method is proposed for deriving necessary conditions of optimality for problems of this kind.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 22–29, 1993.     

Numerical solution of multipoint problems in statics of shells

We purpose an approach to solving multipoint boundary-value problems for a system of ordinary differential equations in the theory of shells. The technique is based on reduction of the original problem to several two-point boundary-value problems, which are solved by a stable numerical method. Examples of calculation of variable-thickness cylindrical shells are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, 58–65, 1988.     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.