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.     

Boundary-value problems for Helmholtz equations in an angular domain. I

The boundary-value problems are investigated that arise when studying the diffraction of acoustic waves on an infinite cylinder with cross-section of an arbitrary shape situated inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory is worked out which enables one to reduce these boundary-value problems to integral equations on a one-dimensional contour — the boundary of the cross-section of this cylinder. The theorems on existence and uniqueness of solutions to the boundary-value problems and the corresponding integral equations are proved. For this case, a principle of limit absorption is established. Effective algorithms for calculating the kernels of the integral operators are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 403–418, March, 1993.     

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 32–35, 1986.     

Determining the internal boundary of a region in the two-dimensional initial–boundary-value problem for the heat equation

The article investigates the reconstruction of the internal boundary of a two-dimensional region in the two-dimensional initial–boundary-value problem for the homogeneous heat equation. The initial values for the determination of the internal boundary are provided by a boundary condition of second kind on the external boundary and the solution of the initial–boundary-value problem at finitely many points inside the region. The inverse problem is reduced to solving a system of integral equations nonlinear in the function describing the sought boundary. An iterative numerical procedure is proposed involving linearization of integral equations.     

Determining the boundary of a two-dimensional region from the solution of the external initial boundary-value problem for the heat equation

We determine the boundary of a two-dimensional region using the solution of the external initial boundary-value problem for the nonhomogeneous heat equation. The initial values for the boundary determination include the right-hand side of the equation and the solution of the initial boundary-value problem given for finitely many points outside the region. The inverse problem is reduced to solving a system of two integral equations nonlinear in the function defining the sought boundary. An iterative procedure is proposed for numerical solution of the problem involving linearization of integral equations. The efficiency of the proposed procedure is investigated by a computer experiment.     

Discretizing a system of parabolic nonlinear equations with a discontinuous solution

A new initial— boundary-value problem is posed for a system of two nonlinear parabolic equations with convective transport terms that allow a discontinuity in the solution. Discretization on classes of discontinuous functions is proposed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 33–42, 1992.     

Matrix-differential methods of solving boundary-value problems for the nonlinear one-dimensional heat-conduction equation

Matrix numerical differentiation algorithms are applied to construct numerical-analytical methods for approximate solution of boundary-value problems for the nonlinear one-dimensional equation of heat conduction. The problems are reduced to a system of differential equations for the values of the sought approximate solution in the interior grid nodes and also to numerical formulas for the solution values at the boundary nodes. A numerical experiment is conducted. The error relative to grid spacing is established.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 37–43, 1986.     

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.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

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.     

Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

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.     

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.     

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.     

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.     

On the algebraic classification of Fredholm integral operators

Using the well-known and specific connections between Fredholm integral equations, two-point boundary-value problems, and linear dynamics-quadratic cost control processes, we present a complete, independent set of algebraic invariants suitable for classifying a wide range of Fredholm integral operators with respect to a certain group of transformations. The group, termed theRiccati group, is naturally suggested by the control theoretic setting, but seems nonintuitive from a purely integral-equations point of view. Computational considerations resulting from this classification are discussed.     

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-analytical algorithm to solve the equation of particle transport in plane geometry

We describe a numerical-analytical algorithm to solve the boundary-value problem for the integrodifferential equation of particle transport in a plane homogeneous medium. The general scheme of approximate solution of this problem is based on its reduction to the solution of some integral equation by summator operators of function approximation theory. Solvability conditions are established for the approximate equations and the algorithm errors are estimated. Working formulas are presented for the algorithm implemented in the form of a computer program. The summator operators in this algorithm are the algebraic interpolation operators with nodes at the extremal points of Chebyshev polynomials of first kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 69–76, 1987.     

Exact solution to the behavior of the electron plasma in a metal layer in an alternating electric field

Exact expressions are obtained for the first time for the response of the conduction electrons in a metal layer to an external alternating electric field. Case's well-known method of separation of the variables is modified, and the methods of singular integral equations and boundary-value problems of complex analysis are used. The solutions can be used to determined the screened field and also the absorption in a metal layer in a wide range of frequencies.N. K. Krupskaya Moscow Region Pedagogical Institute; Moscow Institute of Chemical Mechanical Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 179–189, February, 1992.     

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.