Mixed boundary value problem of potential theory in toroidal coordinates

A new approach is presented for solving the title problems. The method presents further extension of previously obtained results to the case of toroidal coordinates. The method allows us to solve non-axisymmetric problems exactly and in closed form, with no integral transform or special function expansions involved. Some integrals of fundamental value, involving distances between several points, are established.     

Solution of certain boundary value problems of potential theory and elasticity theory for a half-space with a paraboloidal cavity

A method for solving boundary value problems for the Laplace equation in a half space with a paraboloidal cavity or a paraboloidal segment is suggested. Using formulas for the re-expansion of the fundamental solutions of the Laplace equation from a cylindrical to a paraboloidal coordinate system and their inverses, the basic and certain mixed problems are reduced to Fredholm integral equations or systems of equations of the second kind with completely continuous operators in a certain Hilbert space. The problem of torsion of an elastic half-space with a paraboloidal cavity by a stamp linked to part of the surface of the paraboloid and the problem of distribution of electricity on a paraboloidal segment located in the half-space are considered.Translated from Dinamicheskie Sistemy, No. 4, pp. 33–40, 1985.     

An outer layer method for solving boundary value problems of elasticity theory

In this paper, an algorithm for solving boundary value problems of elasticity theory suitable for solving contact problems and those whose deformation domain contains thin layers is presented. The solution is represented as a linear combination of auxiliary and fundamental solutions to the Lame equations. The singular points of the fundamental solutions are located in an outer layer of the deformation domain near the boundary. The linear combination coefficients are determined by minimizing deviations of the linear combination from the boundary conditions. To minimize the deviations, a conjugate gradient method is used. Examples of calculations for mixed boundary conditions are presented.     

Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads

We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.     

External boundary value problems in the quasi static theory of elasticity for triple porosity materials

In this paper the quasi static linear theory of elasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Power series solutions to basic stationary boundary value problems of elasticity

The four basic stationary boundary value problems of elasticity for the Lamé equation in a bounded domain of ³ are under consideration. Their solutions are represented in the form of a power series with non-positive degrees of the parameter =1/(1–2), depending on the Poisson ratio . The coefficients of the series are solutions of the stationary linearized non-homogeneous Stokes boundary value problems. It is proved that the series converges for any values of outside of the minimal interval with the center at the origin and of radiusr1, which contains all of the Cosserat eigenvalues.Dedicated to Prof. I.Gohberg on the occasion of his 70th birthday     

A new version of boundary integral equations and their application to dynamic three-dimensional problems of the theory of elasticity

A version of boundary integral equations of the first kind in dynamic problems of the theory of elasticity is proposed, based on an investigation of the analytic properties of the Fourier transformant of the displacement vector, rather than on fundamental solutions. A system of three boundary integral equations of the first kind with Fredholm kernels is constructed, and the equivalence of the initial boundary-value problem on the vibrations of a bounded region and the system of boundary integral equations obtained is investigated. A version of the numerical realization, which combines the ideas of the classical method of boundary elements and the Tikhonov regularization method, is proposed. The results of numerical experiments are given.     

Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity

In this paper the Aifantis' theory of elasticity for solids with double porosity is considered and the 2D boundary value problem (BVP) of static is investigated. The uniqueness theorem of the internal BVP is proved. The explicit solution the BVP is constructed in the form of absolutely and uniform convergent series for a circle. The numerical solution of the BVP for a circle is obtained. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Certain problems of the theory of difference boundary value problems

This is an author's abstract of a dissertation for the degree of doctor of physicomathematical sciences. The dissertation was defended on December 2,1969,at a session of the Scientific Council of the Institute of Applied Mathematics, Academy of Sciences of the USSR. The official opponents were: Doctor of Physicomathematical Sciences Prof. K. I. Babenko, Doctor of Physicomathematical Sciences Prof. N. S. Bakhvalov, and Doctor of Physicomathematical Sciences E. A. Volkov.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 655–663, May, 1970.     

Integral equations of plane static boundary value problems in the moment elasticity theory of inhomogeneous isotropic media

Boundary value problems in the plane moment and simplified moment elasticity theory of inhomogeneous isotropic media are reduced to Riemann-Hilbert boundary value problems for a quasianalytic vector. Uniquely solvable integral equations over a domain are derived. As a result, weak solutions for composite inhomogeneous elastic media can be determined straightforwardly.     

A new approach to solve a set of nonlinear split boundary value problems

In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms.     

Some basic boundary value problems for plane theory of elasticity of porous Cosserat media with triple-porosity

The static equilibrium of porous elastic materials with triple-porosity is considered in the case of an elastic Cosserat medium. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and three solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with triple-porosity. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an approach to the study of the solvability of boundary value problems for the system of differential equations of the 3D theory of elasticity

We study the solvability of a boundary value problem for a system of second-order linear partial differential equations. A theorem on the existence of a solution of the problem is proved. The method used in the study is to reduce the original system of equations to a system of 3D singular integral equations, whose solvability can be proved with the use of the notion of symbol of a singular operator.