Boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe

We formulate the plane two-dimensional static boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe under the action of a temperature field and compressive normal stresses that are uniformly distributed on its lateral surfaces and present its approximate solution. We assume that the mechanical and thermophysical properties of the materials are temperature-independent, plastic strains arise on the interior lateral surface of the two-layer pipe and completely envelop it, and the material of the pipe is perfectly elastoplastic, incompressible in the domain of plasticity, and satisfies the Tresca-Saint-Venant plasticity condition. I. Franko L'viv State University, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 57–66, April–June, 1998.     

Instability of thermoelastic interaction between a half-space and a rigid base through a thin liquid layer

We investigate the instability of thermoelastic interaction between elastic and rigid half-spaces through a liquid interlayer under the conditions of heat transfer across the interfaces. Due to the small thickness of the liquid layer, its influence on the temperature field is taken into account by the thermal resistance of the contact between the bodies, which depends on the normal displacement of the boundary of the elastic body. The pressure inside the liquid is equal to the external pressure applied to the bodies. We determined the critical value of the external heat flow for which the instability becomes possible in such a system and studied the dependence of this value on the parameters of the elastic half-space, the thickness of the liquid layer, and its thermal conduction. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 76–82, April–June, 1998.     

On a method for constructing one-frequency solutions of a nonlinear wave equation

A method for constructing one-frequency solutions of nonlinear wave equations is suggested. This approach is based on a modified representation of asymptotic expansions by using special periodic Atebfunctions. This method makes it possible to obtain approximate solution of the problem under consideration without difficulty. L'viv Polytechnic Institute, L'viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 782–784, June 1994.     

On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations

We present the results of an investigation and some applications of fundamental solutions of the Cauchy problem for a new class of parabolic equations. In these equations: (i) there exist three groups of spatial variables, one basic and two auxiliary, (ii) different weights of spatial variables from the basic group with respect to the time variable are admitted, (iii) degeneracies in variables from the auxiliary groups are present, (iv) a degeneracy on the initial hyperplane is present. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Ternopil' Academy of Economics, Ternopil'. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 13–19, April–June, 1998.     

One-dimensional wave processes in a magneto-elastic medium

We construct different solutions of one-dimensional magnetoelastic problems. We analyze the process of propagation of disturbances in a magnetoelastic half-space with finite conductivity when a uniformly distributed force load and magnetic field intensity are prescribed on the surface. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 65–69.     

Determination of the boundary of the plastic zone in a mineral vein weakened by a three-dimensional hole

We propose a method of computing the boundary of the plastic zone formed in a neighborhood of the hole during the mining of a mineral. The problem is studied in a three-dimensional formulation. The boundary of the plastic zone is determined from the condition of continuity of the vertical normal stresses acting on the surface of contact of an elastic half-space and an elastoplastic layer. The computation is carried out for a hole having a parallelepipedal shape. One figure. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 6–10, 1991.     

The Gold Partition Conjecture for 6-Thin Posets

We consider the Gold Partition Conjecture (GPC) that implies the 1/3–2/3 Conjecture. We prove the GPC in the case where every element of the poset is incomparable with at most six others. The proof involves the extensive use of computers. This paper contains results obtained using computer resources of the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), University of Warsaw.     

Insulated antenna in a layered medium

The article presents a generalization of the integral equation of an insulated linear antenna immersed in a cylindrically layered lossy dielectric medium. The insulation is provided by a lossless dielectric layer. The kernel of the integral equation is represented as a superposition of the fundamental solutions of the wave equation with equivalent propagation constants for the given media. A generalization to a plane-layered medium is proposed. The problem of a vertical radiator above a layered half-space is considered. Translated from Chislennye Metody v Matematicheskoi Fiziki, Published by Moscow University, Moscow, 1996, pp. 80–88.     

Effect of overload on the integral parameters of a concentric annular die

The non-axisymmetric contact problem in the theory of elasticity is solved for a smooth concentric annular die on a uniform elastic half-space when an overload normal to the boundary of the half-space acts outside the die. A Hankel-Fourier integral transform and triple integral equations are used to reduce the problem to a quasi-regular system of algebraic equations which is solved by a perturbation method and, in general, by a reduction method. The effect of a lumped force on the integral parameters of the die is examined. Graphs are shown which characterize the degree to which the parameters of the problem influence the total force and moment, displacement, and inclination of the die. Kharkov State Economics University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 111–117, 1999.     

Solution of problems of the theory of elasticity for a half-space with planar boundary cracks

We propose a method of solving three-dimensional problems of the theory of elasticity for a half-space containing planar boundary cracks. The problem is reduced to a system of integro-differential equations for determining the functions that characterize the opening of the crack during deformation of the halfspace. The kernels of the equations, besides having poles, also have a fixed singularity at the points of intersection of the surface of the crack with the boundary of the half-space. The equations obtained are solved numerically for the case of cracks that are part of a circular region. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 58–63.     

Generalized theorem on positive definiteness of energy in the general relativity theory

We introduce the notion of the energy of gravitational and material fields with respect to an arbitrary time-like vector field and a space-like hypersurface as an integral over a nonholonomic hypersurface. In the case of an asymptotically Minkowskian space, by developing the approach of Witten and Nester, we obtain an expression for the energy functional in terms of the spinors associated with a differential-geometric distribution. By applying the Sen-Witten generalized equation, we prove the nonnegativity of this functional. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences. L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 26–34, April–June, 1998.     

Maximal L p -regularity for a linear three-phase problem of parabolic–elliptic type

We prove maximal L _p -regularity for a three-phase problem consisting of strongly coupled parabolic–elliptic equations with inhomogeneous data. This problem is related to a nonlinear problem which arises in chemically reacting systems incorporating electromigration. Particular features are a transmission condition and a jump condition on the boundary, which couple all unknowns. By means of localization the problem is reduced to model problems in full and half-space. To solve model problems, we make use of Dore–Venni Theory, real interpolation and the Mikhlin multiplier theorem in the operator-valued version. Here it is crucial to find conditions on the data that are necessary and sufficient for maximal L _p -regularity of the respective solution.     

Mathematical model of an internal crack in an elastoplastic cylindrical shell

We propose a mathematical model that makes it possible to reduce the problem of the stressed state and limit equilibrium of a cylindrical anisotropic elastoplastic shell with internal crack to a system of nonlinear singular integral equations with discontinuous functions on the right-hand sides. We construct an algorithm for numerical solution of such systems together with the conditions of plasticity and boundedness of stresses near the crack. For a transversally isotropic shell, we carry out a numerical analysis of the dependence of the opening of the internal crack front on the load and geometric and mechanical parameters. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 111–116, April–June, 1998     

Stress distribution in a transversely isotropic body with a thermally insulating parabolic crack subjected to a uniform heat flux

An explicit static thermoelastic solution is constructed for an infinite transversely isotropic body containing a thermally insulating parabolic crack in the plane of isotropy. The surface of the crack is free of stress. A uniform thermal flux is incident on the crack perpendicular to its surface. Formulas are obtained for the stress intensity factors near the tip of the crack. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Catholic University, Portugal. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 54–66, 1999.     

Exact linearization of the sliding problem for a dilute gas in the Bhatnagar-Gross-Krook model

We consider the nonlinear Boltzmann equation in the Bhathnagar-Gross-Krook model for the gas flow in a half-space (the Kramers problem). The problem can be exactly linearized, and its solution can be reduced to a linear integral equation with an addition-difference kernel and a simple nonlinear relation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 339–342, November, 2000.     

The contact problem of electroelasticity for a plane elliptic die

We obtain an exact solution of the problem of the stress-strain state of an elastic piezoelectric half-space acted on by a rigid elliptic die with a flat base. The axis of symmetry of the body coincides with the direction of the field of preliminary polarization of the body. The solution is confined to the case of translational displacement of the die. We determine the quantities that characterize the mechanical and electric fields that arise in the region of contact of the die with the half-space. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 40–52.     

Constructing a model transport equation for a massless bose gas and its analytic solution

A model kinetic equation is constructed for the transport of a massless Bose gas. This equation is applied to solve the boundary value problem for the transport of radiation in the half-space occupied by a dispersive medium that is in local thermal equilibrium with the radiation. It is shown that the difference in temperature between the dispersive medium and the incident radiation substantially depends on the character of the scattering properties of the particles in the medium. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 462–472, June, 1997.     

Branching of solutions of the problem of synthesis of a linear antenna with given directional amplitude diagram in a Hölder space

We investigate the branching of solutions of a nonlinear integral equation of the Hammerstein type which arises in the problem of synthesis of a linear antenna with given directional amplitude diagram. Systems of transcendental equations for determination of branching points of various types are deduced, and the number of branched solutions and their qualitative characteristics are analyzed. Pidtryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 35–44, April–June, 1998.     

Quasi-optimal control of axisymmetric temperature displacements in a given section of a body using internal heat sources

For a thermoelastic half-space we study the problem of constructing in the space of continuous functions a quasi-optimal control of the axisymmetric vertical displacements in a given section parallel to the boundary surface in the form of a sequence that minimizes the original optimality criterion. For a half-space heated over a circular region by a heat flux of constant intensity we carry out a numerical analysis of the behavior of the elements of the minimizing sequence. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 104–109.     

On the efficiency of the WKB–Galerkin method in differential equations with variable coefficients

We have proposed an algorithm for the solution of inhomogeneous singular second-order differential equations with variable coefficients, based on a model of the hybrid WKB–Galerkin method. The efficiency of this approach is illustrated in the solution of an applied problem describing heat removal through a radiator of variable geometry. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 82–87, January–March, 2008.