Theory Of Boundary Value Problems Research Articles

Numerical Solution of the Two-Dimensional Elasticity Problem in Strains

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.

Dirichlet problem with measurable data for quasilinear Poisson equations

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb D$ goes to the famous dissertation of Luzin, see e.g. its reprint \cite{L}. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. Following this way, we proved in \cite{GNR1} Theorem 7 on the solvability of the Dirichlet problem for the Poisson equations $\triangle\, U=G$ with sources in classes $G\in L^p,$ $ p>1$, in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. There we assumed that the domains satisfy the quasihyperbolic boundary condition by Gehring--Martio, generally speaking, without the known $(A)-$condition by Ladyzhenskaya--Ural'tseva and, in particular, without the outer cone condition that were standard for boundary-value problems in the PDE theory. Note that such Jordan domains cannot be even locally rectifiable. With a view to further development of the theory of boundary value problems for semi-linear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is first constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations $\triangle\, U=G$ with the sources $G\in L^p,$ $ p>1$. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form $\triangle\, U(z)=H(z)\cdot Q(U(z))$ for multipliers $H\in L^p$ with $ p>1$ and continuous functions $Q: \mathbb R\to\mathbb R$ with $Q(t)/t\to 0$ as $t\to \infty$. As consequences, we give applications to some concrete quasilinear equations of mathematical physics, arising under modelling various phy\-si\-cal processes such as diffusion with absorption, plasma states, stationary burning etc. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

FINITE DIFFERENCE METHOD FOR SOLVING THE ELASTOPLASTIC PROBLEMS OF ANISOTROPIC BODIES

Usually, for the numerical solution of elastoplastic boundary value problems based on deformation theory of plasticity is used an elastic solutions method proposed by A.A. Ilyushin. In literatures the method of elastic solutions relative to boundary value problems of the theory of plastic flow is called the method of initial stresses or the method of initial deformations. In this paper, to solve the boundary value problems of the deformation theory of plasticity of transversally isotropic bodies, it is used relatively “simple” finite-difference method considered in combination with the iterative method, that is, the elastic solution method. The essence of the method is to construct symmetric finite-difference equations, separately for internal and boundary nodes of the area under consideration, and to solve them with respect to “central” or boundary node displacements and the organization of the iterative process. Elastoplastic problems are solved numerically for isotropic and transversely isotropic parallelepipeds under various boundary and boundary conditions. The obtained results are consistent with the known solutions, which shows the validity of the applied methodology. It is explored the spreading of the zone of plasticity and the effect of anisotropy on their distribution.

Splitting of Initial Boundary Value Problems in Anisotropic Linear Elasticity Theory

The splitting of initial boundary value problems in the theories of elasticity is considered for some anisotropic media. In particular, the initial boundary value problems of the micropolar classical theory of elasticity are represented using the tensor-block matrix operators (or tensor operators). In the case of isotropic micropolar elastic media known also as isotropic or transversally isotropic classical media, we propose the tensor-block matrix operators of algebraic cofactors corresponding to the tensor-block matrix operators of the initial boundary value problems, which allows us to split these problems.

Boundary problems on a manifold with edge

We establish a calculus of boundary value problems (BVPs) on a manifold [Formula: see text] with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.

Nonsmooth Boundary Value Problems in Theory of Rods, Plates, and Shells

We study the stability of elastic systems with one-sided constraints on displacements. We analytically solve the contact problem for rods and propose a new method for solving contact problems with a free boundary. We present numerical results concerning the stability of elastic systems with one-sided constraints on displacements. Bibliography: 10 titles.

On One Class of Boundary Value Problems of the Steady-State Filtration Theory in Porous Grounds

:In the framework of Darcy's law of filtration the investigation results of one class of boundary value problems of the steady-state filtration theory in porous ground base are presented. The plane mixed bounadry value problems on the structural analysis of hydrotechnical con­struction of dam type on filtrating ground base in the form of a layer of finite or infinite thickness are considered. The coefficient of filtration is assumed to be constant, piecewise constant, or changing by the depth of base according to the exponential law, the property of anisotropy of filtration is also taken into account. Axis-symmetric and three-dimentional boundary value problems of the theory of steady-state fluid filtration in a three-dimentional layer of a finite or infinite thickness are discussed. These problems are of the type of Lamb well-known hydrodynamic problems in the theory of steady-state flow of the ideal fluid, when through the circular or rectangular openeing of a rigid screen on the upper bound of the layer the liquid with a definite vertical velocity or with a definite pressure is injected into porous ground base. Here, the fields of velocities and pressures in the layer, as well as flow rates of liquid through the certain sections of the ground base are determined.

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.

Boundary-Value Problems in a Cut Plane

We present a survey of the theory of boundary-value problems in the plane with curvilinear cuts. The problem of linear conjugation, the Riemann-Hilbert problem, and the Riemann-Hilbert-Poincare problem are considered in detail both in the classical setting and for a cut plane, with an emphasis on problems with shifts. The main focus is on the solvability conditions and index formulas in various function classes.

On Fučik Spectra and Periodic Solutions

In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fucik spectra.

On Shifts along Trajectories

The extension of the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand sides based on the construction of a new version of the method of shifts along trajectories is continued.

A Small-Parameter Problem from the Theory of Boundary-Value Problems with a Small Nonlinearity

We consider a system of nonlinear equations with a small (in a certain sense) nonlinearity. The existence of a solution is investigated in the presence of a perturbation parameter of small absolute value, and the perturbed solution is required to approach the original solution of the limiting linear system. The sufficient conditions are presented in geometrical and analytical forms. The problem considered in this article arises in connection with nonlinear boundary-value problems for systems of differential equations with a small (in a certain sense) nonlinearity.

Solvability and Regularity of Solutions for Some Classes of Elliptic Functional-Differential Equations

1. This review is devoted to the theory of boundary-value problems for some classes of elliptic functionaldifferential equations. Boundary-value problems for elliptic functional-differential equations have some astonishing properties. For example, unlike elliptic differential equations, the smoothness of generalized solutions can be broken in a bounded domain and is preserved only in some subdomains. The symbol of a self-adjoint semibounded functional-differential operator can change its sign. Boundary-value problems for elliptic functional-differential equations are closely related to nonlocal elliptic problems. On the other hand, elliptic functional-differential equations have important applications to nonlinear optics, elasticity theory, control theory, and diffusion processes. Elliptic functional-differential equations containing transformations of arguments were studied by A. B. Antonevich [2], D. Przeworska-Rolewicz [36], and V. S. Rabinovich [37]. The authors assumed that the transformations of arguments map a domain onto itself. Therefore, their results were similar to the well-known results for elliptic differential equations. The situation changes if the equation has these shifts in the highest derivatives, and the shifts map the points of the boundary into the domain. For an introduction to the theory of these equations, see A. L. Skubachevskii [67]. 2. In this paper, we mainly study boundary-value problems for elliptic differential-difference equations and for elliptic functional-differential equations with contractions and expansions. In order to illustrate the properties of elliptic differential-difference equations, we consider the following example:

Principle of correspondence of static boundary-value problems of nonlinear viscoelasticity with aging to boundary-value problems of the theory of elasticity

The principle of correspondence of boundary-value problems of the nonlinear nonuniform anisotropic theory of viscoelasticity to boundary-value problems of the theory of elasticity is formulated. The correspondence is established by means of integral transforms with previously unknown kernels. A class of viscoelastic materials for which these transforms can be reduced to boundary-value problems of fictitious elasticity is determined.

Determination of the frequencies and forms of oscillations of non-uniform distributed systems with boundary conditions of the third kind

An effective method of constructing a solution of the eigenvalue and eigenfunction problem for an essentially non-uniform oscillatory system with variable distributed parameters is developed. The proposed approach is based on a combination of a variational approach, the theory of boundary-value problems and perturbation methods. An original determination of the small parameter of the problem is given and a recurrence algorithm for the successive refinement of the eigenvalues and eigenfunctions is proposed, which leads to accelerated (quadratic) convergence of the Newton's “method of tangents” type. A commentary is made on the method and is illustrated by the solution of model examples.

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.

Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.

A Green's function method of determining the residual stresses in cylindrical glass shells

We propose a way of modifying the mathematical model of the nondestructive theoretical-experimental method of determining the residual stresses for a specific collection of technological conditions for welding two cylindrical glass shells. The method is based on the general theory of boundary-value problems for ordinary differential equations.

Boundedness of the generalized solutions of elliptic equations with nonpower nonlinearities

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