Sobolev Space Methods for Dual Integral Equations in Axialsymmetric Screen Problems

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

The Numerical Solution of Integral Equations on the Half-Line

A flat annular crack in a magnetoelectroelastic layer subjected to mechanical, electric and magnetic loadings is investigated under magnetoelectrically impermeable boundary condition on the crack surface. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. With the aid of Gauss-Chebyshev integration technique, the integral equations are further reduced to a system of algebraic equations. The field intensity factor and energy release rate are determined. Numerical results reveal the effects of electric and magnetic loadings and crack configuration on crack propagation and growth.

The effectiveness of dual integral equations for handling mixed boundary value problems has established them as an important tool for applied mathematicians. Their many applications in mathematical physics have prompted extensive research over the last 25 years, and many researchers have made significant contributions to the methodology of solving and to the applications of dual integral equations. However, until now, much of this work has been available only in the form of research papers scattered throughout different journals.In Advances in Dual Integral Equations, the authors systematically present some of the recent developments in dual integral equations involving various special functions as kernel. They examine dual integral equations with Bessel, Legendre, and trigonometric functions as kernel plus dual integral equations involving inverse Mellin transforms. These can be particularly useful in studying certain mixed boundary value problems involving homogeneous media in continuum mechanics. However, when dealing with problems involving non-homogenous media, the corresponding equations may have different kernels. This application prompts the authors to conclude with a discussion of hybrid dual integral equations-mixed kernels with generalized associated Legendre functions and mixed kernels involving Bessel functions.Researchers in the theory of elasticity, fluid dynamics, and mathematical physics will find Advances in Dual Integral Equations a concise, one-stop resource for recent work addressing special functions as kernel.

Previous article Next article Optimal Control Problems in Sobolev Spaces with WeightsClaudia SimionescuClaudia Simionescuhttps://doi.org/10.1137/0314011PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe consider an optimal control problem in certain Sobolev spaces with weights used systematically by F. Trèves in [1], [2]. The notations and definitions are the same as in [2] and [3].[1] François Trèves, Domination et problèmes aux limites de type mixte, C. R. Acad. Sci. Paris, 245 (1957), 2454–2457 MR0098912 (20:5362) 0079.33202 Google Scholar[2] François Trèves, Relations de domination entre opérateurs différentiels, Acta Math., 101 (1959), 1–139 MR0125322 (23:A2625) CrossrefISIGoogle Scholar[3] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Avant propos de P. Lelong, Dunod, Paris, 1968xiii+426 MR0244606 (39:5920) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Bibliography6 October 2017 Cross Ref Linearization by means of optimal control for a model in plasma physicsInternational Journal of Engineering Science, Vol. 18, No. 2 Cross Ref Control for an hereditary systemMathematics and Computers in Simulation, Vol. 21, No. 2 Cross Ref Notes On a Mathematical Model in Plasma Physics Cross Ref Optimal control problems in sobolev spaces with weights. Numerical approaches applications to plasma optimal control and time delay problems21 May 2005 Cross Ref Numerical methods for a generalized optimal control problem Cross Ref Volume 14, Issue 1| 1976SIAM Journal on Control and Optimization History Submitted:27 June 1974Published online:03 August 2006 InformationCopyright © 1976 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0314011Article page range:pp. 137-143ISSN (print):0363-0129ISSN (online):1095-7138Publisher:Society for Industrial and Applied Mathematics

It is shown that a class of axially symmetric boundary value problems in potential theory for the region external to a lens may be reduced to problems in two-dimensional potential theory. The solution for the general axially symmetric Dirichlet problem is obtained and it is shown that the Stokes stream function for an arbitrary flow may also be reduced to a Dirichlet problem in potential theory. The Green's function for the lens is also derived and the extension of the approach to non-axially symmetric problems is considered.

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

The aim of the present work is to consider a mixed boundary value problem for the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges y=0,y = h having clamped conditions on intervals |x|≥a and hinged support conditions for |x|<a. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of solution of system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the dual integral equation to infinite system of linear algebraic equations is also proposed.

The processes of reflection of three-dimensional electromagnetic waves by locally irregular media interfaces are investigated. The boundary value problem for the system of Maxwell equations in an infinite half-space is reduced to the solution of two systems of singular integral equations. To construct a numerical algorithm for solving these systems of singular integral equations, the approximation and collocation method is used. Special attention is focused on calculation of the kernels of these equations. Solutions of systems of singular integral equations are used to calculated the far-field pattern for the scattered field in the far zone.

Axisymmetric dynamic contact problem for a viscoelastic half-space in the presence of adhesion under the stamp: PMM vol. 37, n≗4, 1973, pp. 731–738

The problem of determination of the stationary temperature field in a body with plane sheet of heat sources is reduced to the solution of an integral equation of the first kind. A method for the determination of the set of solutions to this equation is proposed. The components of the vector of elastic displacements and the components of the tensor of temperature stresses are found according to the known temperature field by using the equations of thermoelasticity. We study the distributions of thermal displacements and stresses for any axially symmetric distribution of temperature in a circular domain. Numerous elements of contemporary engineering constructions operate under conditions of nonuniform heating that promote the formation of temperature gradients and thermal stresses. For the comprehensive analysis of the strength of structures, it is necessary to know the levels of thermal stresses and the character of their action. The investigation of the thermoelastic state of heat-generating elements is of high importance for practical purposes [9]. The presence of thin heat-generating inclusions in a body leads to the formation of tensile or compressive stresses. In the first case, the action of these stresses, together with the action of the other factors, may lead to crack initiation and propagation. In the second case, the action of stresses may decrease the intensity of stresses in the vicinity of the existing crack under force loading and promote the formation of conditions inhibiting the process of crack growth [10]. In the investigation of the stressed state of a body with thermally active cracks (with temperature or heat fluxes given on these cracks), the determination of stresses acting at the sites of the cracks is an intermediate stage and then the stresses acting upon the crack surfaces are removed. The major part of investigations of the stress-strain state of bodies with circular thermally active or heatinsulated cracks are performed for axially symmetric problems by the method of integral Hankel transformation and dual integral equations [11–16]. In [3], a method of two-dimensional boundary integral equations based on the theory of the Newton potential is developed for the solution of three-dimensional problems of stationary heat conduction and thermoelasticity for bodies with cracks. According to this method, the densities of potentials in the problem of heat conduction are the intensities of heat sources at the sites of the cracks determined, for given temperatures on the cracks, from the integral equations with polar kernels. The temperature field and stressed state of a body with given temperature or heat flow in a circular domain described by polynomials of the third degree are investigated in [4]. In the axially symmetric case, for polynomials of any degree, this problem is studied in [5]. According to the classic statement of the problems of stationary heat conduction with thermally active inclusions [4, 5], the presence of heat sources is postulated solely in the domain of the inclusion, which leads to singular distributions of the heat flows on its edges.

This paper gives an integral equation, the solution of which is a solution of a classical problem in potential theory: Given a region with boundary B \mathcal {B} , what distribution of charge on B \mathcal {B} will produce a potential having specified values on B \mathcal {B} ? The paper also indicates briefly how the integral equation is useful in simplifying certain proofs and extending certain theorems in potential theory.

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ℝ2 coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with 2×2 operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.

The thermoelectromechanical fracture of a piezoelectric material strip with two parallel cracks of different lengths under electric and thermal loadings is considered. The crack faces are supposed to be insulated thermally and electrically. Fourier transform techniques are used to reduce the mixed boundary value problems to two systems of singular integral equations. The singular integral equations are solved by using the Gauss-Jacobi integration formula. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the parameters of the geometric parameters on the stress and electric displacement intensity factors.

Partial differential equations and boundary value problems: By Viorel Barbu. Kluwer Academic Publishers, Dordrecht. (1998). 277 pages. $129.00, NLG 240.00, GBP 82.00