System Of Fredholm Integral Equations Research Articles

A Direct Boundary Integral Method for a Mobility Problem

Abstract The problem of a Stokes flow in the presence of a solid particle, a rigid wall and a viscous cell is formulated as a system of Fredholm integral equations of the second kind, with the surface force on the boundary of the solid particle and the velocity on the interface as unknowns. The particularity of the problem consists in the fact that the total force and the total torque of the flow on the solid particle are zero. The existence and the uniqueness result of solution is obtained when the boundaries are curves of the class C 2.

Influence of initial stress on fracture of a halfspace containing a penny-shaped crack under radial shear

In this study, an axisymmetrical problem for a penny-shaped crack under radial shear is considered. The crack is located parallel to the surface of a halfspace, which is subjected to initial stress parallel to the crack plane. An approach proposed by Guz (1983) in the framework of the three-dimensional linearised solid mechanics is used. Analysis involves reducing the problem to a system of Fredholm integral equations of the second kind, where the solutions are identified with harmonic potential functions. The representations of the stress intensity factors KI and KII near the crack edges are obtained. These stress intensity factors are both influenced by the initial stress.

A class of boundary-value problems in the dynamic theory of elasticity

A class of boundary-value problems of the dynamic theory of elasticity, which has not been studied to any great extent, is investigated. In these problems all the components of the displacement vector and the stress vector are specified on part of the boundary of the body (nothing is known about the components of these fields on the remaining part of the boundary). A uniqueness theorem is proved. The problem is investigated by reducing the boundary-value problem to a system of Fredholm integral equations of the first kind with smooth kernels. A scheme for the numerical determination of the unknown fields is proposed, based on a combination of the boundary-element method and the Tikhonov regularization method. A number of model examples in establishing the wave fields for an anisotropic body are considered.

Transient analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions

This paper introduces an efficient numerical algorithm for transient analysis of deterministic and stochastic Petri nets (DSPNs) and other discrete-event stochastic systems with exponential and deterministic events. The proposed approach is based on the analysis of a general state space Markov chain (GSSMC) whose state equations constitute a system of multidimensional Fredholm integral equations. Key contributions of this paper constitute the observations that the transition kernel of this system of Fredholm equations is piece-wise continuous and separable. Due to the exploitation of these properties, the GSSMC approach shows great promise for being effectively applicable for the transient analysis of large DSPNs with concurrent deterministic transitions. Moreover, for DSPNs without concurrent deterministic transitions the proposed GSSMC approach requires three orders of magnitude less computational effort than the previously known approach based on the method of supplementary variables.

Vertical dynamic response of a disk on a saturated poroelastic half-space

This paper considers the vertical dynamic response of a disk on a saturated poroelastic half-space. Firstly the pressure-solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. These equations are solved by a new method. Then the mixed boundary value problem for the vertical harmonic vibration of a disk on a poroelastic half-space is studied. The two types of drainage conditions at the surface of the poroelastic half-space are considered: (a) the surface of the poroelastic half-space is assumed to be completely pervious both within and exterior to the plate; (b) The interface between the plate and the poroelastic half-space is assumed to be impervious and the exterior region is assumed to be pervious. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind by Abel transform.

Macrocrack-Microcrack Interaction in Piezoelectric Materials, Part I: Basic Formulations and J-Analysis

The macrocrack-microcrack interaction problem in transversely isotropic piezoelectric materials is studied. The microcracks near a macrocrack tip in the process zone are assumed to be parallel to the latter, while the poling direction of the piezoelectric materials is assumed to be perpendicular to the cracks. Three kinds of elementary solutions with different crack configurations and under different loading conditions are given, from which the interaction problem is reduced to a system of Fredholm integral equations by using the pseudo-traction electric displacement method (abbreviated PTED). After the equations are solved numerically, the traditional mode I and mode II stress intensity factors and the electric displacement intensity factor are evaluated. In order to confirm the proposed method as well as the numerical results, a consistency check is proposed which is based on the J-integral analysis and provides a powerful tool to examine the numerical results. Thus, any mistakes are avoided since they would certainly lead to unsatisfied numerical results contrary to the check. It is concluded also that the disturbance of the near-tip electric field provides another source of shielding.

The Nyström method for solving a class of singular integral equations and applications in 3D-plate elasticity

The paper consists of two parts. In the first part we investigate a Nystrom- or product integration method for second kind singular integral equations. We prove an asymptotically optimal error estimate in the scale of Sobolev Hilbert spaces. Although the result can also be obtained as a special case of a discrete iterated collocation method our proof is more direct and uses the Nystrom interpolation. In the second part of this paper we consider the Dirichlet problem for thin elastic plates with transverse shear deformation. The boundary value problem is transformed into a 3 x 3 system of singular Fredholm integral equations of second kind. After discussing existence and uniqueness of the solution to the integral equations in a Sobolev space setting, we apply the Nystrom method to solve the integral equations numerically.

The vertical vibration of an elastic circular plate on a fluid-saturated porous half space

In this paper, the vertical vibration of an elastic circular plate on fluid-saturated porous half space is studied by a new analytical method. The contact between the elastic circular plate and the poroelastic half space is assumed to be smooth. The drainage conditions at the surface of the poroelastic half space are considered as either completely drained, or partially drained. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind. Numerical examples are given at the end of the paper.

Tomographic reconstruction from arbitrary directions using ridge functions

In this paper tomographic reconstruction based on the concept of ridge functions (Logan and Shepp) is considered. A reconstruction approach for the ridge functions from a finite number of arbitrary projections is suggested within the framework of parallel beam geometry. The method deals with images that can be presented as a sum of ridge functions. This assumption results in a consistent linear system of Fredholm integral equations of the third kind in terms of one-dimensional functions. We derive a formula to calculate the ridge functions from the set of arbitrary projections. In the case of equally spaced projections formulae for the analytical inversion of matrices encountered in the calculation of ridge functions are derived.

Investigation of macrocrack-microcrack interaction problems in anisotropic elastic solids - Part I : General solution to the problem and application of the J-integral

A general solution is derived for the plane problem of multiple microcracks near the tip in process zone of a semi-infinite macrocrack in an anisotropic elastic solid. The pseudo-traction method, addressed thoroughly in isotropic cases, is extended to anisotropic cases. A system of Fredholm integral equations, with difficulty in evaluation of the singular integrals, is solved by invoking the asymptotic behavior of the pseudo-tractions on the macrocrack faces. The interaction effect of the release of residual stresses due to near-tip microcracking is then evaluated. The J-integral analysis is also performed to give a consistency check of the solution and some useful conclusions.

Convective cooling in contact problems of thermoelasticity with frictional heat generation

Contact problems with frictional heat generation are, as a rule, studied under idealized conditions of perfect thermal insulation of the surfaces of bodies outside the region of contact. We solve an axially symmetric contact problem of stationary thermoelasticity for semibounded bodies of revolution by taking into account the effect of convective cooling of the free surfaces. The problem is reduced to a system of Fredholm integral equations. An approximate solution of these equations is obtained in terms of simple analytic relations. We determine the conditions under which the influence of convective heat exchange on the level and distribution of contact stresses can be neglected.

Counter-rotating dust rings around a static black hole

Relativistic configurations describing counter-rotating dust discs with a central Schwarzschild black hole are analysed. It is shown that the corresponding solutions to the Einstein equations which belong to the Weyl class can be obtained from a boundary value problem for the Laplace equation where the boundary data are given at the two-dimensional dust ring. Using a result of Poole, one can prove existence and uniqueness of a solution to this problem. In adapted coordinates, the potential can be represented via eigenfunctions of the Lamé equation in the `flat ring' case which are not known explicitly. Following Kanwal, one can reformulate the boundary value problem in a system of Fredholm integral equations. As an example, a rigidly (counter-)rotating dust ring is considered. In the case of a dust ring stretching to infinity, it is possible to relate this problem via a Kelvin transformation to the case of a disc. An explicit solution is given for an isochrone dust ring.

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution.

Elastic field for eccentrically loaded rigid plate on multilayered solids

This paper presents an analytical investigation of the contact problem of a multilayered elastic solid subjected to the eccentric indentation of a rigid circular plate in the framework of classical elasticity. The total number of the dissimilar layers is an arbitrary integer n. These elastic layers can rest either on a dissimilar elastic halfspace or on a rigid rock base. The rigid circular plate is in smooth contact with the multilayered elastic solid. The classical theory of Fourier integral transforms is employed to solve the partial differential equations governing the behaviour of the multilayered elastic solid subjected to surface loading. Systems of standard Fredholm integral equations of the second-kind are then developed to govern the interaction between the rigid plate and the multilayered elastic solid. Explicit solution expression is further presented for the elastic field in the multilayered solid due to the eccentric indentation of the rigid plate. Closed-form results are respectively obtained for the singular stress field at the rigid plate edge in the multilayered elastic solid and for the elastic field in a homogeneous elastic halfspace induced by the eccentric indentation of the rigid plate. In particular, an asymptotic technique is utilized to overcome the difficulty associated with the convergence and singularity of the solution near or at the surface of the multilayered solid. Numerical results are presented to verify the techniques adopted in the paper and to illustrate the effect of layering material non-homogeneity on the elastic field induced by the eccentrically loaded rigid plate. The solution can be applied to the interpretation of non-destructive testing of layered materials such as highway and airport pavements.

A numerical analysis of the data inversion of particle sizing instruments

The retrieval of an unknown desired function from a set of physical measurements taken from an instrument constitutes an important inverse problem in indirect sensing measurement theory. In general, given a set of discrete fractional measurements and a set of instrument kernel functions, the mathematical formulation reduces to solving a system of discrete Fredholm integral equations of the first kind which is generating an ill-posed problem. In order to deal with the ill-posedness of the inverse problem, a modified first-order regularization procedure, subject to certain physical constraints and bounds on the variables, which does not use any a priori information regarding the parameterised form of the solution, is employed and compared with other available methods. As a practical application the numerical method is illustrated for the determination of the particle size distribution from diffusion battery data. For exact data, it is shown how the degree of non-uniqueness of the inverse problem analysed may be reduced by decreasing the flow rate at which the diffusion battery operates or better by recording additional deposition data corresponding to several flow rates. For noisy data, a detailed numerical investigation with respect to the bounds on which the unknown variables are to be constrained is analysed.

Numerical analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions

This paper introduces an efficient numerical algorithm for transient analysis of deterministic and stochastic Petri nets (DSPNs) and other discrete-event stochastic systems with exponential and deterministic events. The proposed approach is based on the analysis of a general state space Markov chain (GSSMC) whose state equations constitute a system of multidimensional Fredholm integral equations. Key contributions of this paper constitute the observations that the transition kernel of this system of Fredholm equations is piece-wise continuous and separable. Due to the exploitation of these properties, the GSSMC approach shows great promise for being effectively applicable for the transient analysis of large DSPNs with concurrent deterministic transitions. Moreover, for DSPNs without concurrent deterministic transitions the proposed GSSMC approach requires three orders of magnitude less computational effort than the previously known approach based on the method of supplementary variables.

Certain two-punch problems for an elastic layer

This paper examines the normal and tangential indentation problems for two circular punches on the surface of an elastic layer. By using the method proposed for studying offset parallel penny-shaped cracks in an elastic solid, we show that these problems are governed by systems of Fredholm integral equations, which for some special cases, can be solved by iteration. For certain prescribed indentations, asymptotic solutions are presented to illustrate the manner in which the presence of the second punch and thickness of the layer influences the total forces induced on the punches.

A theory for the diffraction of SH waves by randomly rough surfaces in two dimensions

In this paper we shall consider a mathematical model for the diffraction of SH waves by randomly rough surfaces. We use the crack Green function method to formulate the problem (see G. R. Wickham, Proc. R. Soc. A 378 (1981) 241-261, P. A. Lewis and G. R. Wickham, Phil. Trans. R. Soc. A 340 (1992) 503-529). This method gives rise to a Fredholm integral equation of the second kind for the jump in in the potential across the scatterer (also known as the crack opening displacement). To analyse the expectation of the crack opening displacement we use a smoothing method in which we perform an ensemble average over all realizations of a certain class of surfaces. A closure assumption is made which is valid for surfaces with small root mean square height. This gives rise to a coupled system of Fredholm integral equations for the average of the crack opening displacement and the average of the random components of the crack opening displacement and kernel. If we make the further assumption that the root mean square gradients are also small, then this coupled system reduces to a single Fredholm integral of the second kind for the expectation of the crack opening displacement. Numerical results are presented for the average crack opening displacement and the average scattered field.

The Existence and Uniqueness of Solutions by the Covering Domain Method in Linear Elastostatics

AbstractThe paper deals with problems of linear elastostatics with multiply connected solution domains. Generally, a multiply connected domain can be represented in a non‐unique way by the intersection set of n > 1 domains, each of which covering completely the original solution domain. If exact analytical solutions can be found in the covering domains, then, making use of the superposition principle, the solution of the original problem can be constructed by summation of the n solutions of the covering domains. Following this idea, the “covering Domain Method” is developed, by which the original problem is transformed into a system of Fredholm integral equations which finally offers the solution of the problem. In this paper we consider the existence and uniqueness of the solutions for different choice of covering domains. Two theorems are presented, which are useful for the application of the covering domain method in elasticity.

On the elasticT-term of a main crack induced by near tip microcracks

In the expansion form of the stress field near a crack tip, the nonsingular and constant stress acting parallel to the crack plane is called the elasticT-term, which is very important in fracture analysis. The elasticT-term of a main crack induced by near tip microcracks is analyzed. The original problem considered consists of a main crack accompanied by near tip microcracks, which are loaded remotely by the first three terms of the crack tip stress field expansion parameterized by stress intensity factorsKI∞,KII∞, and the nonsingular stress, i.e., theT-term,∞. With the principle of superposition, the problem can be reduced to a system of Fredholm integral equations which can be solved numerically. After obtaining the solution, the induced elasticT-term is evaluated. Several typical numerical examples are given and the results are shown in Figures, from which some useful discussion and conclusions are obtained.