Numerical Solution Of Integral Equations Research Articles

On the numerical solution of integral equations of the second kind with weakly singular kernels

The purpose of this paper is to introduce the (Toeplitz) quadrature method for solving Fredholm integral equations of the second kind with mildly singular kernels. We are presented some numerical examples for the computation of the error estimate using the MathCad package.

Numerical solution of integral equations of first and second kind in scalar and vector problems of diffraction on a cube

We consider the problem of numerical simulation of the scattering of acoustic and electromagnetic waves on a cube whose edge ha s length up to 8 wave lengths of the incident wave. We describe a scheme using a representation of the boundary integral equation in the form of an operator convolution equation on the symmetry group of the cube. We compare the results of numerical solution of integral equations of first and second kind for scalar and vector problems of diffraction of a plane wave on a cube.

Numerical solution of integral equations with finite part integrals

We obtain convergence rates for several algorithms that solve a class of Hadamard singular integral equations using the general theory of approximations for unbounded operators.

Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane

We develop a set of scalar integral equations that govern the electromagnetic scattering from a two-dimensional (2-D) trough in an infinite perfectly conducting ground plane. We obtain accurate and efficient numerical solution to these equations via the method of moments (MoM). Our numerical implementation compares favorably to popular methods such as the finite element/boundary integral (FE/BI) method, generalized network formulation (GNF), and electric field integral equation (EFIE) techniques.

Performance/dependability modelling using stochastic reward models: transient behaviour

Stochastic Reward Models (SRMs) are commonly used for evaluating combined performance and dependability of fault-tolerant systems. An SRM is composed of a stochastic process, describing the evolution of the system, and a superimposed reward structure, reflecting different performance levels of the system. Evaluating combined transient performance/dependability measures using SRMs leads either to differential or integral equations. This paper discusses the use of SRMs for modelling performance/dependability of fault-tolerant systems and proposes an approach for the numerical solution of integral equations which commonly arise. A bound for the error due to the numerical approximation is obtained. As an example, an n-unit parallel system is analysed numerically in transient state.

Numerical solution of integral equations with logarithmic-, Cauchy- and Hadamard‐type singularities

Numerical solution of integral equations with logarithmic-, Cauchy- and Hadamard‐type singularities

Numerical solution of integral equations with logarithmic-, Cauchy- and Hadamard-type singularities

This study presents an extension of the piecewise quadratic polynomial technique to solve singular integral equations with logarithmic- and Hadamard-type singularities. For completeness and continuity, the evaluation of the weights for logarithmic-, Cauchy- and Hadamard-type singularities are given explicitly. Numerical results concerning logarithmic- and Hadamard-type singular integral equations are compared with the known solutions. The solution of Cauchy-type singular integral equations was discussed in detail in a previous study by Miller and Keer. © 1998 John Wiley & Sons, Ltd.

On the numerical solution of integral equations of the first kind

Article On the numerical solution of integral equations of the first kind was published on January 1, 1998 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 13, issue 3).

Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics

The problem of error estimation in the numerical solution of integral equations that arise in electromagnetics is addressed. The direct method (Green's theorem or field approach) and the indirect method (layer ansatz or source approach) lead to well-known integral equations both of the first kind [electric field integral equations (EFIE)] and the second kind [magnetic field integral equations (MFIE)]. These equations are analyzed systematically in terms of the mapping properties of the integral operators. It is shown how the assumption that field quantities have finite energy leads naturally to describing the mapping properties in appropriate Sobolev spaces. These function spaces are demystified through simple examples which also are used to demonstrate the importance of knowing in which space the given data lives and in which space the solution should be sought. It is further shown how the method of moments (or Galerkin method) is formulated in these function spaces and how residual error can be used to estimate actual error in these spaces. The condition number of all of the impedance matrices that result from discretizing the integral equations, including first kind equations, is shown to be bounded when the elements are computed appropriately. Finally, the consequences of carrying out all computations in the space of square integrable functions, a particularly friendly Sobolev space, are explained.

The axisymmetric problem of a disc in a bimaterial sandwich—The (history dependent) scalar potential case

The axisymmetric potential problem of a disc in a bimaterial sandwich is considered. A (history dependent) medium (2) with relaxation modulus G 2 is sandwiched between a medium (1) with relaxation modulus G 1 The two media are welded together. The interaction of a disc shaped region to a remote applied field is considered. In particular the variation of the potential field close to the rim of the disc as a function of the layer thickness and the material properties of the sandwich is investigated. A variety of asymptotic techniques are used to elucidate properties for both very small and very large values of the layer thickness and the results verified and extended to intermediate thickness by numerical solution of integral equations.

Interaction of sounding electromagnetic waves with elongated defects of the material

We develop a constructive method for the investigation of the vector interaction of electromagnetic waves with bodies. As an example, we consider a system of perfectly conducting cylindrical surfaces (screens) and a dielectric cylinder. The arrangement of scatterers and their cross sections are regarded as arbitrary. Problems of this sort are typical of the nondestructive electromagnetic inspection of elongated defects. They are solved by the method of singular integral equations. Integrals with logarithmic singularities, Cauchy-type kernels, and hypersingular integrals (in the sense of Hadamard) were found by using interpolational quadrature formulas. Numerical solutions of integral equations were obtained by the method of mechanical quadratures. We also present some examples of the calculation of the characteristics of diffraction fields in the far-field zone.

Numerical study of Fredholm integral equations

Two numerical methods for solving Fredholm integral equations of the first kind are examined, namely, a direct quadrature method and a boundary‐integral method. These methods are tested out on integral equations with a range of kernels. A regularization technique which replaces ill‐posed equations of the first kind by well‐posed equations of the second kind was employed to produce meaningful results for comparison purposes. In this way orders of error are computed for the various methods and conclusions are drawn. This investigation should prove helpful in the classroom to students studying the numerical solution of integral equations.

Two-Grid Iteration Methods for Linear Integral Equations of the Second Kind on Piecewise Smooth Surfaces in $\mathbb{R}^3 $

The numerical solution of integral equations of the second kind on surfaces in $\mathbb{R}^3 $ often leads to large linear systems that must be solved by iteration. An especially important class of such equations is boundary integral equation (BIE) reformulations of elliptic partial differential equations; and, in this paper BIEs of the second kind are considered for Laplace’s equation. The numerical methods used are based on piecewise polynomial isoparametric interpolation over the surface and the surface is also approximated by such interpolation. Two-grid iteration methods are considered for (1) integral equations with a smooth kernel function, (2) BIEs over smooth surfaces, and (3) BIEs over piecewise smooth surfaces. In the last case, standard two-grid iteration does not perform well, and a modified two-grid iteration method is proposed and examined empirically.

Parallel solution of Fredholm integral equations of the second kind by accelerated projection methods

Conventional projection methods for the numerical solution of integral equations are serial in structure, and require substantial synchronization which seriously degrades their performance in a parallel environment. The present work focuses on the exploitation and implementation of a new class of computational techniques, known as accelerated projection methods, which are distinctly different from conventional methods, capable of high accuracy and which are particularly appropriate for concurrent computing. The methods are implemented in detail on a multiprocessor for a particular linear integral equation of interest. It is shown that accelerated projection methods possess a high degree of parallelism that leads to extremely efficient parallel algorithms.

Discussion of the Paper by Vardi and Lee

Discussion of the Paper by Vardi and Lee

First-generation SQUID-based nondestructive testing system

The paper is an overview of the results of an investigation of a first-generation SQUID NDT system working on the principle of the detection of magnetic inhomogeneities. The system incorporates a superconducting field generation coil—to apply a static magnetic field perpendicular to the test subject—and a differential two-coil planar pick-up coil connected to a SQUID, acting as the detector. These components are operated in a LHe cryostat in a normal (unscreened) laboratory environment. Test subjects are inspected by passing them beneath the cryostat during computer-controlled data acquisition. Typical experimental results are presented, and postprocessing techniques are outlined. Theoretical descriptions of the static field distortion technique have also been developed, based on the numerical solution of integral equations which model flaws as combinations of magnetic dipoles. The equations are presented, and results are compared with those obtained experimentally. Despite significant limitations it is concluded that potential applications for SQUID NDT exist and further investigation is worthwhile.

Discussion of the Paper by Donoho, Johnstone, Hoch and Stern

Discussion of the Paper by Donoho, Johnstone, Hoch and Stern

REVIEW OF CURRENT STATUS AND TRENDS IN THE USE OF INTEGRAL EQUATIONS IN COMPUTATIONAL ELECTROMAGNETICS

Abstract A summary of the state-of-the-art in the formulation and numerical solution of integral equations in electromagnetics is presented. Following a brief summary of the method of moments, database and software requirements are given for specifying and numerically representing geometries and their electrical characteristics and excitation. While concentrating on the formulation of problems in the frequency domain, issues involved in time domain formulations are are also briefly summarized. Basic principles involved in the formulation of integral equations, including the use of Green's functions and symmetry to simplify the solution procedure, are presented. A class of basis and testing functions is defined which subsumes many of those in common use, and their application to compuational electromagnetics is outlined. Finally, matrix solution methods are summarized. For each of the issues considered, an indication of possible new directions in computational electromagnetics is also given.

Numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points

An integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points (ACP). As a first step a Gaussian quadrature of order n (hence of polynomial accuracy 2n− 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x̄k, determined as roots of appropriate transcedental functions, in order to retain the 2n − 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that xk may be arbitrary and yet the high (2n − 1) accuracy of the Gaussian quadrature is retained.

On numerical solution of integral equations of planar and spatial diffraction problems

On numerical solution of integral equations of planar and spatial diffraction problems