L p-approximation method for the numerical solution of singular integral equations

The Numerical Solution of Integral Equations on the Half-Line

In this section, we obtain theorems on the approximation of functions of the complex variable dened on closed contours bounding a simply connected domain in the complex plane; the approximations are constructed in the space of continuous functions and in Lebesgue spaces with the use of Lozinskii interpolation polynomials. These results are used to obtain a theoretical justication of the subdomain method for the approximate solution of singular integral equations dened on closed contours of various smoothness. The solvability of the computational scheme of this method is proved by the technique in [1]. Note that issues related to the approximate solution of singular integral equations have been considered in numerous papers (see [2{10] and the bibliography therein). However, the subdomain method for singular integral equations on closed contours other than the unit circle has not been studied. Let be a closed smooth contour bounding a connected domain F + containing the origin, and let F = CnfF + [ g ,w hereC is the entire complex plane. We say that belongs to the class (m) ,w herem is a positive integer, if the Riemann function t = (w), w2 0 =f : jj =1 g, (1 )= 1, 0 (1 )= a1 (a1 > 0), of this contour has continuous :

Several methods for the numerical solution of Cauchy-type singular integral equations, frequently encountered in two-dimensional elasticity, are established. The well known numerical integration rules are properly combined in order to handle numerically Cauchy-type singular integral equations along the integration interval [-1, 1] and for a large variety of weight functions. In this way, such an equation (or a system of such equations) can be numerically sloved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected points of the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation.

The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type

We present a new method for the numerical solution of singular integral equations on the real axis. The method’s value stems from a new formula for the Cauchy integral of a rational function with an oscillatory exponential factor. The inner product of such functions is also computed explicitly. With these tools in hand, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. In specific cases, ideas from Fredholm theory and Riemann–Hilbert problems are used to motivate preconditioners for these singular integral equations. A significant acceleration in convergence is realized for these examples. This presents a useful link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann–Hilbert problem.

Numerical solution of singular integral equations in stress concentration problems

A new approach for investigating the collocation method for singular integral equations is given. Using Banach algebra techniques and a local principle established by I. Gohberg and N. Krupnik under suitable conditions the convergence of the Lagrange- and Mul thopp-collocation method is proved for one-dimensional singular integral equations and systems of such equations with piecewise continuous coefficients having a countable infinite number of discontinuities.

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss‐Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto‐Chebyshev method for the numerical solution of the same class of equations.

Numerial Methods for Volterra Integral Equations with Singular Kernels

Existing Green functions for a conducting wedge in the form of infinite series of orthogonal eigenfunctions are known to suffer from slow and conditional convergence 1) near the source point, due to the singularity of the Green functions at source points, and 2) when the source point is near the surface of the wedge, due to the presence of a nearby image source. As a result, such expansions are unsuited for the exact solution of singular integral equations wherein values of the Green functions at the source point do appear inside the integral. In this paper, alternative expansions are proposed which converge rapidly for any combination of the source and observation points. The main idea is to extract in closed form both the singular term and the nearby image term together with certain simple asymptotic terms. This helps isolate the inherent singularities and, thus, transform the remaining part of the Green function into a rapidly converging series of elementary terms. Numerical examples and case studies illustrate the stability and high accuracy of the new expansions. Application to the solution of an integral equation associated with wave diffraction by perfectly conducting curved strips in the vicinity of a wedge is exemplified.

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

This paper deals with numerical solutions of singular integral equations of the body force method in interaction problems of notches and holes under general loading conditions. The problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the densities of body forces distributed in the x-and y-directions are to be unknown functions. In order to satisfy the boundary conditions along the notches, several types of fundamental density functions are proposed ; then, the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The accuracy of the present analysis is verified by comparing with the results obtained by the previous method. The present method is found to give rapidly converging numerical results for stress distribution along the notch boundaries.

Stability of the numerical procedure for solution of singular integral equations on semi-infinite interval. Application to fracture mechanics

The well-known weighted Galerkin method for the direct numerical solution of Cauchy type singular integral equations of the first and the second kind, based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials, is shown to be identical to the corresponding method based on the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind and the application to the latter of the weighted Galerkin method. This result permits the direct transfer of the whole set of convergence and rate-of-convergence results for the weighted Galerkin method of numerical solution of Fredholm integral equations with regular kernels to the case of Cauchy type singular integral equations. An application to the case where Chebyshev polynomials are used is also made, and the available convergence results for the direct weighted Galerkin method are rederived with essentially no analysis.