The numerical solution of Cauchy singular integral equations with additional fixed singularities

In this paper we propose a quadrature method for the numerical solution of Cauchy singular integral equations with additional fixed singularities. The unknown function is approximated by a weighted polynomial which is the solution of a finite dimensional equation obtained discretizing the involved integral operators by means of a Gauss-Jacobi quadrature rule. Stability and convergence results for the proposed procedure are proved. Moreover, we prove that the linear systems one has to solve, in order to determine the unknown coefficients of the approximate solutions, are well conditioned. The efficiency of the proposed method is shown through some numerical examples

A Nyström method for integral equations with fixed singularities of Mellin type in weighted Lp spaces

We consider integral equations of the second kind with fixed singularities of Mellin type. According to the behavior of the Mellin kernel, we first determine suitable weighted Lp spaces where we look for the solution. Then, for its approximation, we propose a numerical method of Nyström type based on a Gauss–Jacobi quadratura formula. Actually, a slight modification of the classical procedure is introduced in order to prove convergence results in weighted Lp spaces. Moreover, a preconditioning technique allows us to solve well conditioned linear systems. We show the efficiency of the proposed method through some numerical tests

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described