Weighted polynomial approximation methods for Cauchy singular integral equations in the non-periodic case

Uwe Weber

ISBN 978-3-89722-352-3
115 pages, year of publication: 2000
price: 40.50 €
Weighted polynomial approximation methods for Cauchy singular integral equations in the non-periodic case

Abstract
In the present paper a new approach to the numerical solution of Cauchy singular integral equations on the interval by collocation and Galerkin methods is considered. Both methods are based on weighted orthogonal polynomials. The main advantage of our approach, in particular of the collocation method, is the fact that in contrast to usual methods its construction does not depend on the concrete equation and requires less preprocessing. Furthermore, it can also be applied to the system case. On the basis of Banach algebra methods, necessary and sufficient stability conditions are derived, where from the coefficients of the operator only piecewise continuity is required. In a scale of Sobolev spaces we can prove results on convergence rates. Furthermore, in the case of the collocation method we discuss some computational aspects to derive effective algorithms for the fast solution of the approximate equations and present numerical results. In the case of equations perturbed by an integral operator with smooth kernel, we consider a quadrature method and two fast algorithms that allow to make use of the fast solution of the unperturbed equation by the collocation method.