Numerical solution of systems of Cauchy Singular Integral Equations with constant coefficients ✩ Maria Carmela De Bonis ∗ , Concetta Laurita Department of Mathematics and Computer Sciences, University of Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy Tel.: +39-0971-205859, Fax: +39-0971-205896 Abstract This paper deals with the numerical solution of a class of systems of Cauchy singular integral equations with constant coefficients. The proposed procedure consists of two basic steps: the first one is to consider a modified problem equivalent to the original one under suitable conditions, the second one is to approximate its solution by means of a vector of polynomial functions. Such array is constructed by applying a quadrature type method, based on Gaussian rules, that leads to solve a determined and well conditioned linear system. The convergence and stability of the method are proved in weighted L 2 spaces. Some numerical tests are also shown. Keywords: Cauchy singular integral equation, quadrature method, Lagrange interpolation 2010 MSC: 65R20, 45E05 1. Introduction The solution of a large class of boundary value problems in physics and engineering can be connected to the solution of systems of singular integral equations. The general theory on this topic has been developed in the funda- mental books [20, 21, 22]. Furthermore, a very wide literature concerning the numerical treatment of a Cauchy singular integral equation is available, espe- cially in the case of equations having constant coefficients and index 0 or 1. In the development of direct methods, particular attention was given to those procedures which use polynomials as trial functions (Galerkin, collocation and ✩ This research was supported by University of Basilicata (local funds) and by PRIN Project 2008 “Equazioni integrali con struttura e sistemi lineari” Protocol No. 20083KLJEZ (first and second author); by GNCS Project 2011 “Tecniche numeriche per problemi di propagazione di onde elastiche in multidomini” (first author). * Corresponding author Email addresses: mariacarmela.debonis@unibas.it (Maria Carmela De Bonis), concetta.laurita@unibas.it (Concetta Laurita) Preprint submitted to Applied Mathematics and Computation January 20, 2012