Approximating the singular integrals of Cauchy type with weight function on the interval

It is known that the solutions of characteristic singular integral equations (SIEs) are expressed in terms of singular integrals of Cauchy type with weight functions w(x) = ( 1+x)(nu)(1-x)(mu), where nu = +/- 1/2, mu = +/- 1/2. New quadrature formulas (QFs) are presented to approximate the singular integrals (SIs) of Cauchy type for all solutions of characteristic SIE on the interval left prerpendicular-1, 1right perpendicular. Linear spline interpolation, modified discrete vortex method and product quadrature rule are utilized to construct the QFs. Estimation of errors are obtained in the classes of functions H-alpha(inverted right perpendicular-1, 1inverted left perpendicular, A) and C-1 (left prerpendicular-1. 1right prerpendicular). It is found that the numerical results are very stable even for the cases of semi-bounded and unbounded solutions of singular integral equation of the first kind. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.