Prössdorf, Siegfried; Saranen, J.; Sloan, I. H. A discrete method for the logarithmic-kernel integral equation on an open arc. (English) Zbl 0782.65159 J. Aust. Math. Soc., Ser. B 34, No. 4, 401-418 (1993). The third author and B. J. Burn [J. Integral Equations Appl. 4, No. 1, 117-151 (1992; Zbl 0760.65131)] introduced a family of quadrature methods for solving the singular integral equation \[ (-1/\pi)\int_ \Gamma v(y)\log| x-y| ds_ y = g(x),\quad x\in\Gamma, \] on a closed curve \(\Gamma\). The present paper analyses the adaption of this approach to the case where \(\Gamma\) is an open (smooth) arc: here, the quadrature methods are used, after employing the well-known cosine substitution, to discretize a certain Petrov-Galerkin approximation. It is shown that this family of quadrature methods includes schemes of arbitrary order \(p\). As an illustration, the authors describe a stable \({\mathcal O}(h^ 3)\) method on a uniform mesh. Reviewer: H.Brunner (St.John’s) Cited in 1 ReviewCited in 4 Documents MSC: 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) Keywords:Symm’s integral equation; open arc; cosine substitution; logarithmic-kernel integral equation of the first kind; quadrature methods; singular integral equation; Petrov-Galerkin approximation Citations:Zbl 0760.65131 PDFBibTeX XMLCite \textit{S. Prössdorf} et al., J. Aust. Math. Soc., Ser. B 34, No. 4, 401--418 (1993; Zbl 0782.65159) Full Text: DOI