A rectangular quadrature method for logarithmically singular integral equations of the first kind (Q1195895)

By extracting the logarithmic singularity, the given singular integral equation is first transformed into an equivalent integral equation whose integrand is regular and periodic. The problem is then discretized by using rectangular quadrature rules and collocation at the quadrature nodes. The authors present a detailed analysis of the discretization error and the stability of the method; it is also shown that Richardson extrapolation accelerates the convergence from \(O(h^ 3)\) to \(O(h^ 5)\). An analysis of the rectangular quadrature rule can also be found in a recent paper by \textit{J. Saranen} [J. Integral Equations Appl. 3, No. 4, 575-600 (1991; Zbl 0747.65100)]; the stability of collocation methods for first-kind integral equations with logarithmic singularity has been investigated using a similar approach by the second author [Math. Comput. 54, No. 189, 139-154 (1990; Zbl 0685.65121)] and by \textit{I. G. Graham} and the second author [J. Aust. Math. Soc., Ser. B 33, No. 1, 39-64 (1991; Zbl 0738.65097)].