A quadrature method for the Cauchy singular integral equations (Q1914811)

The author considers a quadrature method for the Cauchy singular integral equation \[ au(t) + b\int^1_0 {u(s) \over s-t} ds=f(t), \quad t \in (0,1), \tag{1} \] with real constant coefficients \(a\) and \(b\). The method is based on the composite rectangular rule and on a mesh grading transformation, which yields fast convergence of the approximate solution by smoothing the singularities of the exact solution. A stability and convergence analysis is given by the use of local Mellin transformation techniques. The stability relies on a strong ellipticity result which, however, is stated as a conjecture and is only partially justified by numerical experiments. The paper presents numerical results, including the case of variable coefficients in equation (1).