A method of numerical solution of cauchy-type singular integral equations with generalized kernels and arbitrary complex singularities

A numerical method is proposed for the approximate solution of a Cauchy-type singular integral equation (or an uncoupled system of such equations) of the first or the second kind and with a generalized kernel, in the sense that, besides the Cauchy singular part, the kernel has also a Fredholm part presenting strong singularities when both its variables tend to the same end-point of the integration interval. In this case any type of real or generally complex singularities in the unknown function of the integral equation may be present near the end-points of the integration interval. The method proposed consists simply in approximating the integrals in the integral equation by using an appropriate numerical integration rule with generally complex abscissas and weights, followed by the application of the resulting approximate equation at properly selected complex collocation points lying outside the integration interval. Although no proof of the convergence of the method seems possible, this method was seen to exhibit good convergence to the results expected in an example treated.