On an approximate solution of integral equations with a strong singularity in the kernel
The author considers a hypersingular integral equation over the interval perturbed by an integral operator with smooth kernel function. Multiplying the integral operator by invertible operators, she obtains two equivalent equations, a Fredholm equation of the second kind and a Cauchy singular equation. She applies the theory of these zero-order operators and formulates the Noether theorem (Fredholm's alternative) for the perturbed hypersingular equation. Finally, she introduces uniform grids, and, discretizing the hypersingular part by piecewise constant collocation and the smooth perturbation by quadrature, she obtains a numerical method for the approximate solution. For this method, she proves convergence.