Fredholm integral equations with non-smooth kernels in weighted spaces: Nyström approximations, stability and convergence

Abstract

This paper deals with the numerical solution of Fredholm integral equations of the second kind with endpoint algebraic singularities and with a kernel of Green’s function type. Novel Nyström schemes employing the Gauss quadrature rule are presented. These methods take into account the lack of smoothness along the diagonal of the kernel and may recover the full convergence rate of smooth kernels. A complete analysis of the stability and convergence is provided, and several numerical tests that illustrate the efficiency and accuracy of various approaches are considered.