Abstract: The high accuracy Nyström methods are presented for solving weakly singular integral equations of the second kind based on the quadrature rules by A.Sidi and M.Israeli in [4]. The asymptotics of the error of the approximations are shown, that is, by using extrapolations, the accuracies can be improved, which corrects F.Chatelin's point of view that the extrapolation for non smooth kernel is no longer founded theoretically.

Key words: weakly singular Fredholm integral equation of the second kind, Nytröm method, periodic transformation