A Nyström method for integral equations of the second kind with fixed singularities based on a Gauss-Jacobi-Lobatto quadrature rule

The Gauss-Lobatto quadrature rule for integration over the interval [−1,1], relative to a Jacobi weight function wα,β (t) = (1−t)α(1+t)β , α,β > −1, is considered and an error estimate for functions belonging to some Sobolev-type subspaces of the weighted space L1 wα,β ([−1,1]) is proved. Then, a Nyström type method based on a modified version of this quadrature formula is proposed for the numerical solution of integral equations of the second kind with kernels having fixed singularities at the endpoints of the integration interval and satisfying proper assumptions. The stability and the convergence of the proposed modified Nyström method in suitable weighted spaces are proved and confirmed through some numerical tests.