Anti-Gauss cubature rules with applications to Fredholm integral equations on the square

Abstract: The purpose of this paper is to develop the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of Fredholm integral equations of the second-kind is also explored. The stability, convergence, and conditioning of the proposed Nystr\"om-type method are studied. The numerical solution of the resulting dense linear system is also investigated and several numerical tests are presented.