Anti-Gauss Cubature Rules with Applications to Fredholm Integral Equations on the Square

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 Nyström-type method are studied. The numerical solution of the resulting dense linear system is also investigated, and several numerical tests are presented.