A mixed collocation scheme for solving second kind Fredholm integral equations in [-1,1]

In this paper we propose a suitable combination of two collocation methods based on the zeros of Jacobi polynomials in order to approximate the solution of Fredholm integral equations on [−1, 1]. One of the main interesting aspects of this procedure is that our approach is cheaper than the usual collocation method based on standard Lagrange interpolation using Jacobi zeros. Moreover, we can successfully manage functions with algebraic singularities at the endpoints. The error of the method is comparable with the error of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method is proved, and some numerical tests, which confirm the theoretical estimates, are provided.