A Nyström method for Volterra-Fredholm integral equations with highly oscillatory kernel

In the present paper, we propose a Nyström method for a class of Volterra-Fredholm integral equations containing a fast oscillating kernel. The approximation tool consists of the ℓ− Iterated Boolean sums of Bernstein operators, also known as Generalized Bernstein (GB) operators, based on equally spaced nodes of the interval [−1, 1]. The corresponding GB polynomials associated with any continuous function depend on the additional parameter ℓ, which can be suitably chosen in order to improve the rate of convergence, as the smoothness of the function increases. Hence, the low degree of approximation by the classical Bernstein polynomials or by piecewise polynomials functions, typically based on equispaced nodes, is overcome in some sense. The numerical method we propose here is stable and convergent in the space of the continuous functions equipped with the uniform norm. Error estimates are proved in Hölder-Zygmund type subspaces and some numerical tests confirm the theoretical error estimates.