We will discuss a class of fast algorithms for linear and nonlinear integral equations. These are two-level algorithms based on the classic Atkinson-Brakhage method from the 1970s. We will present more efficient approach which uses a matrix-free Newton-Krylov iteration on the coarse mesh and does the fine-to-coarse intergrid transfer with an average. We will then apply the approach to the Ornstein-Zernike (OZ) equations for atomic fluids and some extensions of the OZ equations for molecular fluids.