An RBF-based Nyström method for Second–Kind Fredholm Integral Equations

The aim of this paper is to numerically solve Fredholm Integral Equations (FIEs) of the second kind, when the right-hand side term and the kernel are known only at scattered sample points. The proposed method is of the Nyström type and leverages a cubature rule based on Radial Basis Function (RBF) interpolation, with the Leave-One-Out Cross-Validation (LOOCV) technique used to select the optimal RBF shape parameter. The convergence of the method is proven in the space of continuous functions. Finally, numerical tests demonstrate the performance of the method for various RBF choices and provide direct comparisons with other RBF-based techniques for FIEs available in the literature.