On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where integration kernel approximated using quadrature formula. Traditionally, rule used classical polynomial Gauss quadrature. Motivated by observation that given function can be better spline lower degree than single piece higher degree, in this work, we investigate use Gaussian rules for splines method. We show that, continuous kernels, approximate solution linear computed converges exact m→∞, m being number points. Our results also when fixing same points, approximation more accurate rules. non-linear case, considering Hammerstein equations, and present some tests.