Superconvergent Nystr & ouml;m Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations

Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree d and of class Cd-1 on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nystrom method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is (2d+2) if d is odd and (2d+3) if d is even. In case of even degrees, we show that the convergence order of the iterated solution increases to (2d+4). Finally, we conduct numerical tests that validate the theoretical findings.