Projection methods based on spline quasi-interpolation for Urysohn integral equations

TL;DR

This paper introduces high-order projection methods using spline quasi-interpolation for solving nonlinear Urysohn integral equations, demonstrating their effectiveness through theoretical convergence analysis and numerical tests.

Contribution

It develops novel spline quasi-interpolating projection methods with proven high convergence orders for nonlinear integral equations, including implementation and comparison with collocation methods.

Findings

Methods achieve convergence orders of 2d+2 (odd d) and 2d+3 (even d).

Numerical results confirm theoretical convergence rates.

Comparison shows advantages over existing collocation approaches.

Abstract

In this paper we propose projection methods based on spline quasi-interpolating projectors of degree $d$ and class $C^{d-1}$ on a bounded interval for the numerical solution of nonlinear integral equations. We prove that they have high order of convergence $2d+2$ if $d$ is odd and $2d+3$ if $d$ is even. We also present the implementation details of the above methods. Finally, we provide numerical tests, that confirm the theoretical results. Moreover, we compare the theoretical and numerical results with those obtained by using a collocation method based on the same spline quasi-interpolating projectors.