# Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels

## TL;DR

This paper develops discrete modified projection methods using piecewise polynomials and Nyström approximation to solve Urysohn integral equations with Green's function kernels, analyzing convergence and providing numerical results.

## Contribution

It introduces a discrete approach combining orthogonal projection and Nyström approximation for Urysohn equations with Green's function kernels, ensuring optimal convergence.

## Key findings

- Convergence order depends on the quadrature choice.
- Numerical results confirm theoretical convergence rates.
- Method effectively solves integral equations with Green's function kernels.

## Abstract

In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green's function. For $r \geq 0,$ a space of piecewise polynomials of degree $\leq r $ with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr\"{o}m approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.