Acceleration of convergence in approximate solutions of Urysohn integral equations with Green's kernels

Consider a non-linear operator equation $x-K(x)=f$, where $f$ is a given function and $K$ is a Urysohn integral operator with Green’s function type kernel defined on $L^{\mathrm{\infty }}[0,1]$. We apply approximation methods based on interpolatory projections onto the approximating space ${\mathcal{X}}_n$, which is the space of piecewise polynomials of even degree with respect to a uniform partition of $[0,1]$. The approximate solutions obtained from these methods demonstrate enhanced accuracy compared to the classical collocation solution for the same equation. Numerical examples are given to support our theoretical results.