Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 0$, we employ the interpolatory projection at $2r + 1$ collocation points (not necessarily Gauss points) onto a space of piecewise polynomials of degree $\leq 2r$ with respect to a uniform partition of $[0, 1]$. Previous researchers have established that, in the case of smooth kernels with piecewise polynomials of even degree, iteration in the collocation method and its variants improves the order of convergence by projection methods. In this article, we demonstrate the improvement in order of convergence by modified collocation method when the kernel is of Green's function type.

翻译：考虑线性算子方程 $x - Kx = f$，其中 $f$ 已知，$K$ 为定义在 $C[0, 1]$ 上具有格林函数型核的Fredholm积分算子。对于 $r \geq 0$，我们在 $2r + 1$ 个配置点（不一定是高斯点）上采用插值投影，投影到关于 $[0, 1]$ 的一致剖分上次数 $\leq 2r$ 的分段多项式空间。先前的研究者已证明，对于光滑核及偶次分段多项式的情形，配置法及其变体中的迭代能够提升投影方法的收敛阶。本文证明了当核为格林函数类型时，修正配置法同样能实现收敛阶的改进。