A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green's kernels
Gobinda Rakshit, Shashank K. Shukla, Akshay S. Rane
TL;DR
This work addresses solving the Fredholm integral equation $x - Kx = f$ on $X=C[0,1]$ where $K$ has a Green's-function-type kernel. It develops an interpolatory collocation framework using a projection $P_n$ onto piecewise polynomials of degree $\le 2r$ on a uniform grid with $2r+1$ collocation points per subinterval, and analyzes three schemes: collocation, iterated collocation, and modified collocation (and its iteration). By deriving bounds on $K(I-P_n)$ and $K(I-P_n)K(I-P_n)$ and establishing key divided-difference estimates for the Green-type kernel, the paper proves convergence orders: $\|\phi-\phi_n^C\|_\infty=O(h^{2r+1})$, $\|\phi-\phi_n^S\|_\infty=O(h^{2r+2})$, $\|\phi-\phi_n^M\|_\infty=O(h^{2r+2})$ (with $r\ge1$, or $O(h^3)$ if $r=0$), and $\|\phi-\tilde{\phi}_n^M\|_\infty=O(h^{2r+3})$ (or $O(h^4)$ for $r=0$). These results, encapsulated in Theorems t4 and t5, show that the proposed iteration strategies achieve higher-order convergence for Green-type kernels, informing efficient high-order numerical solutions for such integral equations.
Abstract
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.