An iterative method for solving Fredholm integral equations of the first kind

TL;DR

This paper analyzes the convergence of an iterative scheme for stably solving Fredholm integral equations of the first kind with noisy data, using finite-dimensional approximations and a specific iterative formula.

Contribution

It provides a convergence analysis for a new iterative method involving finite-dimensional approximations for solving Fredholm integral equations of the first kind.

Findings

The iterative scheme converges under specified conditions.

Finite-dimensional approximations ensure stability with noisy data.

The method effectively solves integral equations of the first kind.

Abstract

The purpose of this paper is to give a convergence analysis of the iterative scheme: \bee u_n^\dl=qu_{n-1}^\dl+(1-q)T_{a_n}^{-1}K^*f_\dl,\quad u_0^\dl=0,\eee where $T:=K^*K,\quad T_a:=T+aI,\quad q\in(0,1),\quad a_n:=\alpha_0q^n, \alpha_0>0,$ with finite-dimensional approximations of $T$ and $K^*$ for solving stably Fredholm integral equations of the first kind with noisy data.