Minimal-norm solution to the Fredholm integral equations of the first kind via the H-HK formulation

TL;DR

This paper presents a closed-form minimal-norm solution for Fredholm integral equations of the first kind using the H-HK formulation, addressing their ill-posed nature and extending results to non-degenerate cases.

Contribution

It introduces a novel closed-form solution for degenerate kernel equations and demonstrates its extension to non-degenerate equations, linking solutions to matrix equations.

Findings

Solution structure aligns with matrix equations

Method effectively solves degenerate kernel equations

Extensions to non-degenerate equations validated

Abstract

The Fredholm integral equations of the first kind is a typical ill-posed problem, so that it is usually difficult to obtain its analytical minimal-norm solution. This paper gives a closed-form minimal-norm solution for the degenerate kernel equations based on the H-HK formulation. Furthermore, it has been shown that the structure of solutions to degenerate kernel equations and matrix equations are consistent. Subsequently, the obtained results are extended to non-degenerate integral equations. Finally, the validity and applicability of the proposed method are demonstrated by some examples.