Minimal-norm solution to the Fredholm integral equations of the first kind via the H-HK formulation
Renjun Qiu, Ming Xu, Wei Qu
TL;DR
The paper tackles the ill-posed Fredholm integral equations of the first kind by deriving a closed-form minimal-norm solution for degenerate kernels using the H-$H_K$ formulation. It develops an operator-theoretic framework in which $k(x,t)=\sum_{i=1}^n g_i(x) h_i(t)$ leads to a reproducing-kernel Hilbert space structure for $R(L)$ with kernel $K(x,x')=G^T(x) H G(x')$, and establishes an isometric isomorphism between $N(L)^{\perp}$ and $R(L)$. The main result provides $u^{\dagger}(t)=(H^{-1} A^{-1} F)^T H(t)$ (with $\|u^{\dagger}\|^2=(A^{-1}F)^T H^{-1}(A^{-1}F)$) plus a general solution form $u(t)=u^{\dagger}(t)+\sum_i c_i \varphi_i(t)$, and extends the method to non-degenerate kernels (NDKFIE). Through diverse illustrative examples, including non-invertible $A$ and a backward-heat-derived ND problem, the approach is shown to be accurate, consistent with matrix-structure insights, and applicable without regularization in these cases.
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.