Numerical solution of linear Fredholm integral equations of the second kind based on optimal interpolation formulas with derivatives

Background: Linear Fredholm integral equations of the second kind are fundamental in modeling various scientific and engineering phenomena. Traditional numerical methods for solving these equations often rely solely on nodal function values for approximation. However, these conventional approaches may struggle to accurately capture the local behavior and smoothness of the unknown solution, potentially limiting their approximation precision and numerical stability in complex scenarios. Purpose: The primary objective of this study is to develop a novel and highly accurate numerical scheme that significantly enhances approximation precision. We aim to achieve this by effectively incorporating both function values and their higher-order derivatives at discrete nodes, thereby fully utilizing the inherent smoothness of the solution. Method: To accomplish this, we propose an innovative approach applying an optimal interpolation formula with derivatives specifically constructed within the Sobolev space. By utilizing the exact coefficients of this optimal interpolation formula, we systematically transform the continuous integral equation into a finite, closed system of linear algebraic equations. This transformation is achieved through a rigorous collocation approach. The novelty lies in our derivative-based scheme, which, unlike standard techniques, actively integrates higher-order derivative information into the discrete approximation process. Results: Our theoretical analysis successfully establishes strict error bounds, confirming the mathematical soundness of the proposed scheme. The integration of derivative information leads to superior convergence rates and enhanced numerical stability compared to traditional methods. Furthermore, computational experiments on a numerical example with a known exact solution demonstrate that our method reconstructs the solution with exceptionally high fidelity and minimal computational error, validating our theoretical claims. Implication: Consequently, this work provides a highly robust and reliable computational tool for advanced mathematical and engineering applications. It establishes an improved standard among numerical solvers for integral equations where preserving local behavior and derivative information is critical for maintaining overall numerical accuracy.