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Article

Solving Fredholm Integral Equations of the First Kind Using a Gaussian Process Model Based on Sequential Design

School of Computer and Information Engineering, Guizhou University of Commerce, Guiyang 550014, China
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Author to whom correspondence should be addressed.
Mathematics 2025, 13(15), 2407; https://doi.org/10.3390/math13152407
Submission received: 3 July 2025 / Revised: 24 July 2025 / Accepted: 24 July 2025 / Published: 26 July 2025

Abstract

In this study, a Gaussian process model is utilized to study the Fredholm integral equations of the first kind (FIEFKs). Based on the H–Hk formulation, two cases of FIEFKs are under consideration with respect to the right-hand term: discrete data and analytical expressions. In the former case, explicit approximate solutions with minimum norm are obtained via a Gaussian process model. In the latter case, the exact solutions with minimum norm in operator forms are given, which can also be numerically solved via Gaussian process interpolation. The interpolation points are selected sequentially by minimizing the posterior variance of the right-hand term, i.e., minimizing the maximum uncertainty. Compared with uniform interpolation points, the approximate solutions converge faster at sequential points. In particular, for solvable degenerate kernel equations, the exact solutions with minimum norm can be easily obtained using our proposed sequential method. Finally, the efficacy and feasibility of the proposed method are demonstrated through illustrative examples provided in this paper.
Keywords: Fredholm integral equations; Tikhonov regularization; Gaussian process model; ill-posed problem; Moore–Penrose pseudoinverse; H–Hk formulation Fredholm integral equations; Tikhonov regularization; Gaussian process model; ill-posed problem; Moore–Penrose pseudoinverse; H–Hk formulation

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MDPI and ACS Style

Qiu, R.; Xu, J.; Xu, M. Solving Fredholm Integral Equations of the First Kind Using a Gaussian Process Model Based on Sequential Design. Mathematics 2025, 13, 2407. https://doi.org/10.3390/math13152407

AMA Style

Qiu R, Xu J, Xu M. Solving Fredholm Integral Equations of the First Kind Using a Gaussian Process Model Based on Sequential Design. Mathematics. 2025; 13(15):2407. https://doi.org/10.3390/math13152407

Chicago/Turabian Style

Qiu, Renjun, Juanjuan Xu, and Ming Xu. 2025. "Solving Fredholm Integral Equations of the First Kind Using a Gaussian Process Model Based on Sequential Design" Mathematics 13, no. 15: 2407. https://doi.org/10.3390/math13152407

APA Style

Qiu, R., Xu, J., & Xu, M. (2025). Solving Fredholm Integral Equations of the First Kind Using a Gaussian Process Model Based on Sequential Design. Mathematics, 13(15), 2407. https://doi.org/10.3390/math13152407

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