Nyström method for the system of multi-dimensional nonlinear Fredholm integral equations of the second kind by the variable transformation

Yanying Ma; Zhenxing Hao; Hongyan Liu; Guodong Wang; Changqing Wang; Yanying Ma; Zhenxing Hao; Hongyan Liu; Guodong Wang; Changqing Wang

doi:10.3934/math.2025777

AIMS Mathematics

2025, Volume 10, Issue 8: 17381-17402. doi: 10.3934/math.2025777

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Research article

Nyström method for the system of multi-dimensional nonlinear Fredholm integral equations of the second kind by the variable transformation

1.
School of Mathematics and Physics, Anhui Jianzhu University, Hefei, Anhui, 230601, China
2.
Operations Research and Data Science Laboratory, Anhui Jianzhu University, Hefei, Anhui, 230601, China
3.
School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou, 550025, China
4.
School of Biomedical Engineering, Anhui Medical University, Heifei, Anhui, 230000, China

Received: 07 April 2025 Revised: 29 June 2025 Accepted: 23 July 2025 Published: 01 August 2025
MSC : 45B05, 45G15, 45L05, 65R20

An efficient numerical scheme is developed to solve the system of multi-dimensional nonlinear Fredholm integral equations (MDNFIEs) of the second kind in terms of the variable transformation technique of Sidi type in conjunction with the trapezoidal quadrature rule. Using the product Nyström method, the considered integral system is discretized into a set of nonlinear algebraic equations. Additionally, a rigorous convergence analysis of the proposed method is provided, thus demonstrating that compared with the classical trapezoidal quadrature rule approach, this scheme achieves a significantly improved convergence rate. Furthermore, numerical examples are presented to validate the efficiency and accuracy of the described method.
- system of multi-dimensional nonlinear Fredholm integral equations,
- Sidi's transformation,
- trapezoidal quadrature rule,
- Nyström method
Citation: Yanying Ma, Zhenxing Hao, Hongyan Liu, Guodong Wang, Changqing Wang. Nyström method for the system of multi-dimensional nonlinear Fredholm integral equations of the second kind by the variable transformation[J]. AIMS Mathematics, 2025, 10(8): 17381-17402. doi: 10.3934/math.2025777

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Abstract

An efficient numerical scheme is developed to solve the system of multi-dimensional nonlinear Fredholm integral equations (MDNFIEs) of the second kind in terms of the variable transformation technique of Sidi type in conjunction with the trapezoidal quadrature rule. Using the product Nyström method, the considered integral system is discretized into a set of nonlinear algebraic equations. Additionally, a rigorous convergence analysis of the proposed method is provided, thus demonstrating that compared with the classical trapezoidal quadrature rule approach, this scheme achieves a significantly improved convergence rate. Furthermore, numerical examples are presented to validate the efficiency and accuracy of the described method.

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