A Quasi-Newton reproducing kernel method for nonlinear high-order boundary value problems

Chaoyue Guan; Jian Zhang; Chaoyue Guan; Jian Zhang

doi:10.3934/math.2025661

AIMS Mathematics

2025, Volume 10, Issue 6: 14699-14717. doi: 10.3934/math.2025661

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A Quasi-Newton reproducing kernel method for nonlinear high-order boundary value problems

Chaoyue Guan ^1,2,
Jian Zhang ^{1
,
,}

1.
College of Mathematics and Computer Science, GuangDong Ocean University, 524000 Zhanjiang Guangdong, China
2.
School of Mathematical Science, Harbin Normal University, 150025 Harbin Heilongjiang, China

Received: 05 May 2025 Revised: 18 June 2025 Accepted: 19 June 2025 Published: 26 June 2025
MSC : 90C53, 34A34, 46E22

This paper proposes a novel Quasi-Newton reproducing kernel method (QNRKM) for efficiently solving nonlinear fifth-order two-point boundary value problems (BVPs). The proposed scheme innovatively combines the strengths of the Quasi-Newton method (QNM) and the reproducing kernel method (RKM), forming a hybrid framework that addresses the limitations of traditional kernel-based approaches. In particular, it eliminates the need for the computationally intensive Schmidt orthogonalization process required in conventional RKM, which significantly improves numerical efficiency. A comprehensive and rigorous error analysis is performed to evaluate the performance of the proposed numerical scheme. The theoretical results confirm that the method achieves second-order convergence with respect to both the solution and its first derivative. This convergence is measured in the maximum norm. These findings demonstrate the accuracy and reliability of the proposed method for solving the target class of BVPs. Several numerical examples are also presented to validate its accuracy and convergence. The approach is compared with other methods such as the Homotopy Perturbation method and the Variational Iteration method. The results are analyzed through error tables and figures under different grid sizes. They demonstrate that the proposed method is accurate, convergent, and effective. The numerical results consistently show that it achieves superior precision while maintaining low computational cost.
- reproducing kernel method,
- Quasi-Newton method,
- nonlinear differential equations,
- boundary value problem,
- convergence
Citation: Chaoyue Guan, Jian Zhang. A Quasi-Newton reproducing kernel method for nonlinear high-order boundary value problems[J]. AIMS Mathematics, 2025, 10(6): 14699-14717. doi: 10.3934/math.2025661

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Abstract

This paper proposes a novel Quasi-Newton reproducing kernel method (QNRKM) for efficiently solving nonlinear fifth-order two-point boundary value problems (BVPs). The proposed scheme innovatively combines the strengths of the Quasi-Newton method (QNM) and the reproducing kernel method (RKM), forming a hybrid framework that addresses the limitations of traditional kernel-based approaches. In particular, it eliminates the need for the computationally intensive Schmidt orthogonalization process required in conventional RKM, which significantly improves numerical efficiency. A comprehensive and rigorous error analysis is performed to evaluate the performance of the proposed numerical scheme. The theoretical results confirm that the method achieves second-order convergence with respect to both the solution and its first derivative. This convergence is measured in the maximum norm. These findings demonstrate the accuracy and reliability of the proposed method for solving the target class of BVPs. Several numerical examples are also presented to validate its accuracy and convergence. The approach is compared with other methods such as the Homotopy Perturbation method and the Variational Iteration method. The results are analyzed through error tables and figures under different grid sizes. They demonstrate that the proposed method is accurate, convergent, and effective. The numerical results consistently show that it achieves superior precision while maintaining low computational cost.

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