| Summary: | 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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