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An enhanced wavelet-based numerical scheme with higher convergence order for a class of high-order boundary value problems

Author

Listed:
  • Zhang, Ruimin
  • Yang, Lei
  • Zhu, Hui
  • Li, Linghui
  • Wang, Yahong

Abstract

In this paper, a new numerical scheme is proposed to effectively solve a class of high-order boundary value problems (BVPs) with arbitrary orders and three types of boundary conditions. The method can reduce computational cost. By constructing an enhanced wavelet basis based on Legendre polynomials within a defined reproducing kernel space Wm[0,1], the ε-approximate solution of BVPs can be obtained applying the least squares method. These enhanced wavelets maintain compact support, ensuring superior approximation performance compared to classical wavelets. The convergence of the proposed scheme is characterized by its analytical order, while stability and complexity are also investigated. Specifically, for high-order nonlinear BVPs, the Quasi-Newton method is utilized effectively. We present numerical examples of several high-order linear and nonlinear BVPs with various boundary conditions, including Dirichlet, Neumann, and Robin conditions, to assess the stability and efficiency of the method. The numerical results show that our approach provides more accurate approximations, which are consistent with analytical solutions and show a high order of accuracy, outperforming several existing methods in the literature.

Suggested Citation

  • Zhang, Ruimin & Yang, Lei & Zhu, Hui & Li, Linghui & Wang, Yahong, 2025. "An enhanced wavelet-based numerical scheme with higher convergence order for a class of high-order boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 507(C).
    • Handle: RePEc:eee:apmaco:v:507:y:2025:i:c:s0096300325003285
    DOI: 10.1016/j.amc.2025.129602
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    References listed on IDEAS

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    1. Muhammad Aslam Noor & Syed Tauseef Mohyud-Din, 2008. "Variational Homotopy Perturbation Method for Solving Higher Dimensional Initial Boundary Value Problems," Mathematical Problems in Engineering, Hindawi, vol. 2008, pages 1-11, June.
    2. Yahong Wang & Haili Zhou & Liangcai Mei & Yingzhen Lin & Marjan Uddin, 2022. "A Numerical Method for Solving Fractional Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-8, June.
    3. Xu, M.Q. & Lin, Y.Z. & Wang, Y.H., 2016. "A new algorithm for nonlinear fourth order multi-point boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 163-168.
    4. Faezeh Toutounian & Emran Tohidi & Stanford Shateyi, 2013. "A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, April.
    5. Faezeh Toutounian & Emran Tohidi & Stanford Shateyi, 2013. "A Collocation Method Based on the Bernoulli Operational Matrix for Solving High‐Order Linear Complex Differential Equations in a Rectangular Domain," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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