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Eighth-Order Numerov-Type Methods Using Varying Step Length

Author

Listed:
  • Obaid Alshammari

    (Department of Electrical Engineering, College of Engineering, University of Hail, Ha’il 81481, Saudi Arabia)

  • Sondess Ben Aoun

    (Department of Computer Engineering, College of Computer Science and Engineering, University of Hail, Ha’il 81481, Saudi Arabia)

  • Mourad Kchaou

    (Department of Electrical Engineering, College of Engineering, University of Hail, Ha’il 81481, Saudi Arabia)

  • Theodore E. Simos

    (School of Mechanical Engineering, Hangzhou Dianzi University, Er Hao Da Jie 1158, Xiasha, Hangzhou 310018, China
    Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref 32093, Kuwait
    Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
    Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece)

  • Charalampos Tsitouras

    (General Department, National & Kapodistrian University of Athens, Euripus Campus, 34400 Psachna, Greece)

  • Houssem Jerbi

    (Department of Industrial Engineering, College of Engineering, University of Hail, Ha’il 81481, Saudi Arabia)

Abstract

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community.

Suggested Citation

  • Obaid Alshammari & Sondess Ben Aoun & Mourad Kchaou & Theodore E. Simos & Charalampos Tsitouras & Houssem Jerbi, 2024. "Eighth-Order Numerov-Type Methods Using Varying Step Length," Mathematics, MDPI, vol. 12(14), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2294-:d:1440425
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    References listed on IDEAS

    as
    1. Ch. TSITOURAS, 2006. "Explicit Eighth Order Two-Step Methods With Nine Stages For Integrating Oscillatory Problems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 861-876.
    2. Franco, J.M. & Rández, L., 2016. "Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 493-505.
    3. J. M. Franco & L. Rández, 2018. "Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 29(01), pages 1-18, January.
    Full references (including those not matched with items on IDEAS)

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