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Ninth-Order Two-Step Methods with Varying Step Lengths

Author

Listed:
  • Rubayyi T. Alqahtani

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia)

  • Theodore E. Simos

    (Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref 32093, Kuwait)

  • Charalampos Tsitouras

    (General Department, National & Kapodistrian University of Athens, GR34-400 Euripus Campus, Greece)

Abstract

This study investigates a widely recognized ninth-order numerical technique within the explicit two-step family of methods (a.k.a. hybrid Numerov-type methods). To boost its performance, we incorporate an economical step-size control algorithm that, after each iteration, either preserves the current step length, reduces it by half, or doubles it. Any additional off-grid points needed by this strategy are computed using a local interpolation routine. Indicative numerical experiments confirm the substantial efficiency gains realized by this method. It is particularly adept at resolving differential equations with oscillatory dynamics, delivering high precision with fewer function evaluations. Furthermore, a detailed Mathematica implementation is supplied, enhancing usability and fostering further research in the field. By simultaneously improving computational efficiency and accuracy, this work offers a significant contribution to the numerical analysis community.

Suggested Citation

  • Rubayyi T. Alqahtani & Theodore E. Simos & Charalampos Tsitouras, 2025. "Ninth-Order Two-Step Methods with Varying Step Lengths," Mathematics, MDPI, vol. 13(8), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1257-:d:1632708
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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. 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.
    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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

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