IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i14p2294-d1440425.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/14/2294/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/14/2294/
    Download Restriction: no
    ---><---

    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. 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.
    3. 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)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Andrey V. Chukalin & Theodore E. Simos & Charalampos Tsitouras, 2021. "Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits," Mathematics, MDPI, vol. 9(23), pages 1-19, November.
    2. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Tamara V. Karpukhina & Theodore E. Simos & Charalampos Tsitouras, 2021. "Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(21), pages 1-12, October.
    3. Tsitouras, Ch., 2014. "On fitted modifications of Runge–Kutta–Nyström pairs," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 416-423.
    4. Theodore E. Simos, 2024. "A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions," Mathematics, MDPI, vol. 12(4), pages 1-32, February.
    5. Lee, K.C. & Nazar, R. & Senu, N. & Ahmadian, A., 2024. "A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving y′′=f(x,y): Application to Verhulst logistic growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 28-49.
    6. Higinio Ramos & Ridwanulahi Abdulganiy & Ruth Olowe & Samuel Jator, 2021. "A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(7), pages 1-22, March.
    7. Changbum Chun & Beny Neta, 2019. "Trigonometrically-Fitted Methods: A Review," Mathematics, MDPI, vol. 7(12), pages 1-20, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2294-:d:1440425. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.