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Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Author

Listed:
  • Vladislav N. Kovalnogov

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia)

  • Ruslan V. Fedorov

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia)

  • Tamara V. Karpukhina

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia)

  • Theodore E. Simos

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
    College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
    Department of Mathematics, University of Western Macedonia, 52100 Kastoria, Greece
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan)

  • Charalampos Tsitouras

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

Abstract

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2756-:d:668437
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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. 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. & Gómez, I., 2014. "Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 643-657.
    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.
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