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Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions

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  • 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)

  • Andrey V. Chukalin

    (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 Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
    Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, China)

  • Charalampos Tsitouras

    (General Department, Euripus Campus, National & Kapodistrian University of Athens, 34400 Athens, Greece
    Administration of Businesses and Organizations Department, Hellenic Open University, 26335 Patras, Greece)

Abstract

The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated Dormand–Prince pair also belongs. The chosen family comes with coefficients that all depend on five free parameters. These latter parameters are tuned in a way to furnish a new method that performs best on a couple of oscillators. Then, we observe that this trained pair outperforms other well known methods in the relevant literature in a standard set of problems with periodic solutions. This is remarkable since no special property holds such as high phase-lag order or an extended interval of periodicity.

Suggested Citation

  • Vladislav N. Kovalnogov & Ruslan V. Fedorov & Andrey V. Chukalin & Theodore E. Simos & Charalampos Tsitouras, 2021. "Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions," Mathematics, MDPI, vol. 9(18), pages 1-11, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2306-:d:638511
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Arsen Palestini, 2022. "Preface to the Special Issue “Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics”," Mathematics, MDPI, vol. 10(10), pages 1-2, May.

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