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A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems

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)

  • Yuri A. Khakhalev

    (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, National & Kapodistrian University of Athens, GR34400 Euripus Campus, Greece
    Administration of Businesses and Organizations Department, Hellenic Open University, 26335 Patras, Greece)

Abstract

We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge–Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.

Suggested Citation

  • Vladislav N. Kovalnogov & Ruslan V. Fedorov & Yuri A. Khakhalev & Theodore E. Simos & Charalampos Tsitouras, 2021. "A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems," Mathematics, MDPI, vol. 9(16), pages 1-10, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1842-:d:608349
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    Cited by:

    1. Zacharias A. Anastassi & Athinoula A. Kosti & Mufutau Ajani Rufai, 2023. "A Parametric Method Optimised for the Solution of the (2+1)-Dimensional Nonlinear Schrödinger Equation," Mathematics, MDPI, vol. 11(3), pages 1-17, January.

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