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A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

Author

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  • 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
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
    Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, Ulyanovsk 432027, Russia)

Abstract

In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:504-:d:1334438
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    References listed on IDEAS

    as
    1. 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.
    2. T. E. Simos & Jesus Vigo Aguiar, 2001. "On The Construction Of Efficient Methods For Second Order Ivps With Oscillating Solution," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(10), pages 1453-1476.
    3. T. E. Simos, 1998. "An Eighth Order Exponentially Fitted Method for the Numerical Solution of the Schrödinger Equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 271-288.
    4. Chunfeng Wang & Zhongcheng Wang, 2007. "A P-Stable Eighteenth-Order Six-Step Method For Periodic Initial Value Problems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 419-431.
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