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Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs

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  • Franco, J.M.
  • Rández, L.

Abstract

The construction of exponentially fitted (EF) two-step hybrid methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt),exp(−λt)},λ∈height6ptC, or equivalently {sin (ωt), cos (ωt)} when λ=iω,ω∈IR, where λ represents an approximation of the main frequency of the problem. The necessary and sufficient conditions for this class of EF hybrid methods to have algebraic order p are derived. With the help of these order conditions and the EF conditions we construct explicit EF two-step hybrid methods with symmetric nodes and weights and orders six and seven. The numerical experiments carried out with several orbital and oscillatory problems show that the new high order EF two-step hybrid methods are more efficient than other EF and standard codes proposed in the scientific literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:493-505
    DOI: 10.1016/j.amc.2015.10.031
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    Cited by:

    1. 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.
    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. 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.
    4. Rubayyi T. Alqahtani & Theodore E. Simos & Charalampos Tsitouras, 2025. "Ninth-Order Two-Step Methods with Varying Step Lengths," Mathematics, MDPI, vol. 13(8), pages 1-15, April.
    5. Changbum Chun & Beny Neta, 2019. "Trigonometrically-Fitted Methods: A Review," Mathematics, MDPI, vol. 7(12), pages 1-20, December.
    6. 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.

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