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A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

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  • F. F. Ngwane
  • S. N. Jator

Abstract

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Suggested Citation

  • F. F. Ngwane & S. N. Jator, 2017. "A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems," International Journal of Differential Equations, Hindawi, vol. 2017, pages 1-14, January.
    • Handle: RePEc:hin:jnijde:9293530
    DOI: 10.1155/2017/9293530
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    Cited by:

    1. Lee, K.C. & Nazar, R. & Senu, N. & Ahmadian, A., 2024. "A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving y′′=f(x,y): Application to Verhulst logistic growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 28-49.
    2. Higinio Ramos & Ridwanulahi Abdulganiy & Ruth Olowe & Samuel Jator, 2021. "A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(7), pages 1-22, March.

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