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A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

Author

Listed:
  • Higinio Ramos

    (Department of Applied Mathematics, Universidad de Salamanca, 37008 Salamanca, Spain
    These authors contributed equally to this work.)

  • Ridwanulahi Abdulganiy

    (Distance Learning Institute, University of Lagos, Lagos Mainland 101017, Nigeria
    These authors contributed equally to this work.)

  • Ruth Olowe

    (Department of Mathematics, University of Lagos, Lagos Mainland 101017, Nigeria
    These authors contributed equally to this work.)

  • Samuel Jator

    (Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA
    These authors contributed equally to this work.)

Abstract

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:713-:d:524075
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    References listed on IDEAS

    as
    1. Ch. TSITOURAS, 2006. "Explicit Eighth Order Two-Step Methods With Nine Stages For Integrating Oscillatory Problems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 861-876.
    2. Li, Jiyong, 2018. "Trigonometrically fitted three-derivative Runge–Kutta methods for solving oscillatory initial value problems," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 103-117.
    3. 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.
    4. Ramos, Higinio & Rufai, M.A., 2018. "Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 231-245.
    5. Ramos, Higinio & Singh, Gurjinder & Kanwar, V. & Bhatia, Saurabh, 2016. "An efficient variable step-size rational Falkner-type method for solving the special second-order IVP," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 39-51.
    6. F. F. Ngwane & S. N. Jator, 2015. "Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm," International Journal of Differential Equations, Hindawi, vol. 2015, pages 1-15, November.
    7. Li, Jiyong, 2017. "A family of improved Falkner-type methods for oscillatory systems," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 345-357.
    8. 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.
    9. T. E. Simos, 2002. "DISSIPATIVE TRIGONOMETRICALLY-FITTED METHODS FOR SECOND ORDER IVPsWITH OSCILLATING SOLUTION," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 13(10), pages 1333-1345.
    10. Samuel N. Jator & Kindyl L. King, 2018. "Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-15, June.
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