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An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

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  • Ramos, Higinio
  • Singh, Gurjinder
  • Kanwar, V.
  • Bhatia, Saurabh

Abstract

In this paper, firstly a rational one-parameter family of Falkner-type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations. The proposed family of methods has second algebraic order of convergence. Imposing that the principal term of the local truncation error of the proposed family vanishes, we get an expression for the free parameter at the grid point (xn, yn). By substituting this value of the free parameter in the family, a new rational third order method is obtained. Further, by combining the third order method with any member of the second order family, their variable step-size formulation as an embedded pair is considered. Some numerical experiments are given to illustrate the performance and efficiency of the proposed methods.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:291:y:2016:i:c:p:39-51
    DOI: 10.1016/j.amc.2016.06.033
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    References listed on IDEAS

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    1. Ramos, Higinio & Singh, Gurjinder & Kanwar, V. & Bhatia, Saurabh, 2015. "Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in variable step-size formulation," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 796-805.
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    Cited by:

    1. 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.
    2. Denis Butusov, 2021. "Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
    3. Reem Allogmany & Fudziah Ismail, 2020. "Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications," Mathematics, MDPI, vol. 8(10), pages 1-16, October.
    4. Ramos, Higinio & Rufai, M.A., 2019. "A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 139-155.
    5. Singh, Gurjinder & Garg, Arvind & Kanwar, V. & Ramos, Higinio, 2019. "An efficient optimized adaptive step-size hybrid block method for integrating differential systems," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    6. 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.
    7. Higinio Ramos & Samuel N. Jator & Mark I. Modebei, 2020. "Efficient k -Step Linear Block Methods to Solve Second Order Initial Value Problems Directly," Mathematics, MDPI, vol. 8(10), pages 1-17, October.

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