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Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems

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  • Ramos, Higinio
  • Rufai, M.A.

Abstract

This paper is devoted to the development and analysis of a modified family of Falkner-type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty stability regions for k=2,3,4. Some numerical test are presented to illustrate the efficiency of the proposed family.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:333:y:2018:i:c:p:231-245
    DOI: 10.1016/j.amc.2018.03.098
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    References listed on IDEAS

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    1. 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.
    2. Mazzia, Francesca & Nagy, A.M., 2015. "A new mesh selection strategy with stiffness detection for explicit Runge–Kutta methods," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 125-134.
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    Citations

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    Cited by:

    1. Janez Urevc & Miroslav Halilovič, 2021. "Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs," Mathematics, MDPI, vol. 9(2), pages 1-21, January.
    2. Ramos, Higinio & Singh, Gurjinder, 2022. "Solving second order two-point boundary value problems accurately by a third derivative hybrid block integrator," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    3. Denis Butusov, 2021. "Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
    4. 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.
    5. Khalsaraei, Mohammad Mehdizadeh & Shokri, Ali & Ramos, Higinio & Heydari, Shahin, 2021. "A positive and elementary stable nonstandard explicit scheme for a mathematical model of the influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 397-410.
    6. 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.
    7. Kinda Abuasbeh & Sania Qureshi & Amanullah Soomro & Muath Awadalla, 2023. "An Optimal Family of Block Techniques to Solve Models of Infectious Diseases: Fixed and Adaptive Stepsize Strategies," Mathematics, MDPI, vol. 11(5), pages 1-23, February.
    8. 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.
    9. 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.
    10. 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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    1. 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.
    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.
    3. 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.
    4. Amodio, P. & Iavernaro, F. & Mazzia, F. & Mukhametzhanov, M.S. & Sergeyev, Ya.D., 2017. "A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 24-39.
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    6. 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.
    7. 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.

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