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A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems

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  • Ramos, Higinio
  • Rufai, M.A.

Abstract

In this article, a third derivative continuous 2-step block Falkner-type method for the general solution of second order boundary value problems of ordinary differential equations (ODEs) with different types of boundary conditions is developed. The approaches of collocation and interpolation are adopted to derive the new Falkner-type method, which is then implemented in a block mode to get approximations at all the grid points simultaneously. This method is said to be a global method since it simultaneously produces a solution over the entire interval, although it may also be categorized as a boundary value method (see Brugnano and Trigiante (1998)). The order and the convergence analysis of the proposed method are studied. The new Falkner-type scheme is applied to solve linear and non-linear systems of second-order boundary value problems of ODEs considering different types of boundary conditions. Numerical results obtained through the implementation of the scheme are very much close to the theoretical solution and found favourably compared with various existing methods in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:165:y:2019:i:c:p:139-155
    DOI: 10.1016/j.matcom.2019.03.003
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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. Ramos, Higinio & Singh, Gurjinder, 2017. "A tenth order A-stable two-step hybrid block method for solving initial value problems of ODEs," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 75-88.
    3. Manni, Carla & Mazzia, Francesca & Sestini, Alessandra & Speleers, Hendrik, 2015. "BS2 methods for semi-linear second order boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 147-156.
    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.
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    Cited by:

    1. Tomar, Saurabh & Dhama, Soniya & Ramos, Higinio & Singh, Mehakpreet, 2023. "An efficient technique based on Green’s function for solving two-point boundary value problems and its convergence analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 408-423.
    2. 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.
    3. 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).
    4. 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.
    5. Nadirah Mohd Nasir & Zanariah Abdul Majid & Fudziah Ismail & Norfifah Bachok, 2019. "Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy," Mathematics, MDPI, vol. 7(11), pages 1-16, November.
    6. Tafakkori–Bafghi, M. & Loghmani, G.B. & Heydari, M., 2022. "Numerical solution of two-point nonlinear boundary value problems via Legendre–Picard iteration method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 133-159.
    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.
    8. Mohd Nasir, Nadirah & Abdul Majid, Zanariah & Ismail, Fudziah & Bachok, Norfifah, 2021. "Direct integration of the third-order two point and multipoint Robin type boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 411-427.
    9. Mufutau Ajani Rufai, 2022. "An Efficient Third-Derivative Hybrid Block Method for the Solution of Second-Order BVPs," Mathematics, MDPI, vol. 10(19), pages 1-15, October.

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