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Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy

Author

Listed:
  • Nadirah Mohd Nasir

    (Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia)

  • Zanariah Abdul Majid

    (Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia)

  • Fudziah Ismail

    (Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia)

  • Norfifah Bachok

    (Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia)

Abstract

This study is intended to evaluate numerically the solution of second order boundary value problems (BVPs) subject to mixed boundary conditions using a direct method. The mixed set of boundary conditions is subsumed under Type 1: mixed boundary conditions of Dirichlet and Robin and Type 2: mixed boundary conditions of Robin and Neumann. The direct integration procedure will compute the solutions at two values concurrently within a block with a fixed step size. The shooting technique adapted to the derivative free Steffensen method is employed as the iterative strategy to generate the new initial estimates. Four numerical examples are given to measure the efficiency and effectiveness of the developed numerical scheme of order six. The computational comparison indicates that the proposed method gives favorably competitive performance compared to the existing method in terms of accuracy, total function calls, and time saving.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1075-:d:284910
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    References listed on IDEAS

    as
    1. 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.
    2. Nadirah Mohd Nasir & Zanariah Abdul Majid & Fudziah Ismail & Norfifah Bachok, 2018. "Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-12, October.
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