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Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Author

Listed:
  • Reem Allogmany

    (Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah P.O. Box 344, Saudi Arabia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia)

  • Fudziah Ismail

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

Abstract

Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1771-:d:427643
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    References listed on IDEAS

    as
    1. Lee Ken Yap & Fudziah Ismail & Norazak Senu, 2014. "An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-9, May.
    2. 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.
    3. 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.
    4. 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.
    5. M. Mechee & N. Senu & F. Ismail & B. Nikouravan & Z. Siri, 2013. "A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, June.
    6. D. Jikantoro, Y. & Ismail, F. & Senu, N. & Ibrahim, Z.B., 2018. "Hybrid methods for direct integration of special third order ordinary differential equations," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 452-463.
    Full references (including those not matched with items on IDEAS)

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