IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i10p1771-d427643.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/10/1771/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/10/1771/
    Download Restriction: no
    ---><---

    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 & 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.
    4. 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.
    5. 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.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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.
    3. 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.
    4. 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.
    5. Emilio Defez & Javier Ibáñez & José M. Alonso & Michael M. Tung & Teresa Real-Herráiz, 2021. "On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem," Mathematics, MDPI, vol. 9(18), pages 1-18, September.
    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. 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).
    8. 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.
    9. Denis Butusov, 2021. "Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
    10. Modebei, Mark I. & Adeniyi, Rapheal B. & Jator, Samuel N. & Ramos, Higinio, 2019. "A block hybrid integrator for numerically solving fourth-order Initial Value Problems," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 680-694.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. Michael M. Tung & Emilio Defez & Javier Ibáñez & José M. Alonso & Julia Real-Herráiz, 2022. "A Matrix Spline Method for a Class of Fourth-Order Ordinary Differential Problems," Mathematics, MDPI, vol. 10(16), pages 1-18, August.
    18. Y. D. Jikantoro & F. Ismail & N. Senu & Z. B. Ibrahim, 2018. "A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(1), pages 151-167, March.
    19. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1771-:d:427643. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.