IDEAS home Printed from https://ideas.repec.org/a/hin/complx/9407656.html
   My bibliography  Save this article

Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs

Author

Listed:
  • Fayyaz Ahmad
  • Shafiq Ur Rehman
  • Malik Zaka Ullah
  • Hani Moaiteq Aljahdali
  • Shahid Ahmad
  • Ali Saleh Alshomrani
  • Juan A. Carrasco
  • Shamshad Ahmad
  • Sivanandam Sivasankaran

Abstract

In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is , where is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.

Suggested Citation

  • Fayyaz Ahmad & Shafiq Ur Rehman & Malik Zaka Ullah & Hani Moaiteq Aljahdali & Shahid Ahmad & Ali Saleh Alshomrani & Juan A. Carrasco & Shamshad Ahmad & Sivanandam Sivasankaran, 2017. "Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs," Complexity, Hindawi, vol. 2017, pages 1-30, May.
    • Handle: RePEc:hin:complx:9407656
    DOI: 10.1155/2017/9407656
    as

    Download full text from publisher

    File URL: http://downloads.hindawi.com/journals/8503/2017/9407656.pdf
    Download Restriction: no
    File URL: http://downloads.hindawi.com/journals/8503/2017/9407656.xml
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2017/9407656?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Ullah, Malik Zaka & Serra-Capizzano, Stefano & Ahmad, Fayyaz, 2015. "An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 249-259.
    2. Eman S. Alaidarous & Malik Zaka Ullah & Fayyaz Ahmad & A.S. Al-Fhaid, 2013. "An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-11, November.
    3. Javidi, M. & Golbabai, A., 2008. "Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 309-313.
    4. Ullah, Malik Zaka & Serra-Capizzano, S. & Ahmad, Fayyaz & Al-Aidarous, Eman S., 2015. "Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 972-987.
    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. Howk, Cory L., 2016. "A class of efficient quadrature-based predictor–corrector methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 394-406.
    2. R. H. Al-Obaidi & M. T. Darvishi, 2022. "Constructing a Class of Frozen Jacobian Multi-Step Iterative Solvers for Systems of Nonlinear Equations," Mathematics, MDPI, vol. 10(16), pages 1-13, August.
    3. Golbabai, A. & Javidi, M., 2009. "A spectral domain decomposition approach for the generalized Burger’s–Fisher equation," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 385-392.
    4. Ahmad, F. & Soleymani, F. & Khaksar Haghani, F. & Serra-Capizzano, S., 2017. "Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 199-211.
    5. Mozafar Rostami & Taher Lotfi & Ali Brahmand, 2019. "A Fast Derivative-Free Iteration Scheme for Nonlinear Systems and Integral Equations," Mathematics, MDPI, vol. 7(7), pages 1-11, July.
    6. Javidi, M. & Golbabai, A., 2009. "A new domain decomposition algorithm for generalized Burger’s–Huxley equation based on Chebyshev polynomials and preconditioning," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 849-857.
    7. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2152-2159.
    8. Ullah, Malik Zaka & Serra-Capizzano, S. & Ahmad, Fayyaz & Al-Aidarous, Eman S., 2015. "Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 972-987.
    9. Ramandeep Behl & Ioannis K. Argyros, 2020. "A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems," Mathematics, MDPI, vol. 8(2), pages 1-21, February.
    10. Vikash Kumar Sinha & Prashanth Maroju, 2023. "New Development of Variational Iteration Method Using Quasilinearization Method for Solving Nonlinear Problems," Mathematics, MDPI, vol. 11(4), pages 1-11, February.

    More about this item

    Statistics

    Access and download statistics

    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:hin:complx:9407656. 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: Mohamed Abdelhakeem (email available below). General contact details of provider: https://www.hindawi.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.