| Peer-Reviewed

A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations

Received: 25 April 2020     Accepted: 18 May 2020     Published: 23 September 2020
Views:       Downloads:
Abstract

This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.

Published in Mathematics Letters (Volume 6, Issue 2)
DOI 10.11648/j.ml.20200602.12
Page(s) 13-31
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Derivation, Analysis, Fourth-order Ordinary Differential Equations, New numerical Method, Hybrid Techniques, Convergence, Zero Stability, Consistency, Taylor Series, Order 10, Integrators

References
[1] Familua, A. B. and Omole, E. O. (2017) Five points Mono Hybrid point Linear Multistep Method for solving nth Order Ordinary Differential Equations Using Power Series function. Asian Journal of Research and Mathematics, 3 (1), 1-17. http://dx.doi.org/10.9734/ARJOM/2017/31190.
[2] Kayode, S. J, Ige, S. O, Obarhua, F. O. and Omole, E. O. (2018) An Order Six Stormer-cowell-type Method for Solving Directly Higher Order Ordinary Differential Equations. Asian Research Journal of Mathematics, 11 (3), 1-12. http://dx.doi.org/10.9734/ARJOM/2018/44676.
[3] Adoghe, L. O, Ogunware, B. G. and Omole, E. O. (2016) A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations. Journal of Theoretical Mathematics & Application, 6 (3): 67-84.
[4] Areo, E. A. and Omole, E. O. (2015) HALF-Step symmetric continuous hybrid block method for the numerical solutions of fourth order ordinary differential equations, Archives of Applied Science Research, 7 (10): 39-49. www.scholarsresearchlibrary.com.
[5] Sunday, J. Skwame Y. and Huoma I. U. (2015) Implicit One-Step Legendre Polynomial Hybrid Block Method for the Solution of First-Order Stiff Differential Equations. British Journal of Mathematics and Computer Science. 8 (6), 482-491, https://doi.org/10.9734/BJMCS/2015/16252.
[6] Al-Mazmumy, M. Al-Mutairi, A. and Al-Zahrani, K. (2017). An Efficient Decomposition Method for Solving Bratu’s Boundary Value Problem. American Journal of Computational Mathematics, 7, 84-93. https://doi.org/10.4236/ajcm.2017.71007
[7] Atkinson, K. E. (1989) An Introduction to Numerical Analysis,” 2nd Edition, John Wiley and Sons, New York, pp- 284.
[8] M. I. Bhatti, and Bracken, P. (2007) Solutions of Differential Equations in a Bernstein Polynomial Basis. Journal of Computational and Applied Mathematics, Vol. 205, No. 1, pp. 272-280. doi: 10.1016/j.cam.2006.05.002
[9] Adeniran, A. O. & Longe, I. O. (2019) Solving directly second order initial value problems with Lucas polynomial. Journal of Advances in Mathematics and Computer science, 32 (4), 1-7. http://dx.doi.org/10.9734/jamcs/2019/v32i430152.
[10] R. L. Burden, and J. D. Faires, (1992) Numerical Analysis,” Brooks/Cole Publishing Co., Pacific Grove.
[11] Fatunla, S. O. (1991) Block Method for Second Order IVPs. International Journal of Computer Mathematics, 41, 55-63, http://dx.doi.org/10.1080/00207169108804026.
[12] Buckmire, R. (2003) Investigations of Nonstandard, Mickens-Type, Finite-Difference Schemes for Singular Boundary Value Problems in Cylindrical or Spherical Coordinate. Numerical Methods for Partial Differential Equations, 19, 380-398. https://doi.org/10.1002/num.10055.
[13] Lambert, J. D.(1973) Computational Methods in ODEs. John Wiley & Sons, New York.
[14] Harier, and G. Wanner, (1996) Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer-Verlag, New York. doi: 10.1007/978-3-642-05221-7.
[15] Kasim H, Fudziah I. and Norazak S. (2015) Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations, Journal of Mathematical Problems in Engineering, Vol. 15, Article ID 196595, 12 pages.
Cite This Article
  • APA Style

    Ezekiel Olaoluwa Omole, Luke Azeta Ukpebor. (2020). A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Mathematics Letters, 6(2), 13-31. https://doi.org/10.11648/j.ml.20200602.12

    Copy | Download

    ACS Style

    Ezekiel Olaoluwa Omole; Luke Azeta Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Math. Lett. 2020, 6(2), 13-31. doi: 10.11648/j.ml.20200602.12

    Copy | Download

    AMA Style

    Ezekiel Olaoluwa Omole, Luke Azeta Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Math Lett. 2020;6(2):13-31. doi: 10.11648/j.ml.20200602.12

    Copy | Download

  • @article{10.11648/j.ml.20200602.12,
      author = {Ezekiel Olaoluwa Omole and Luke Azeta Ukpebor},
      title = {A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations},
      journal = {Mathematics Letters},
      volume = {6},
      number = {2},
      pages = {13-31},
      doi = {10.11648/j.ml.20200602.12},
      url = {https://doi.org/10.11648/j.ml.20200602.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20200602.12},
      abstract = {This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.},
     year = {2020}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations
    AU  - Ezekiel Olaoluwa Omole
    AU  - Luke Azeta Ukpebor
    Y1  - 2020/09/23
    PY  - 2020
    N1  - https://doi.org/10.11648/j.ml.20200602.12
    DO  - 10.11648/j.ml.20200602.12
    T2  - Mathematics Letters
    JF  - Mathematics Letters
    JO  - Mathematics Letters
    SP  - 13
    EP  - 31
    PB  - Science Publishing Group
    SN  - 2575-5056
    UR  - https://doi.org/10.11648/j.ml.20200602.12
    AB  - This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.
    VL  - 6
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics & Statistics, College of Natural Sciences, Joseph Ayo Babalola University, Osogbo, Nigeria

  • Department of Mathematics, Faculty of Physical Sciences, Ambrose Alli University, Ekpoma, Nigeria

  • Sections