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Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations

Received: 21 April 2020     Accepted: 30 April 2020     Published: 19 May 2020
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Abstract

Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 6, Issue 1)

This article belongs to the Special Issue Computational Mathematics

DOI 10.11648/j.ijamtp.20200601.12
Page(s) 7-13
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

Accuracy, Adomian Decomposition Method, Convergence, Differential Equation, Efficiency, Initial Value Problem, Successive Approximation Method

References
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[2] Y. Ansari, A. Shaikh, and S. Qureshi. Error bounds for a numerical scheme with reduced slope evaluations, J. Appl. Environ. Biol. Sci., 8 (7), 2018.
[3] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.
[4] M. E. Davis, Numerical methods and modeling for chemical engineers, Courier Corporation, 2013.
[5] S. E. Fadugba, Numerical technique via interpolating function for solving second order ordinary differential equations, Journal of Mathematics and Statistics, 1 (2): 1-6, 2019.
[6] S. E. Fadugba and A. O. Ajayi, Comparative study of a new scheme and some existing methods for the solution of initial value problems in ordinary differential equations, International Journal of Engineering and Future Technology, 14: 47-56, 2017.
[7] S. Fadugba and B. Falodun, Development of a new one-step scheme for the solution of initial value problem (IVP) in ordinary differential equations, International Journal of Theoretical and Applied Mathematics, 3: 58–63, 2017.
[8] S. E. Fadugba and J. T. Okunlola, Performance measure of a new one-step numerical technique via interpolating function for the solution of initial value problem of first order differential equation, World Scientific News, 90: 77–87, 2017.
[9] S. E. Fadugba and T. E. Olaosebikan, Comparative study of a class of one-step methods for the numerical solution of some initial value problems in ordinary differential equations, Research Journal of Mathematics and Computer Science, 2: 1-11, 2018, DOI: 10.28933/rjmcs-2017-12-1801.
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[19] Y. Q. Hasan and L. M. Zhu, Modified Adomian decomposition method for singular initial value problems in the second order ordinary differential equations. Surveys in Mathematics and its Applications, 3: 183-193, 2008.
[20] S. E. Fadugba and J. O. Idowu, Analysis of the properties of a third order convergence numerical method derived via transcendental function of exponential form, International Journal of Applied Mathematics and Theoretical Physics, 5: 97-103, 2019.
[21] J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley & Sons, Inc., New York, 1991.
[22] S. Qureshi and S. E. Fadugba, Convergence of a numerical technique via interpolating function to approximate physical dynamical systems, Journal of Advanced Physics, 7: 446-450, 2018.
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Cite This Article
  • APA Style

    Fadugba Sunday Emmanuel, Adebayo Kayode James, Ogunyebi Segun Nathaniel, Okunlola Joseph Temitayo. (2020). Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations. International Journal of Applied Mathematics and Theoretical Physics, 6(1), 7-13. https://doi.org/10.11648/j.ijamtp.20200601.12

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    ACS Style

    Fadugba Sunday Emmanuel; Adebayo Kayode James; Ogunyebi Segun Nathaniel; Okunlola Joseph Temitayo. Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations. Int. J. Appl. Math. Theor. Phys. 2020, 6(1), 7-13. doi: 10.11648/j.ijamtp.20200601.12

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    AMA Style

    Fadugba Sunday Emmanuel, Adebayo Kayode James, Ogunyebi Segun Nathaniel, Okunlola Joseph Temitayo. Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations. Int J Appl Math Theor Phys. 2020;6(1):7-13. doi: 10.11648/j.ijamtp.20200601.12

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  • @article{10.11648/j.ijamtp.20200601.12,
      author = {Fadugba Sunday Emmanuel and Adebayo Kayode James and Ogunyebi Segun Nathaniel and Okunlola Joseph Temitayo},
      title = {Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations},
      journal = {International Journal of Applied Mathematics and Theoretical Physics},
      volume = {6},
      number = {1},
      pages = {7-13},
      doi = {10.11648/j.ijamtp.20200601.12},
      url = {https://doi.org/10.11648/j.ijamtp.20200601.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20200601.12},
      abstract = {Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations
    AU  - Fadugba Sunday Emmanuel
    AU  - Adebayo Kayode James
    AU  - Ogunyebi Segun Nathaniel
    AU  - Okunlola Joseph Temitayo
    Y1  - 2020/05/19
    PY  - 2020
    N1  - https://doi.org/10.11648/j.ijamtp.20200601.12
    DO  - 10.11648/j.ijamtp.20200601.12
    T2  - International Journal of Applied Mathematics and Theoretical Physics
    JF  - International Journal of Applied Mathematics and Theoretical Physics
    JO  - International Journal of Applied Mathematics and Theoretical Physics
    SP  - 7
    EP  - 13
    PB  - Science Publishing Group
    SN  - 2575-5927
    UR  - https://doi.org/10.11648/j.ijamtp.20200601.12
    AB  - Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.
    VL  - 6
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria

  • Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria

  • Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria

  • Department of Mathematical and Physical Sciences, Afe Babalola University, Ado Ekiti, Nigeria

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