American Journal of Applied Mathematics

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A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE)

Received: Aug. 24, 2023    Accepted: Oct. 08, 2023    Published: Nov. 21, 2023
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Abstract

Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations.

DOI 10.11648/j.ajam.20231106.12
Published in American Journal of Applied Mathematics ( Volume 11, Issue 6, December 2023 )
Page(s) 106-118
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), 2024. Published by Science Publishing Group

Keywords

Euler’s Method (EM), Runge-Kutta Method of Order Four (RK-4), Runge-Kutta Method of Order Six (RK-6), Adamsh-Bashforth Moulton Method (ABMM)

References
[1] Richard L. Burden, J. Douglas Faires, “Numerical Analysis”, Ninth Edition, ISBN-10: 0-538-73351-9.
[2] David Houcque, “Applications of MATLAB: Ordinary Differential Equation” Robert R. Mc. Cormic School of Engineering & Applied Science – Northwestern University.
[3] Dennis G. Zill, Warren S. Wright, “Advanced Engineering Mathematics”, Fourth Edition/ ISBN: 978-93-80108-92-6.
[4] Didier Gonze, 2013. “Numerical methods for Ordinary Differential Equations”.
[5] Islam, Md. A. (2015) Accuracy Analysis of Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE). IOSR Journal of Mathematics, 11, 18-23.
[6] Islam, Md. A. (2015) Accurate Solutions of Initial Value Problems for Ordinary Differential Equations with Fourth Order Runge Kutta Method. Journal of Mathematics Research, 7, 41-45.
[7] Vishal V. Mehtre, Kanishka Gupta, 2022, An Analysis of Numerical Solutions and Errors with Euler's Method, International Journal of Innovative Research in Electrical, Electronics, Instrumentation and Control Engineering, Vol. 10, Issue 2.
[8] R. B. Ogunrinde, K. S. Olayemi, I. O. Isah & A. S. Salawu, 2019, A Numerical Solver for First Order Initial Value Problems of Ordinary Differential Equation Via the Combination of Chebyshev Polynomial and Exponential Function, Journal of Physical Sciences, Vol. 1, Issue No. 1, PP 1 – 16.
[9] Courtney Remani, (2012-2013), “Numerical Methods for Solving Systems of Nonlinear Equations”, Lakehead University, Canada.
[10] Ogunrinde, R. B., Fadugba, S. E. and Okunlola, J. T. (2012) On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations. IOSR Journal of Mathematics (IOSRJM) ISSN: 2278-5728 Volume 1, Issue 3, PP 25-31.
[11] J. C. Butcher, “Numerical methods for ordinary differential equations in the 20th century”, The university of Auckland, Department of mathematics, Private Bag 92019, Auckland, New Zealand, 1999, Journal of Computational and Applied Mathematics 125 (2000).
[12] S. Amen, P. Bilokon, A. Brinley Codd, M. Fofaria, T. Saha. Supervised by Professor Jeff Cash, 2004 “Numerical Solution of Differential Equation”, Imperial College London.
[13] Shepley L. Ross, “Introductory to Ordinary Differential Equation”, Fourth Edition, ISBN: 0-471-09881-7.
[14] S. S. Sastry, 2009-2010, Introductory “Methods of Numerical Analysis”. Fourth Edition, ISBN 81-203-2761-6.
[15] Steven C. Chapra, Raymond P. Canale, “Numerical Methods for Engineers” Sixth Edition, ISBN 978-007-126759-5.
[16] Sunday Fadugba, Bosede Ogunrinde, Tayo Okunlola, “Euler’s Method for Solving IVP in ODE” Ekiti State University – Nigeria.
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  • APA Style

    Sumon, M. M. I., Nurulhoque, M. (2023). A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). American Journal of Applied Mathematics, 11(6), 106-118. https://doi.org/10.11648/j.ajam.20231106.12

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    ACS Style

    Sumon, M. M. I.; Nurulhoque, M. A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). Am. J. Appl. Math. 2023, 11(6), 106-118. doi: 10.11648/j.ajam.20231106.12

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    AMA Style

    Sumon MMI, Nurulhoque M. A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). Am J Appl Math. 2023;11(6):106-118. doi: 10.11648/j.ajam.20231106.12

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  • @article{10.11648/j.ajam.20231106.12,
      author = {Md. Monirul Islam Sumon and Md. Nurulhoque},
      title = {A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE)},
      journal = {American Journal of Applied Mathematics},
      volume = {11},
      number = {6},
      pages = {106-118},
      doi = {10.11648/j.ajam.20231106.12},
      url = {https://doi.org/10.11648/j.ajam.20231106.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20231106.12},
      abstract = {Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations.
    },
     year = {2023}
    }
    

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    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    UR  - https://doi.org/10.11648/j.ajam.20231106.12
    AB  - Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations.
    
    VL  - 11
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Author Information
  • Department of Business Administration, Z. H. Sikder University of Science and Technology, Shariatpur, Bangladesh

  • Department of Mathematics, Dhaka Commerce College, Dhaka, Bangladesh

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