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

Fadugba Sunday Emmanuel; Adebayo Kayode James; Ogunyebi Segun Nathaniel; Okunlola Joseph Temitayo

doi:doi:10.11648/j.ijamtp.20200601.12

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

Fadugba Sunday Emmanuel, Adebayo Kayode James, Ogunyebi Segun Nathaniel, Okunlola Joseph Temitayo

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

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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[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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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

Fadugba Sunday Emmanuel

Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Adebayo Kayode James

Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Ogunyebi Segun Nathaniel

Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Okunlola Joseph Temitayo

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

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Plain Text BibTeX RIS

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}
}

Copy | Download

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  -

Copy | Download