### 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
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.

Keywords

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

References
• 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

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

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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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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T2  - International Journal of Applied Mathematics and Theoretical Physics
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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.
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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

• Sections