Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

Mohammad Asif Arefin; Biswajit Gain; Rezaul Karim

doi:10.21467/ias.10.1.118-133

Authors

Mohammad Asif Arefin Department of Mathematics, Jashore University of Science and Technology https://orcid.org/0000-0002-2892-1683
Biswajit Gain Department of Mathematics, Jashore University of Science and Technology, Jashore-7408, Bangladesh https://orcid.org/0000-0003-3651-6220
Rezaul Karim Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail-1902, Bangladesh https://orcid.org/0000-0002-3986-2068

DOI:

https://doi.org/10.21467/ias.10.1.118-133

Abstract

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

Keywords:

Eulerâ€™s method, Modified Euler method, Fourth-order Runge-Kutta method

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References

M. A. Islam, “Accuracy Analysis of Numerical solutions of initial value problems (IVP) for ordinary differential equations (ODE),” IOSR J. Math. Ver. III, vol. 11, no. 3, pp. 2278–5728, 2015, doi: 10.9790/5728-11331823.

M. A. Islam, “Accurate Solutions of Initial Value Problems for Ordinary Differential Equations with the Fourth Order Runge Kutta Method,” J. Math. Res., vol. 7, no. 3, pp. 41–45, 2015, doi: 10.5539/jmr.v7n3p41.

A. Hasan, “Numerical Computation of Initial Value Problem by,” vol. 1, no. 1, pp. 19–32, 2018.

M. Babul Hossain, “A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP),” Appl. Comput. Math., vol. 6, no. 6, p. 243, 2017, doi: 10.11648/j.acm.20170606.12.

S. Ivp, “Method for an Extension of the Runge-Iwita loo,” vol. 42, no. 4, pp. 551–553, 1992.

O. Abraham, “Improving the Modified Euler Method,” Leonardo J. Sci., vol. 6, no. 10, pp. 1–8, 2007.

M. H. Afshar and M. Rohani, “Embedded modified Euler method: An efficient and accurate model,” Proc. Inst. Civ. Eng. Water Manag., vol. 162, no. 3, pp. 199–209, 2009, doi: 10.1680/wama.2009.162.3.199.

G. D. Hahn, “A modified Euler method for dynamic analyses,” Int. J. Numer. Methods Eng., vol. 32, no. 5, pp. 943–955, 1991, doi: 10.1002/nme.1620320502.

I. Journal, “Modified Euler Method for Finding Numerical Solution of Intuitionistic Fuzzy Differential Equation under Generalized Differentiability Concept,” no. February, 2018.

J. Zhang and B. Jiang, “Enhanced Euler’s method to a free boundary porous media flow problem,” Numer. Methods Partial Differ. Equ., vol. 28, no. 5, pp. 1558–1573, 2012, doi: 10.1002/num.20691.

A. B. M. Hame, I. Yuosif, I. A. Alrhama, and I. San, “Open Access The Accuracy of Euler and modified Euler Technique for First Order Ordinary Differential Equations with initial condition,” no. 9, pp. 334–338, 2017.

J. Demsie Abraha, “Comparison of Numerical Methods for System of First Order Ordinary Differential Equations,” Pure Appl. Math. J., vol. 9, no. 2, p. 32, 2020, doi: 10.11648/j.pamj.20200902.11.

M. A. Islam, “A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods,” Am. J. Comput. Math., vol. 05, no. 03, pp. 393–404, 2015, doi: 10.4236/ajcm.2015.53034.

Z. H. Maibed and M. S. Mechee, “A Compression Study of Multistep Iterative Methods for Solving Ordinary Differential Equations,” no. February, 2020.

D. Houcque and R. R. Mccormick, “Applications of MATLAB: Ordinary Differential Equations (ODE),” pp. 101–323, 2008.

Burden, R. L., and J. D. Faires. "Numerical Analysis (Thomson, Brooks/Cole, Belmont, CA, USA)." (2005).