INFORMATION CHANGE THE WORLD

### International Journal of Mathematical Sciences and Computing(IJMSC)

ISSN: 2310-9025 (Print), ISSN: 2310-9033 (Online)

IJMSC Vol.7, No.2, Jun. 2021

#### Accuracy Analysis for the Solution of Initial Value Problem of ODEs Using Modified Euler Method

Full Text (PDF, 1370KB), PP.31-41

#### Author(s)

Mohammad Asif Arefin, Nazrul Islam, Biswajit Gain, Md. Roknujjaman

#### Index Terms

Modified Euler method; Initial Value Problems; Estimation of Error; and Accuracy Analysis.

#### Abstract

There exist numerous numerical methods for solving the initial value problems of ordinary differential equations. The accuracy level and computational time are not the same for all of these methods. In this article, the Modified Euler method has been discussed for solving and finding the accurate solution of Ordinary Differential Equations using different step sizes. Approximate Results obtained by different step sizes are shown using the result analysis table. Some problems are solved by the proposed method then approximated results are shown graphically compare to the exact solution for a better understanding of the accuracy level of this method. Errors are estimated for each step and are represented graphically using Matlab Programming Language and MS Excel, which reveals that so much small step size gives better accuracy with less computational error. It is observed that this method is suitable for obtaining the accurate solution of ODEs when the taken step sizes are too much small.

#### Cite This Paper

Mohammad Asif Arefin, Nazrul Islam, Biswajit Gain, Md. Roknujjaman," Accuracy Analysis for the Solution of Initial Value Problem of ODEs Using Modified Euler Method ", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.7, No.2, pp. 31-41, 2021. DOI: 10.5815/ijmsc.2021.02.04

#### Reference

[1]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.

[2]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.

[3]L. F. Shampine and H. A. Watts, “Comparing Error Estimators for Runge-Kutta Methods,” Math. Comput., vol. 25, no. 115, p. 445, 1971, doi: 10.2307/2005206.

[4]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.

[5]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.

[6]H. Y. Lee, “Calculation of global error for initial value problem of ordinary differential equations,” Int. J. Comput. Math., vol. 74, no. 2, pp. 237–245, 2000, doi: 10.1080/00207160008804937.

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

[8]N. M. M. Yusop, M. K. Hasan, and M. Rahmat, “Comparison New Algorithm Modified Euler in Ordinary Differential Equation Using Scilab Programming,” Lect. Notes Softw. Eng., vol. 3, no. 3, pp. 199–202, 2015, doi: 10.7763/lnse.2015.v3.190.

[9]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.

[10]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.

[11]N. Samsudin, N. M. M. Yusop, S. Fahmy, and A. S. N. binti Mokhtar, “Cube arithmetic: Improving euler method for ordinary differential equation using cube mean,” Indones. J. Electr. Eng. Comput. Sci., vol. 11, no. 3, pp. 1109–1113, 2018, doi: 10.11591/ijeecs.v11.i3.pp1109-1113.

[12]Burden, R. L., & Faires, J. D. (1997). Numerical Analysis, Brooks. Cole, Belmont, CA.