A COMPARATIVE EXPLORATION ON DIFFERENT NUMERICAL METHODS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS
Authors:Mohammad Asif Arefin,Biswajit Gain,Rezaul Karim,Saddam Hossain,
In this paper, the initial value problem of Ordinary Differential Equations has been solved by using different Numerical Methods namely Euler’s method, Modified Euler method, and Runge-Kutta method. Here all of the three proposed methods have to be analyzed to determine the accuracy level of each method. By using MATLAB Programming language first we find out the approximate numerical solution of some ordinary differential equations and then to determine the accuracy level of the proposed methods we compare all these solutions with the exact solution. It is observed that numerical solutions are in good agreement with the exact solutions and numerical solutions become more accurate when taken step sizes are very much small. Lastly, the error of each proposed method is determined and represents them graphically which reveals the superiority among all the three methods. We fund that, among the proposed methods Runge-Kutta 4th order method gives the accurate result and minimum amount of error.
Keywords:Initial Value Problems (IVP),Euler’s Method,Modified Euler Method,Fourth-order Runge-Kutta Method,Error Estimation,
I. Akanbi, M. A. (2010). Propagation of Errors in Euler Method, Scholars Research Library. Archives of Applied Science Research, 2, 457-469.
II. Hahn, G. D. (1991). A modified Euler method for dynamic analyses. International Journal for Numerical Methods in Engineering, 32(5), 943-955.
III. Hamed, A. B., Yuosif, I., Alrhaman, I. A., & Sani, I. (2017). The accuracy of Euler and modified Euler technique for first order ordinary differential equations with initial condition. Am. J. Eng. Res., 6, 334-338.
IV. Hong-Yi, L. (2000). The calculation of global error for initial value problem of ordinary differential equations. International journal of computer mathematics, 74(2), 237-245.
V. Hossain, M. B., Hossain, M. J., Miah, M. M., & Alam, M. S. (2017). A comparative study on fourth order and butcher’s fifth order runge-kutta methods with third order initial value problem (IVP). Applied and Computational Mathematics, 6(6), 243-253.
VI. Hossain, M. J., Alam, M. S., & Hossain, M. B. (2017). A study on the Numerical Solutions of Second Order Initial Value Problems (IVP) for Ordinary Differential Equations with Fourth Order and Butcher’s Fifth Order Runge-Kutta Mthods. American Journal of Computational and Applied Mathematics, 7(5), 129-137.
VII. Islam, M. A. (2015). Accuracy Analysis of Numerical solutions of initial value problems (IVP) for ordinary differential equations (ODE). IOSR Journal of Mathematics, 11(3), 18-23.
VIII. Islam, M. A. (2015). Accurate solutions of initial value problems for ordinary differential equations with the fourth order Runge Kutta method. Journal of Mathematics Research, 7(3), 41.
IX. Islam, M. A. (2015). A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods. American Journal of Computational Mathematics, 5(03), 393.
X. Kockler, N. (1994). Numerical Method for Ordinary Systems of Initial Value Problems.
XI. Lambert, J. D. (1973). Computational methods in ordinary differential equations.
XII. Mathews, J.H. (2005) Numerical Methods for Mathematics, Science and Engineering. Prentice-Hall, India. 
XIII. Ntouyas, S. K., & Tsamatos, P. C. (1997). Global existence for semilinear evolution integrodifferential equations with delay and nonlocal conditions. Applicable Analysis, 64(1-2), 99-105.
XIV. Ogunrinde, R. B., Fadugba, S. E., & Okunlola, J. T. (2012). On some numerical methods for solving initial value problems in ordinary differential equations. IOSR Journal of Mathematics, 1(3), 25-31.
XV. Samsudin, N., Yusop, N. M. M., Fahmy, S., & binti Mokhtar, A. S. N. (2018). Cube Arithmetic: Improving Euler Method for Ordinary Differential Equation Using Cube Mean. Indonesian Journal of Electrical Engineering and Computer Science, 11(3), 1109-1113.
XVI. Shampine, L. F., & Watts, H. A. (1971). Comparing error estimators for Runge-Kutta methods. Mathematics of computation, 25(115), 445-455.View Download