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A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods

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DOI: 10.4236/ajcm.2015.53034    5,751 Downloads   7,798 Views   Citations

ABSTRACT

This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Islam, M. (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, 393-404. doi: 10.4236/ajcm.2015.53034.

References

[1] Islam, Md.A. (2015) Accuracy Analysis of Numerical solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE). IOSR Journal of Mathematics, 11, 18-23.
[2] Islam, Md.A. (2015) Accurate Solutions of Initial Value Problems for Ordinary Differential Equations with Fourth Order Runge Kutta Method. Journal of Mathematics Research, 7, 41-45.
http://dx.doi.org/10.5539/jmr.v7n3p41
[3] Ogunrinde, R.B., Fadugba, S.E. and Okunlola, J.T. (2012) On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations. IOSR Journal of Mathematics, 1, 25-31.
http://dx.doi.org/10.9790/5728-0132531
[4] Shampine, L.F. and Watts, H.A. (1971) Comparing Error Estimators for Runge-Kutta Methods. Mathematics of Computation, 25, 445-455.
http://dx.doi.org/10.1090/S0025-5718-1971-0297138-9
[5] Eaqub Ali, S.M. (2006) A Text Book of Numerical Methods with Computer Programming. Beauty Publication, Khulna.
[6] Akanbi, M.A. (2010) Propagation of Errors in Euler Method, Scholars Research Library. Archives of Applied Science Research, 2, 457-469.
[7] Kockler, N. (1994) Numerical Method for Ordinary Systems of Initial value Problems. John Wiley and Sons, New York.
[8] Lambert, J.D. (1973) Computational Methods in Ordinary Differential Equations. Wiley, New York.
[9] Gear, C.W. (1971) Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Upper Saddle River.
[10] Hall, G. and Watt, J.M. (1976) Modern Numerical Methods for Ordinary Differential Equations. Oxford University Press, Oxford.
[11] Hossain, Md.S., Bhattacharjee, P.K. and Hossain, Md.E. (2013) Numerical Analysis. Titas Publications, Dhaka.
[12] Balagurusamy, E. (2006) Numerical Methods. Tata McGraw-Hill, New Delhi.
[13] Sastry, S.S. (2000) Introductory Methods of Numerical Analysis. Prentice-Hall, India.
[14] Burden, R.L. and Faires, J.D. (2002) Numerical Analysis. Bangalore, India.
[15] Gerald, C.F. and Wheatley, P.O. (2002) Applied Numerical Analysis. Pearson Education, India.
[16] Mathews, J.H. (2005) Numerical Methods for Mathematics, Science and Engineering. Prentice-Hall, India.

  
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