Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

  • Abhinandan Chowdhury Savannah State University
  • Sammie Clayton Savannah State University
  • Mulatu Lemma Savannah State University
Keywords: Embedded Runge-Kutta Methods, Duffing Oscillator, Adaptive Time-Stepping Schemes

Abstract

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

Downloads

Download data is not yet available.

References

M. T. Ahmadian, M. Mojahedi, and H. Moeenfard. Free vibration analysis of a nonlinear beam using homotopy

and modified Lindstedt-Poincare methods. J. Solid Mechanics, 1:29–36, 2009.

J. C. Butcher. On Fifth and Sixth Order Explicit Runge-Kutta Methods: Order Conditions and Order Barriers.

Can. Appl. Math. Q., 17(3):433–445, 2009.

J. R. Cash and A. H. Karp. A Variable Order Runge-Kutta Method for Initial-value Problems with Rapidly Varying

Right-hand Sides. ACM Transactions on Mathematical Softwares, 16:201–222, 1990.

P. K. Das, D. Singh, and M. M. Panja. An Efficient Scheme for Accurate Closed-form Approximate Solution of

Some Duffing- and Lienard-type Equations. Journal of Advances in Mathematics, 16:8213–8225, 2019.

E. Fehlberg. Low-order classical Runge-Kutta Formulas with stepsize control. Technical report, 1969.

J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems and bifurcations of vector fields.

Springer-Verlag, 1983.

J. Kiusalaas. Numerical methods in engineering with Python 3. Cambridge University Press, 3rd edition, 2013.

S. Nourazar and A. Mirzabeigy. Approximate solution for nonlinear Duffing oscillator with damping effect using

the modified differential transform method. Sci. Iran B, 20(2):364–368, 2013.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in Fortran 77 - The Art of

Scientific Computing. Cambridge University Press, second edition.

O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk. Optimization of the Split-Step Fourier Method in

Modeling Optical-fiber Communications Systems. J. Lightwave Technol., 21(1):61–68, 2003.

E. Yusufoglu. Numerical Solution of Duffing Equation by the Laplace Decomposition Algorithm. Appl. Math.

Comput., 177(2):572–580, 2006.

Published
2019-09-16
How to Cite
Chowdhury, A., Clayton, S., & Lemma, M. (2019). Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method. JOURNAL OF ADVANCES IN MATHEMATICS, 17, 147-154. https://doi.org/10.24297/jam.v17i0.8408
Section
Articles