An Accurate Method for Solving the Undamped Duffing Equation with Cubic Nonlinearity
In this paper, we study the strongly nonlinear undamped duffing equation for undamped oscillators. We present the physical and the mathematical model of nonlinear Duffing equation for undamped oscillators. The reproducing kernal Hilbert space method (RKHSM) is employed to compute an approximation to the solution of this problem. The validity of the RKHSM is ascertained by comparing our results with numerical results and other methods in the literature. The results reveal that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. The existences of the solution is proved. In addition, the uniformly convergent of the proposed method is investigated.
KeywordsDuffing equation Nonlinear boundary value problem Reproducing kernal Hilbert space method
Mathematics Subject Classification76A05 76W05 76Z99 65L05
Compliance with Ethical Standards
Conflicts of interest
The authors declare that there is no conflict of interests regarding the publication of the paper.
- 2.Abraham, R.H., Shaw, C.D.: Dynamics—The Geometry of Behavior, Part I. Aerial Press, Santa Cruz (2000)Google Scholar
- 10.Wazwaz, A.M., Rach, R., Duan, J.S.: Solving the two-dimensional Lane–Emden type equations by the Adomian decomposition method. J. Appl. Math. Stat. 3(1), 15–26 (2016)Google Scholar
- 15.Moiseev, N.N.: Asimptoticheskie metodi nelinejnoj mehaniki. Nauka, Moscow (1981)Google Scholar
- 25.Sibanda, P., Khidir, A.: A New Modification of the HPM for the Duffing Equation with Cubic Nonlinearity, Recent Researches in Applied and Computational Mathematics, pp. 139–145, ISBN: 978-1-61804-002-2Google Scholar