A local differential transform approach to the cubic nonlinear Duffing oscillator with damping term

Abstract

Nonlinear behaviour of various problems can be described by the Duffing model interpreted as a forced oscillator with a spring, which has a restoring force. In this paper, a new numerical approximation technique based on the differential transform method is introduced for the nonlinear cubic Duffing equation with and without damping effect. Since exact solutions to the corresponding equation for all initial guesses do not exist in the literature, an exact solution is produced first for specific parameters using the Kudryashov method to measure the accuracy of the suggested method. The innovative approach is compared with the semi-analytic differential transform and the fourth-order Runge-Kutta methods. Although the semi-analytic differential transform method is valid only at small-time intervals, it is proved that the innovative approach has the ability to capture nonlinear behaviour of the process even at long-time intervals. The computations indicate that the present technique produces more accurate and computationally more economic results than the rival methods do. (C) 2019 Sharif University of Technology. All rights reserved.