Document Type : Research Note

**Authors**

Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul 34220, Turkey.

**Abstract**

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

**Keywords**

- Duffing oscillator
- Differential transform
- Nonlinear behaviour
- Initial value problem
- exact solution
- Runge-Kutta method

**Main Subjects**

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Transactions on Mechanical Engineering (B)

March and April 2019Pages 879-886