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ARTICLE

Damped and Divergence Exact Solutions for the Duffing Equation Using Leaf Functions and Hyperbolic Leaf Functions

Kazunori Shinohara^{1, *}

1 Daido University, 10-3 Takiharu-cho, Minami-ku, Nagoya 457-8530, Japan.

* Corresponding Author: Kazunori Shinohara. Email: -tokyo.ac.jp.

Computer Modeling in Engineering & Sciences 2019, 118(3), 599-647. https://doi.org/10.31614/cmes.2019.04472

Abstract

According to the wave power rule, the second derivative of a function x(t) with respect to the variable t is equal to negative n times the function x(t) raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations are presented. In the ordinary differential equation, in the positive region of the variable x(t), the second derivative d²x(t)/dt² becomes negative. Therefore, in the case that the curves vary with the time t under the condition x(t)>0, the gradient dx(t)/dt constantly decreases as time t increases. That is, the tangential vector on the curve of the graph (with the abscissa t and the ordinate x(t)) changes from the upper right direction to the lower right direction as time t increases. On the other hand, in the negative region of the variable x(t), the second derivative d²x(t)/dt² becomes positive. The gradient dx(t)/dt constantly increases as time t decreases. That is, the tangent vector on the curve changes from the lower right direction to the upper right direction as time t increases. Since the behavior occurring in the positive region of the variable x(t) and the behavior occurring in the negative region of the variable x(t) alternately occur in regular intervals, waves appear by these interactions. In this paper, I present seven types of damped and divergence exact solutions by combining trigonometric functions, hyperbolic functions, hyperbolic leaf functions, leaf functions, and exponential functions. In each type, I show the derivation method and numerical examples, as well as describe the features of the waveform.

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