用叶函数求三次-五次Duffing方程的精确解
Exact Solutions of the Cubic-Quintic Duffing Equation Using Leaf Functions
摘要：
仅用叶函数给出了三次Duffing方程和三次-五次Duffing方程的精确解。在以前的研究中，三次Duffing方程的精确解已经提出了使用在三角函数的相位中积分叶函数的函数。由于这些解不简单，因此转换精确解的过程复杂且不方便。叶函数的一阶导数可以作为根导出。这个导数可以分解。这些因子或因子的乘积是Duffing方程的精确解。这些精确解中的某些解与文献中报道的三次Duffing方程是同一类型的。其中某些精确解满足三次-五次高非线性Duffing方程的精确解。在这项研究中，这些精确解的参数和Duffing方程的各项系数之间的关系被澄清。我们数值分析这些精确解，绘制波形，并讨论波形的周期性和振幅。
英文摘要：
The exact solutions of both the cubic Duffing equation and cubic-quintic Duffing equation are presented by using only leaf functions. In previous studies, exact solutions of the cubic Duffing equation have been proposed using functions that integrate leaf functions in the phase of trigonometric functions. Because they are not simple, the procedures for transforming the exact solutions are complicated and not convenient. The first derivative of the leaf function can be derived as the root. This derivative can be factored. These factors or multiplications of factors are exact solutions to the Duffing equation. Some of these exact solutions are of the same type as the cubic Duffing equation reported in previously. Some of these exact solutions satisfy the exact solutions of the cubic--quintic Duffing equations with high nonlinearity. In this study, the relationship between the parameters of these exact solutions and the coefficients of the terms of the Duffing equation is clarified. We numerically analyze these exact solutions, plot the waveform, and discuss the periodicity and amplitude of the waveform.
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