自由振动下三次Duffing方程的叶函数精确解
Exact Solutions to Cubic Duffing Equation by Leaf Functions Under Free Vibration
摘要：
在三次Duffing方程中给出了初始条件下的精确解。这些精确解用叶函数和三角函数表示。叶函数$r=sleaf_n(t) $或$ r=cleaf_n(t)$满足常微分方程：$ dx^2/dt^2=-nr^{2n-1}$。叶函数的二阶微分等于$-n$乘以叶函数的函数的$(2n-1)$次幂。利用Leaf函数，在一定条件下，可以得到三次Duffing方程的精确解.本文基于叶函数给出了七种类型的精确解。本文揭示了七个精确解的推导过程。最后，将基于精确解的波的数值结果直观地显示在图形中
英文摘要：
Exact solutions with the initial conditions are presented in the cubic duffing equation. These exact solutions are expressed in terms of the leaf function and the trigonometric function. The leaf functions: $r=sleaf_n(t) $ or $ r=cleaf_n(t)$ satisfy the ordinary differential equation: $ dx^2/dt^2=-nr^{2n-1}$. The second order differential of the leaf function is equal to $-n$ times the function raised to the $(2n-1)$ power of the leaf function. By using the Leaf functions, the exact solutions of the cubic Duffing equation can be derived under several conditions. In this paper, seven types of the exact solutions are presented based on the leaf functions. This paper reveals the derivation of the seven exact solutions. Finally, the waves based on the exact solutions are visualized by the numerical results in the graphs
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