一类非线性Jerk方程的改进两变量展开法

A modified two-variable expansion method for a nonlinear Jerk equation

应用改进的两变量展开法求解非线性含有三次非线性项的三阶微分方程的近似频率和近似解析周期解。该方法结合了Lindstedt-Poincare方法与两变量展开法不仅可以适用于弱非线性振动问题的求解而且还可以适用于强非线性振动问题的求解。以一个不含速度线性项的非线性Jerk方程作为例子分析并得到二阶近似周期和二阶近似解析周期解,与数值方法给出的"精确"周期解比较,二阶近似解析周期解比一阶近似解析周期解要精确得多。结果表明,改进的两变量展开法能够适用于求解非线性Jerk方程。而且在Jerk方程不含速度线性项时该方法仍然有效。

A modified two-variable expansion method was used to determine approximate frequencies and approximate analytical periodic solutions to a third-order differential equation with cubic nonlinearterms. This method combining Lindstedt-Poincare technique and the two-variable expansion method was not only valid for weakly nonlinear oscillations but also for strongly nonlinear oscillations. Here, a nonlinear Jerk equation excluding linear part of velocity term as an example was calculated. Its second-order approximate period and second-order approximate analytical periodic solution were obtained. A comparison of the first and second approximate analytical periodic solutions with the numerically exact solutions showed that the second order approximate analytical periodic solution is much more accurate than the first one. The result showed that the modified two-variable expansion method is suitable for solving a nonlinear Jerk equation, moreover, when the Jerk equation doesn＇t have linear part of velocity term, this method is still effective.