A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

中文翻译：

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分数导数性质对非线性振荡周期解的影响

基于分数导数的黎曼-刘维尔定义，推导出了时间分数非线性振荡器的周期解。在这种方法中，找到了分数摄动方程的特定积分。开发了一种增强的微扰方法来研究受迫非线性 Duffing 振子。在该提案中应用了具有两个扩展参数和一个附加辅助参数的修正同伦方程。增强方法的基本思想是应用湮没算子构造一个简化的周期力自由度方程。这种方法使强制问题的求解过程更加简单。由此产生的方程对于研究所有类型的可能的共振状态都是有效的。