In this study, a harmonic oscillator with position-dependent mass is investigated. Firstly, as an introduction, we give a full description of the system by constructing its classical Lagrangian; thereupon, we derive the related classical equations of motion such as the classical Euler–Lagrange equations. Secondly, we fractionalize the classical Lagrangian of the system, and then we obtain the corresponding fractional Euler–Lagrange equations (FELEs). As a final step, we give the numerical simulations corresponding to the FELEs within different fractional operators. Numerical results based on the Caputo and the Atangana-Baleanu-Caputo (ABC) fractional derivatives are given to verify the theoretical analysis.

中文翻译：

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具有位置相关质量的谐振子的分数特征

在这项研究中，研究了具有位置相关质量的谐振子。首先，作为介绍，我们通过构造它的经典拉格朗日函数对系统进行了全面的描述；于是，我们推导出相关的经典运动方程，如经典的欧拉-拉格朗日方程。其次，我们对系统的经典拉格朗日量进行分数化，然后我们得到相应的分数欧拉-拉格朗日方程（FELEs）。作为最后一步，我们给出了对应于不同分数算子内的 FELE 的数值模拟。给出了基于 Caputo 和 Atangana-Baleanu-Caputo (ABC) 分数阶导数的数值结果来验证理论分析。