This paper suggests a fractal mathematical model for Fangzhu's water collection by taking into account its nanoscale surface morphology. A fractal variational principle is established by the semi-inverse method, which elucidates the conservation laws in the fractal space. Additionally, the variational formulation can also reveal the structure of the solution. An approximate analytical solution is obtained via the two-scale transform method and He's frequency formula. Some special cases of the model are discussed numerically, showing the method is remarkably simple and capable to solve nonlinear oscillators with fractal derivatives.

中文翻译：

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方珠纳米级表面形貌对集水的影响

本文通过考虑其纳米级表面形态，提出了方珠集水的分形数学模型。用半逆方法建立了分形变分原理，阐明了分形空间的守恒定律。此外，变分公式还可以揭示解的结构。通过两尺度变换方法和赫氏频率公式得到近似解析解。对模型的一些特殊情况进行了数值讨论，表明该方法非常简单，能够求解具有分形导数的非线性振子。