The problem of the fractal Swift–Hohenberg model (FSHM) with variable coefficient is considered in this work based on the fractal derivative. First, the fractal variational principle (FVP) of the FSHM with variable coefficient is successfully established by employing the fractal semi-inverse method (FSIM), which is very helpful to investigate the structure of the analytical solution. Second, the fractal two-scale variational method (FTSVM) is established by combining the FVP and fractal two-scale transform method (FTSTM). Finally, an example is presented to illustrate the proposed method which is efficient and accurate. The proposed fractal two-scale variational method sheds new light on the nonlinear fractal models.

本文基于分形导数考虑了具有变系数的分形 Swift-Hohenberg 模型 (FSHM) 问题。首先，利用分形半逆法（FSIM）成功地建立了变系数FSHM的分形变分原理（FVP），这对研究解析解的结构非常有帮助。其次，将FVP与分形双尺度变换方法（FTSTM）相结合，建立了分形双尺度变分法（FTSVM）。最后，给出一个例子来说明所提出的方法是有效和准确的。所提出的分形两尺度变分方法为非线性分形模型提供了新的思路。