The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform.

中文翻译：

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具有分形导数的一维对流-色散方程的变分原理

对流-扩散方程一直是研究污染物迁移模型的经典方程。由于介质的杂质和边界不光滑的存在，科学研究存在一定的偏差。本文介绍了分形空间中具有分形导数的一维对流-频散方程，并通过半逆法建立了该方程的分形变分公式。我们得到的分形变分公式可以提供分形空间中能量形式的守恒定律以及给定方程的可能解结构。通过两尺度变换法和拉普拉斯变换得到解析解。