The Korteweg–de Vries (KdV) equation is an essential model to characterize shallow water waves in fluid mechanics. Here, we investigated the generalized time and time-space fractional KdV equation with fractional derivative of Riemann–Liouville. At the beginning of, we applied the fractional Lie symmetry scheme to derive their symmetry, respectively. We found that the vector fields of these considered equations decrease as the independent variables fractionalize. Subsequently, the one-parameter Lie transformation groups of these concerned models were yielded. At the same time, they can be reduced into fractional order ordinary differential equations with the Erdélyi–Kober fractional operators. Finally, by obtaining the nonlinear self-adjointness, conservation laws of the generalized time-space fractional KdV equation were also found. These good results provide a basis for us to further understand the phenomenon of shallow water waves.

Korteweg–de Vries (KdV) 方程是流体力学中描述浅水波特征的重要模型。在这里，我们研究了具有黎曼-刘维尔分数阶导数的广义时间和时空分数 KdV 方程。首先，我们应用分数李对称格式分别推导了它们的对称性。我们发现，这些考虑的方程的矢量场随着自变量的细分而减小。随后，得到了这些相关模型的单参数李变换群。同时，可以使用 Erdélyi-Kober 分数算子将它们简化为分数阶常微分方程。最后，通过获得非线性自共性，还得到了广义时空分数阶KdV方程的守恒定律。这些良好的结果为我们进一步认识浅水波浪现象提供了基础。