In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

中文翻译：

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广义空间和时间分数 Korteweg-de Vries 方程的对称性分析

在本文中，我们研究了 Riemann-Liouville 分数导数意义上的广义时空分数 Korteweg-de Vries (KdV) 方程。最初，通过对称分析方法获得了该考虑方程的对称性。接下来，产生了一个单参数的点变换李群。然后，这个考虑的分数模型可以通过 Erdélyi-Kober 分数微分算子和 Erdélyi-Kober 分数积分算子转换为分数阶的常微分方程。最后，借助非线性自伴随性，可以得到广义时空分数KdV方程的守恒定律。这种方法可以为我们研究分数阶导数的时空微分方程提供一种新的方案。