In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg–de Vries, modified Korteweg–de Vries, Burgers, and modified Burgers equations with conformable fractional time and space derivatives. For each equation, all the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations in terms of solutions of ordinary differential equations. In particular, it is shown that the fractional Korteweg–de Vries can be reduced to the first Painlevé equation and to the fractional second Painlevé equation. In addition, a solution of the fractional modified Korteweg–de Vries is given in terms of solutions of the fractional second Painlevé equation.

中文翻译：

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某些相容分数阶偏微分方程的李对称性分析

在本文中，使用李对称性分析来研究某些具有分数时间和空间导数的非线性分数阶偏微分方程的不变性。该分析适用于具有一致分数时间和空间导数的Korteweg-de Vries，修正的Korteweg-de Vries，Burgers和修正的Burgers方程。对于每个方程，都获得了所有矢量场和Lie对称性。此外，根据常微分方程的解给出了这些方程的精确解。特别是，它表明分数Korteweg-de Vries可以简化为第一Painlevé方程和分数第二Painlevé方程。此外，