In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear [Formula: see text]-dimensional time-fractional Kramers equation via Riemann–Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear [Formula: see text]-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov’s method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.

中文翻译：

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非线性时间分数克莱默斯方程李变换的群形式、精确解和守恒定律

在本文中，我们将专注于通过 Riemann-Liouville 和 Caputo 导数找到非线性 [公式：见文本] 维时间分数 Kramers 方程的李点对称性的系统研究。利用李群分析方法，给出了时间分数Kramers方程的不变性和对称性约简。结果表明，通过使用基础方程的对称性之一，可以将其转换为具有新因变量和 Erdélyi-Kober 意义上的导数的非线性 [公式：见文本] 维分数微分方程。此外，我们使用不变子空间方法构造了时间分数克莱默斯方程的一些精确解。此外，采用伊布拉吉莫夫的方法，使用诺特恒等式、诺特算子和形式拉格朗日，