The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation of an improved form of the Noether’s theorem. It is indicated that invariant solutions, reduced equations, perturbed or unperturbed symmetries and conservation laws can be obtained by applying a nonlinear self-adjoint notion. The method is applied to the time fractional-order Fokker–Planck equation. We obtained new results in a highly efficient and elegant manner.

中文翻译：

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分数阶福克-普朗克方程的新精确解和守恒定律

本文的主要目的是提出一种新方法来实现包含分数阶微分方程的参数的解析解。使用非线性自伴随概念，这些方程的近似解、守恒定律和对称性也可以通过 Noether 定理的改进形式的新公式获得。表明应用非线性自伴随概念可以获得不变解、简化方程、扰动或未扰动的对称性和守恒定律。该方法应用于时间分数阶 Fokker-Planck 方程。我们以高效和优雅的方式获得了新的结​​果。