Finding exact solutions for Riemann–Liouville (RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.

中文翻译：

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用一般的变量分离方法精确求解一些典型的黎曼-刘维尔分数模型

寻找黎曼-刘维尔 (RL) 分数方程的精确解非常困难。我们提出了一种分离变量的通用方法来研究这个问题。我们获得了几个一般结果，作为应用，我们给出了一些典型 RL 分数方程的非平凡精确解，例如分数 Kadomtsev-Petviashvili 方程和分数 Langmuir 链方程。特别是，我们获得了一种 RL 时间分数反应扩散方程的非幂函数解。此外，我们发现变量分离方法更适合处理高维非线性RL分数方程，因为我们有更多的自由选择未定函数。