First, we show that the system consisting of integer-order partial differential equations (PDEs) and time-fractional PDEs with the Riemann–Liouville fractional derivative has an elegant local symmetry structure. Then with the symmetry structure we consider two particular cases where one is the pure time-fractional PDEs whose symmetry invariant condition is divided into two parts of integer-order and time-fractional, the other is the linear system of time-fractional PDEs, which always admits an infinite-dimensional infinitesimal generator. Second, by considering the composition rules of fractional derivatives we establish a theoretical framework of potential symmetry and construct three potential systems to study potential symmetries of the time-fractional PDEs possessing a divergence form. In particular for a single time-fractional PDE the existence condition of potential symmetries via one typical potential system is presented by means of the local symmetry structure. Finally, local symmetry structure and potential symmetries of a class of time-fractional diffusion equations are studied in detail. Several explicit solutions are constructed by means of the potential symmetries.

中文翻译：

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时间分数阶偏微分方程的局部对称结构和势对称

首先，我们表明由整数阶偏微分方程 (PDE) 和具有 Riemann-Liouville 分数阶导数的时间分数 PDE 组成的系统具有优雅的局部对称结构。然后用对称结构我们考虑两种特殊情况，一种是对称不变条件分为整数阶和时间分数两部分的纯时间分数偏微分方程，另一种是线性时间分数偏微分方程，其中总是允许无限维无穷小生成器。其次，通过考虑分数阶导数的组成规则，我们建立了位对称性的理论框架，构建了三个位势系统来研究具有发散形式的时间分数阶偏微分方程的位势对称性。特别是对于单个时间分数 PDE，通过一个典型的势系统的势对称性的存在条件是通过局部对称结构来呈现的。最后，详细研究了一类时间分数扩散方程的局部对称结构和势对称性。几个显式解决方案是通过潜在的对称性构造的。