Local symmetry structure and potential symmetries of time‐fractional partial differential equations

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.