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

摘要

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.