Dynamical system method for investigating existence and dynamical property of solution of nonlinear time-fractional PDEs

Abstract

In this paper, based on the dynamical system method, a new approach for investigating solutions of nonlinear time-fractional partial differential equations (PDEs) is introduced. By proposing a novel technique with separation variables, the phase portraits of system derived from the nonlinear time-fractional PDEs are analyzed, and the issue of existence for the solution of the time-fractional PDEs is considered. Moreover, the dynamical properties of the solution of the time-fractional PDEs are studied in detail. As examples, three nonlinear time-fractional models such as the reaction–diffusion model, the biology population model and the fluid model are studied by using this new approach. In some special parametric conditions, exact solutions of these models are obtained. The dynamical properties of some exact solutions are illustrated by graphs. Compared with the results in current studies, the results obtained in this paper are new.