Separation method of semi-fixed variables together with dynamical system method for solving nonlinear time-fractional PDEs with higher-order terms

Abstract

It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach combined with the separation method of semi-fixed variables and dynamical system method is introduced. As an example, a time-fractional reaction-diffusion equation with higher-order terms is studied under two different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional model are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed, and the graphs of them are illustrated.