Separation Method of Semi-Fixed Variables for Solving a Generalized Time-Fractional Reaction-Diffusion Equation

It is well known that investigations on exact solutions of nonlinear fractional partial differential equations (PDEs) are very difficult compare with those investigations on integer-order nonlinear PDEs. In this paper, a new method called the separation method of semi-fixed variables together with the dynamical system method is introduced. The connection and difference between the classical separation method of variables and the separation method of semi-fixed variables are compared in details. As example, a generalized nonlinear time-fractional reaction-diffusion equation with higher-order terms is studied under the definition of Riemann-Liouville fractional derivative. In different parametric regions, different kinds of phase portraits of the system derived from the generalized model are presented. Existence and dynamic properties of solutions of the generalized model are investigated. In some special parametric conditions, some exact solutions of the generalized equation are obtained. In the absence of the reaction term, a very strange and interesting phenomenon of the model is found.