EXACT SOLUTIONS OF A GENERALIZED TIME-FRACTIONAL KDV EQUATION UNDER RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL OPERATORS

It is well known that searching exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work. In this paper, based on a modified separation method of variables and the dynamic system method, a combinational method is proposed in order to develop new methods for solving nonlinear time-fractional PDEs. Compared with the traditional separation method of variables, the modified separation method of variables has some advantages in reducing nonlinear time-fractional PDEs. As an example for the application of this combinational method, a generalized nonlinear time-fractional KdV equation is studied under the Riemann-Liouville and Caputo fractional differential operators, respectively. In different parametric regions, different kinds of phase portraits of the dynamic systems derived from the generalized time-fractional KdV equation are presented. Existence and dynamic properties of solutions of the generalized time-fractional KdV equation are investigated. In some special parametric conditions, many exact solutions are obtained, some of them are parametric form. Such solutions of parametric form are usually unable to be obtained by other methods, which also shows an advantage of dynamic system method.