Symmetry properties and explicit solutions of the nonlinear time fractional KdV equation

The time fractional KdV equation in the sense of the Riemann-Liouville derivatives is considered. The symmetry properties of the time fractional KdV equation is investigated by using the Lie group analysis method. On the basis of the point symmetry, the vector fields of the time fractional KdV equation are presented. And then, the symmetry reductions are constructed. By right of the obtained Lie point symmetries, it is shown that this equation could transform into a nonlinear ordinary differential equation of fractional order with the new independent variable xi = xt(-alpha/ 3). The derivative is an Erdelyi-Kober derivative depending on a parameter a. At last, by means of the sub-equation method, some exact and explicit solutions of the time fractional KdV equation are constructed.