Idea of invariant subspace combined with elementary integral method for investigating exact solutions of time-fractional NPDEs

In this paper, inspired by the idea of invariant subspace method and combined with elementary integral method, we introduced a novel approach for investigating exact solutions of a time-fractional nonlinear partial differential equation (NPDE). Based on hypothetical structure of solution of separated variable, a time-fractional NPDE defined by time and space variables can be reduced to a nonlinear ordinary differential equation (NODE) or NODEs defined by space variable alone, and then using the elementary integral method to solve the NODE or NODEs, different kinds of exact solutions of a time-fractional NPDE are obtained finally. As examples, the time-fractional Hunter-Saxton equation and time-fractional Li-Olver equation were studied. Different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated.