Extended Separation Method of Semi-Fixed Variables Together With Analytical Method for Solving Time Fractional Equation

Abstract

It is well known that investigation of exact solutions on nonlinear fractional partial differential equations (PDEs) is a very difficult work. In this paper, the separation method of semi-fixed variables is extended. Based on the original separation method, two kinds of new structures of the solutions in a hypothetical way are proposed. With the extended separation method of semi-fixed variables and the mapping method of Riccati equation combined, a new approach for searching exact solutions on time-fractional PDEs is introduced. To demonstrate the effect of the extended method, the time-fractional porous medium equation, time-fractional Hunter-Saxton equation, and time-fractional Fornberg-Whitham equation are solved under the Riemann-Liouville fractional differential operator. Different kinds of new exact solutions of the above three equations are obtained. In order to intuitively show the dynamic property of these exact solutions, the 3D-graphs of some solutions are illustrated as examples. Compared to the previous method, more abundant results can be obtained on solving some complex nonlinear time-fractional PDEs by use of this extended method.