The main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgersí equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgersí equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgersí equation is linearized by using the Cole- Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L2 and L1. All of obtained results are discussed with an example of the Burgersí equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study.