In a recent paper (Zeb et al. in Appl Math Model 37(7):5326-5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.