We present a one-dimensional model of the formation and viscous evolution of protoplanetary disks. The formation of the early disk is modeled as the result of the gravitational collapse of an isothermal molecular cloud. The disk's viscous evolution is integrated according to two parameterizations of turbulence: the classical α representation and a β parameterization, representative of non-linear turbulence driven by the keplerian shear. We apply the model to DM Tau and GM Aur, two classical T-Tauri stars with relatively well-characterized disks, retrieving the evolution of their surface density with time. We perform a systematic Monte-Carlo exploration of the parameter space (i.e. values of the α-β parameters, and of the temperature and rotation rate in the molecular cloud) to find the values that are compatible with the observed disk surface density distribution, star and disk mass, age and present accretion rate. We find that the observations for DM Tau require $0.001<\alpha<0.1$ or $2\times 10^{-5}<\beta<5\times 10^{-4}$. For GM Aur, we find that the turbulent viscosity is such that $4\times 10^{-4}<\alpha<0.01$ or $2\times 10^{-6}<\beta<8\times 10^{-5}$. These relatively large values show that an efficient turbulent diffusion mechanism is present at distances larger than ~$10\,$AU. This is to be compared to studies of the variations of accretion rates of T-Tauri stars versus age that mostly probe the inner disks, but also yield values of $\alpha\sim 0.01$. We show that the mechanism responsible for turbulent diffusion at large orbital distances most probably cannot be convection because of its suppression at low optical depths.