The authors consider, in one space dimension, the advection-diffusion equation, but with Canuto-type fractional time derivative and Riesz-type fractional space derivatives. They develop difference approximations for the nonlocal operator corresponding to the equation and its Cauchy resp. initial-boundary value problem. They prove conditional stability in the explicit case and unconditional stability for the implicit approximation and get first-order convergence (and slightly better results for the implicit case) for sufficiently smooth (in the sense of C-spaces) solutions. Their numerical results show that Richardson extrapolation is very helpful in improving the accuracy whereas a technique to reduce the computational effort, the short-memory principle, is less effective.