The author discusses the numerical solution of some space-time fractional order partial differential equations. One practical implicit numerical method is proposed to solve a class of initial-boundary value space-time fractional convection-diffusion equations with variable coefficients. A new shifted version of the usual Grünwald finite difference approximation [see M. M. Meershaert, J. Mortensen
and H. P. Scheffler
, Fract. Calc. Appl. Anal. 7, No. 1, 61–81 (2004; Zbl 1084.65024
)] is used for the non-local fractional derivative operator and it is proved that the method is first-order consistent and unconditionally stable, for the equation with Dirichlet boundary conditions. The convergence and error estimates of the scheme, are also discussed. One numerical example, with known exact solution, is presented.