  The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$
are first order approximations for the Caputo fractional derivative of the
function $y(x)$ with lower limit at the point $b$. We obtain second and third
order approximations for the Gr\"unwald and shifted Gr\"unwald formulas with
weighted averages of Caputo derivatives when sufficient number of derivatives
of the function $y(x) $ are equal to zero at $b$, using the estimate for the
error of the shifted Gr\"unwald formulas. We use the approximations to
determine implicit difference approximations for the sub-diffusion equation
which have second order accuracy with respect to the space and time variables,
and second and third order numerical approximations for ordinary fractional
differential equations.
