Abstract
We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative. We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results.