Algebraic construction of higher order difference approximations for fractional derivatives and applications

A generalization of the Gr\"{u}nwald difference approximation for fractional derivatives in terms of a real sequence and its generating function is presented. Properties of the generating function are derived for consistency and order of accuracy for the approximation corresponding to the generator. Using this generalization, some higher order Gr\"{u}nwald type approximations are constructed and tested for numerical stability by using steady state fractional differential problems. These higher order approximations are used in Crank-Nicolson type numerical schemes to approximate the solution of space fractional diffusion equations. Stability and convergence of these numerical schemes are analyzed and are supported by numerical examples.