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A generalization of the Gr\"{u}nwald difference approximation for fractionalderivatives in terms of a real sequence and its generating function ispresented. Properties of the generating function are derived for consistencyand order of accuracy for the approximation corresponding to the generator.Using this generalization, some higher order Gr\"{u}nwald type approximationsare constructed and tested for numerical stability by using steady statefractional differential problems. These higher order approximations are used inCrank-Nicolson type numerical schemes to approximate the solution of spacefractional diffusion equations. Stability and convergence of these numericalschemes are analyzed and are supported by numerical examples.

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