A fourth-order approximation of fractional derivatives with its applications

A new fourth-order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted Grünwald formulae combining the compact technique. The properties of proposed fractional difference quotient operator are presented and proved. Then the new approximation formula is applied to solve the space fractional diffusion equations. By the energy method, the proposed quasi-compact difference scheme is proved to be unconditionally stable and convergent in L2 norm for both 1D and 2D cases. Several numerical examples are given to confirm the theoretical results.