ANALYSIS OF FOURTH-ORDER DIFFERENCE APPROXIMATION SCHEMES FOR FRACTIONAL DIFFUSION EQUATIONS

Abstract Two fourth-order difference approximations for fractional derivatives based on Lubich-type second-order approximation with different shifts are derived. These approximations are applied to the space fractional diffusion equation with the Crank–Nicolson scheme. Here, we analyse the stability and convergence of these schemes and prove that they are unconditionally stable and convergent for a fractional order $\alpha $ ranging from $1$ to $2$ . Numerical examples are presented to show that both schemes converge, and we obtain the correct convergence rates and unconditional stability.