Algebraic construction of a third order difference approximations for fractional derivatives and applications

Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence. References