Fast solution methods for space-fractional diffusion equations

We develop fast solution methods for a shifted Grünwald finite difference method for steady-state and time-dependent space-fractional diffusion equations. These methods reduce the memory requirement of the finite difference scheme from O(N2) to O(N) and the computational complexity from O(N3) to O(Nlog2N). Preliminary numerical example runs show the utility of these methods over the traditional direct solvers of the finite difference methods, in terms of computational cost and memory requirements.