Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions

Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O(N3)O(N3) of computational work per time step and O(N2)O(N2) of memory to store where N is the number of spatial grid points in the discretization.In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O(N)O(N) of memory and O(Nlog⁡N)O(Nlog⁡N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.