In this thesis, high dimensional two-sided space fractional diffusion equations, derived from the fractional Fick’s law, and with monotonic variable diffusion coef- ficients, are solved by alternating direction implicit method. Each linear system corresponding to each spatial direction thus result is solved by Krylov subspace method. The method is accelerated by applying an approximate inverse preconditioner, where under certain conditions we showed that the normalized preconditioned matrix equals to a sum of identity matrix, a matrix with small norm, and a matrix with low rank, such that the preconditioned Krylov subspace method converges superlinearly. We also briefly present some fast algorithms which computational cost for solving the linear systems is O(n log n), where n is the matrix size. The results are illustrated by some numerical examples.