In this thesis, we solve the two-sided space fractional diffusion equations (SFDEs) associated with an initial condition and zero Dirichlet boundary conditions. We use the Crank-Nicolson (C-N) technique with a second order discretization fractional operators for solving the two-sided SFDEs, and we prove that the scheme is unconditionally stable and second order in time and space. Moreover, we prove the circulant and skew-circulant splitting (CSCS) iteration method is employed to solve the unsymmetric Toeplitz system, which is discretized from SFDEs problem effectively. We also prove that the two splitting (circulant and skew-circulant) matrices are positive stable for any choice of time step and space step, so that the CSCS iteration method converges to the unique solution of the Toeplitz system. Numerical results are provided to verify the accuracy and efficiency of the CSCS iteration method. In addition, a second order finite difference scheme for two-sided SFDEs problem (1.1) with variable diffusion coefficients is also studied in this thesis. The scheme is proved to be unconditionally stable and convergent if α ∈ (α0, 2) where α0 ≈ 1.5545. Numerical examples are shown to support the theoretical results.