fractional diffusion equation, shifted Grünwald formula, stability, Fractional partial differential equations, fast Fourier transform, damped-Jacobi method, implicit finite difference scheme, Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs, Initial-boundary value problems for second-order parabolic equations, Toeplitz matrices, Finite difference methods for initial value and initial-boundary value problems involving PDEs, multigrid method, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, numerical experiments, Numerical methods for discrete and fast Fourier transforms