In this paper, an efficient numerical technique is presented to solve the partial fractional space equations with variable coefficients on a finite domain. This technique based on nodal Galerkin method. The fractional derivatives are described in the Caputo sense. Also, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In this paper, the problems can be reduced to a set of linear algebraic equations by using the Chebyshev nodal Galerkin method. The existence and uniqueness of the solution for the linear semi discrete weak form solutions are proved. And the stability analysis for the linear semi and fully discrete schemes are discussed. Numerical solutions obtained by this method are in excellent agreement and efficient to use with those obtained by previous work in the literature.