Abstract
The main aim of this paper is to construct an efficient Galerkin-Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre-Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of min{kappa theta, 2-theta} order convergence in time and of spectral accuracy in space, where theta is the order of fractional derivative and kappa is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.