  In the present paper, the Complex Ginzburg-Landau-Schr\"odinger (CGLS)
equation with the Riesz fractional derivative has been treated by a reliable
implicit finite difference method (IFDM) of second order and furthermore for
the purpose of a comparative study, and also for the investigation of the
accuracy of the resulting solutions another effective spectral technique viz.
time-splitting Fourier spectral (TSFS) technique has been utilized. In the case
of the finite difference discretization, the Riesz fractional derivative is
approximated by the fractional centered difference approach. Further the
stability of the proposed methods has been analysed thoroughly and the TSFS
technique is proved to be unconditionally stable. Moreover the absolute errors
for the solutions of |\chi(x, t)|^2 obtained from both the techniques for
various fractional order have been tabulated. Further the L^2 and L^infinity
error norms has been displayed for |\chi(x, t)|^2 and the results are also
graphically depicted.
