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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. Furthermore for the purpose of a comparative study and 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 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 |Ψ(x, t)|^2 obtained from both the techniques for various fractional orders, have been tabulated. Further the L2 and L∞ error norms have been displayed for |Ψ(x, t)|^2 and the results are also graphically depicted.

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