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In this paper, a modified nonlinear Schr\"{o}dinger equation with spatio-temporal dispersion is formulated in the senses of Caputo fractional derivative and conformable derivative. A new generalized double Laplace transform coupled with Adomian decomposition method has been defined and applied to solve the newly formulated nonlinear Schr\"{o}dinger equation with spatio-temporal dispersion. The approximate analytical solutions using the proposed generalized method in the sense of Caputo fractional derivative and conformable derivatives are obtained and compared with each other graphically., Comment: 19 pages, 10 figures, 1 table, submitted to Optik journal, Elsevier