Two effective methods for solving nonlinear coupled time-fractional Schrödinger equations

The objective of this work is to implement two efficient techniques, namely, the Laplace Adomian decomposition method (LADM) and the modified generalized Mittag–Leffler function method (MGMLFM) on a system of nonlinear fractional partial differential equations (NFPDEs) to get an analytic-approximate solution. The nonlinear time-fractional Schrödinger equation (TFSE) and coupled fractional order Schrödinger-Korteweg-de Vries (Sch-KdV) equation are found in various areas such as quantum mechanics and physics. These equations describe different types of wave propagation like dust-acoustic waves, Langmuir and electromagnetic waves in plasma physics. Using the proposed methods, a convenient solution is established for the considered nonlinear fractional order models. The obtained analytic-approximate travelling-waves solutions and the effect of the fractional order α on the behaviour of these projected solutions are presented in some figures and tables along with the exact solution. We compare the approximate values with their corresponding values of the known exact solution and compute the absolute error. Consequently, we can deduce that the used methods are very efficient, reliable and simple to construct a series form that rapidly convergent to the exact solution, which indicates the advantages of the methods.