This manuscript is devoted to implementing spectral numerical solutions to two kinds of fractional space-time advection-
dispersion problems governed by certain constraints conditions. The collocation and tau spectral methods are utilized for
obtaining the proposed spectral solutions. A double Legendre expansion is proposed as an approximate solution. The main
idea of the algorithm is basically depend on converting the equation with its constraints conditions into linear or nonlinear
systems of algebraic equations which can be eciently solved with the aid of suitable numerical solvers. Some illustrative
examples are displayed aiming to conrm robustness, eciency and accuracy of the proposed spectral solutions.