In this work we present the GILTT method for the solution of the advection–diffusion equation utilizing an eddy diffusivity depending on source distance. With this new approach, where no approximation is made along the solution derivation so that it is an exact solution except for the round-off error due to the stepwise approximation of the eddy diffusivity in the x variable, it is possible to simulate the pollutant dispersion in the Planetary Boundary Layer. The present study reinforces that the inclusion of the memory effect, important in near regions from an elevated continuous point source, improves the description of the turbulent transport process of atmospheric pollutants.