New approach to solving the atmospheric pollutant dispersion equation using fractional derivatives

Abstract

In atmospheric environments, traditional differential equations do not adequately describe the problem of turbulent diffusion because the usual derivatives are not well defined in the non-differentiable behaviour introduced by turbulence, where the fractional calculation has become a very useful tool for studying anomalous dispersion and other transportation processes. Considering a new direction, this paper presents an analytical series solution of a three-dimensional advection–diffusion equation of fractional order, in the Caputo sense, applied to the dispersion of atmospheric pollutants. The solution is obtained by applying the generalised integral transform technique (GITT), solving the transformed problem by the Laplace decomposition method (LDM), and considering the lateral and vertical turbulent diffusion dependence on the longitudinal distance from the source, as well as a fractional parameter. The fractional solution is more general than the traditional solution in the sense that consideration of the integer order of the fractional parameter yields the traditional solution. The solution considers the memory effect in eddy diffusivity and in the fractional derivative, and it is simple, easy to implement, and converges rapidly. Numerical simulations were conducted to compare the performance of the proposed fractional solution to the traditional solution using an experimental dataset and other models, which also made it possible to find a better parametrisation for use in Gaussian models. The best results are obtained with the fractional order of the derivative.