LAUSR

Abstract In this paper, we present an analytical solution of the fractional two-dimensional advection–diffusion equation with a bi-flow evolutionary model, which represents a modification of Fick's law applied to the dispersion of pollutants in the planetary boundary layer. The solution is obtained using Laplace decomposition to derive the Mittag–Leffler function, which is intrinsic to the solution of fractional differential equations. The solutions exhibit fast convergence and are in good agreement with data from the traditional Copenhagen (moderately unstable) and Hanford (stable to neutral) experiments in terms of the influence of the fractional derivative and the fourth-order term representing the retention phenomenon in the bi-flow evolutionary model. For the Copenhagen experiment, the best results are achieved with parameter values of α  = 0.95 (fractional order of derivative) and β  = 0.50 (retention control), with a normalized mean square error of 0.10 and 100% of results within a factor of two of the observed values. For the Hanford experiment, values of α  = 1.00 and β  = 1.00 are optimal, giving a normalized mean square error of 0.15 and 83% of results within a factor of two of the observed values, and exhibiting some dependence on the atmospheric stability.