THREE – DIMENSIONAL TIME –FRACTIONAL ADVECTION-DIFFUSION MODELING MODELING OF POLLUTANT DISPERSION: ANALYTICAL AND NUMERICAL FRAMEWORK

Abstract

In the study of atmospheric pollutant transport, traditional diffusion models fall short in accurately characterizing sub-diffusive behavior, where particles demonstrate non-Gaussian dispersion and extended retention due to memory effects and anomalous dynamics. To address this limitation, we have developed both analytical and numerical solutions for a three-dimensional time-fractional advection-diffusion equation in the Caputo sense, employing Dirichlet boundary conditions. This approach effectively captures the anomalous transport behaviors and memory-dependent dynamics present in the atmosphere. The analytical solution, obtained through eigenfunction expansion and Laplace transform, results in a closed form that incorporates Mittag–Leffler functions to represent temporal memory effects. Furthermore, a finite-difference method based on the Grunwald–Letnikov discretization was devised to numerically solve the fractional derivative. The proposed scheme’s consistency, stability, and convergence were thoroughly evaluated. The simulation results demonstrate that fractional orders within the range α = 0.5 to 0.8 significantly improve the precision of the model, achieving absolute error reductions of 42 %- 89% compared to the classical case (α = 1.0), particularly at vertical levels below 500 m. Under sub-diffusive conditions, the persistence of pollutant was extended by approximately 30 %- 50%. These findings underscore the efficacy of fractional modeling in precisely describing atmospheric pollutant dispersion.