A time fractional advection-diffusion approach to air pollution: Modeling and analyzing pollutant dispersion dynamics

In this work, we investigate the dynamics of pollutant dispersion using a one-dimensional time-fractional advection-diffusion equation with the Caputo fractional derivative to predict air pollution levels. The focus is on pollutants such as ${\mathrm{NH}}_3$, $\mathrm{CO}$, and ${\mathrm{CO}}_2$, Dirichlet boundary conditions applied in homogeneous and heterogeneous environments. Numerical simulations are performed using the Grünwald–Letnikov method to discretize the fractional derivative, and analytical solutions are obtained through eigenfunction expansion. Results demonstrate that both numerical and analytical approaches accurately capture pollutant behavior, graphical visualizations illustrate concentration profiles and the impact of varying diffusivities. This work enhances the understanding of contaminant dispersion by addressing complex boundary conditions, integrating variable diffusivity, and employing fractional time derivatives. The combination of these methodologies highlights the benefits of using fractional models while visual analysis underscores their utility for improved pollution control and environmental management.