Fractional Advection–Diffusion Framework for Modeling Anomalous Pollutant Transport in Complex Environments

Abstract

Classical advection–diffusion models often fail to capture anomalous transport processes observed in complex systems, such as urban atmospheres, where particle dispersion deviates from Gaussian profiles and exhibits memory-dependent dynamics. To overcome these limitations, we develop a dimensionally consistent space–time fractional advection–diffusion equation (FADE). To maintain dimensional consistency, two scaling parameters, σ_x and σ_t, are introduced to characterize the fractional contributions in space and time. The parameters are related, with space–time solutions expressed via the Mittag–Leffler function in terms of β and γ. An exact analytical solution is derived using the separation of variables to generalize the classical advection–diffusion equation, rigorously ensuring the existence, uniqueness, and convergence. This formulation provides a comprehensive analytical framework for understanding anomalous transport and offers a reliable benchmark for validating fractional models of pollutant dispersion.