An analytical framework for modeling urban pollutant dispersion with fractional transport dynamics

Conventional advection–diffusion models often struggle to represent pollutant dispersion in complex urban settings such as Kathmandu, Nepal, because they cannot adequately capture memory effects and anomalous diffusion generated by turbulent airflow over intricate terrain. To address this limitation, we develop a two-dimensional time-fractional advection–diffusion equation using the Caputo fractional derivative to embed temporal memory and non-classical diffusion behavior into the transport formulation. The model incorporates a fractional-order parameter that controls subdiffusive behavior and memory effects, thereby extending the classical diffusion framework. To support this formulation, Analytical solutions are derived via eigenfunction expansions involving sine functions and Mittag–Leffler terms under Dirichlet boundary conditions. For practical implementation and verification, numerical solutions are obtained using the forward-time central-space method combined with the standard fractional approximation for the fractional derivative. Simulations demonstrate that smaller values of the fractional order lead to stronger pollutant retention, whereas the classical-order case recovers the uniform spreading behavior characteristic of classical diffusion. These findings show how the fractional order mediates the transition between anomalous and classical regimes and illustrate the distinct dispersion patterns generated by fractional-order dynamics. Overall, the framework provides a flexible methodological basis for investigating pollutant transport under anomalous diffusion conditions and offers a path toward future coupling with realistic wind fields and topographic data. The present study does not undertake the representation of the full atmospheric and terrain complexity of Kathmandu but sets up a basic framework that can be extended with meteorological inputs to study dispersion processes in complex urban basins. • Develop a fractional model to simulate pollutant dispersion in urban environments. • Capture sub-diffusion effects and memory-driven transport for better forecasting. • Combine analytical and numerical techniques to improve simulation accuracy. • Link model parameters to practical air quality mitigation decisions. • Offer insights for policymakers through predictive urban airflow analysis.