Nonlocal Modeling and Inverse Parameter Estimation of Time-Varying Vehicular Emissions in Urban Pollution Dynamics

Muratkan Madiyarov, Nurlana Alimbekova (), Aibek Bakishev, Gabit Mukhamediyev and Yerlan Yergaliyev
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Muratkan Madiyarov: Department of Mathematics, Higher School of Information Technology and Natural Sciences, SarsenAmanzholov East Kazakhstan University, Oskemen 070000, Kazakhstan
Nurlana Alimbekova: Department of Mathematics, Higher School of Information Technology and Natural Sciences, SarsenAmanzholov East Kazakhstan University, Oskemen 070000, Kazakhstan
Aibek Bakishev: Department of Mathematics, Higher School of Information Technology and Natural Sciences, SarsenAmanzholov East Kazakhstan University, Oskemen 070000, Kazakhstan
Gabit Mukhamediyev: Department of Mathematics, Higher School of Information Technology and Natural Sciences, SarsenAmanzholov East Kazakhstan University, Oskemen 070000, Kazakhstan
Yerlan Yergaliyev: Department of Mathematics, Higher School of Information Technology and Natural Sciences, SarsenAmanzholov East Kazakhstan University, Oskemen 070000, Kazakhstan

Mathematics, 2025, vol. 13, issue 17, 1-26

Abstract: This paper investigates the dispersion of atmospheric pollutants in urban environments using a fractional-order convection–diffusion-reaction model with dynamic line sources associated with vehicle traffic. The model includes Caputo fractional time derivatives and Riesz fractional space derivatives to account for memory effects and non-local transport phenomena characteristic of complex urban air flows. Vehicle trajectories are generated stochastically on the road network graph using Dijkstra’s algorithm, and each moving vehicle acts as a mobile line source of pollutant emissions. To reflect the daily variability of emissions, a time-dependent modulation function determined by unknown parameters is included in the source composition. These parameters are inferred by solving an inverse problem using synthetic concentration measurements from several fixed observation points throughout the area. The study presents two main contributions. Firstly, a detailed numerical analysis of how fractional derivatives affect pollutant dispersion under realistic time-varying mobile source conditions, and secondly, an evaluation of the performance of the proposed parameter estimation method for reconstructing time-varying emission rates. The results show that fractional-order models provide increased flexibility for representing anomalous transport and retention effects, and the proposed method allows for reliable recovery of emission dynamics from sparse measurements.

Keywords: convection–diffusion-reaction equation; Navier–Stokes equations; Caputo fractional derivative; Riesz fractional derivative; stochastic differential equation; inverse problem; parameter estimation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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