A Unified Three-Dimensional Extended Fractional Analytical Solution for Air Pollutants Dispersion

Abstract

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

Fractional advection diffusion asymmetry equation, derivation, solution and application

“…The non-Markovian characteristic of CTRW becomes apparent, especially when the disorder is introduced into the system, as observed in the sense of ensemble average [10]. Fractional kinetic equations [4,[24][25][26][27][28][29][30] are a popular tool used to describe various anomalous phenomena [31,32]. When the bias of the system is a constant, the fractional time diffusion equation follows [33,34]…”

Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer

“…Refs. [3,[5][6][7][8][9] discuss the relevance of using the fractional derivative in studying dispersion contaminants in the planetary boundary layer (PBL). The authors of [5,10], solving a simple fractional derivative equation, describing the steady-state spatial distribution of a non-reactive pollutant and comparing the results with those of real experiments, show some advantageous traits of the fractional model.…”

“…Refs. [3,[5][6][7][8]18] developed the modeling of pollutant transport under turbulence with Caputo and Riemann-Liouville fractional derivatives that present the anomalous diffusion phenomena. Here, we also include realistic forms of the vertical diffusion coefficient and windfield.…”

“…Fractional atmospheric models are introduced and analyzed in [3,[5][6][7][8]18,31]. In [3,5,6], the fractional differential equation in the longitudinal and vertical directions is used to derive the concentration distribution of pollutants in the PBL.…”

“…The authors compare the results with the one obtained by the traditional Gaussian model and report a very good fitting of the fractional model to the experimental data. Closed-form solutions for solving the fractional dispersion problem under a variable atmospherical hypothesis in a vertical non-homogeneous PBL is proposed in [8]. Better performance of the fractional-order model in comparison with that of the integer order is observed.…”