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Three-Dimensional Analytical Solution of the Fractional Atmospheric Pollutant Dispersion Equation Considering Caputo and Conformable Derivatives

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Abstract

In this work, two analytical solutions for the 3D steady-state fractional advection–diffusion equation are obtained in order to study the dispersion of air pollutants. The first solution was obtained considering Caputo's derivative, and the other considered a conformable derivative. The Laplace decomposition method (LDM), which generates a fast convergence series solution, was used to obtain the solutions. The solution generated from the Caputo derivative (nonlocal derivative) is represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable derivative (local derivative) generates an exponential function. The solutions were only identical when they were of integer order (\(\alpha\) = 1). In addition, sensitivity analyses were performed on two explicit parameters obtained in the solutions: fractional parameter \(\alpha\), which controls the order of the fractional derivative, and parameter \(\varphi\), responsible for maintaining the correct dimensionality of the equations. When Caputo and conformable solutions utilized data from the traditional Copenhagen experiment with \(\varphi = 1\) m and \(\alpha\) = 0.98, the values of statistical indexes NMSE = 0.10, COR = 0.90, and FAT2 = 0.91 were shown in both with only small differences in FB and FS, and these results are better than some of the more complex integer-order equation models already existing in the literature. In practical terms, using experimental data moderately convective, it is clear that the models did not show statistically significant differences in the concentrations of pollutants simulated at ground level under conditions of low fractionality, even when considering the dimensional parameter ranging from unity to the smallest scale of atmospheric turbulence (\(\varphi \simeq 10^{ - 3}\) m, Kolmogorov length scale). However, when fractionality increased, the microscale dimensional parameter became more effective, and the results quickly deteriorated.

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Acknowledgements

This work was supported in part by CNPq and FAPESB.

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  1. SENAI CIMATEC, Salvador, BA, Brazil

    José Humberto de Souza Prates, José Roberto Dantas da Silva, Filipe Milani de Souza & Davidson Martins Moreira

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de Souza Prates, J.H., da Silva, J.R.D., de Souza, F.M. et al. Three-Dimensional Analytical Solution of the Fractional Atmospheric Pollutant Dispersion Equation Considering Caputo and Conformable Derivatives. Pure Appl. Geophys. 179, 3411–3426 (2022). https://doi.org/10.1007/s00024-022-03114-9

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