A Model Using Fractional Derivatives with Vertical Eddy Diffusivity Depending on the Source Distance Applied to the Dispersion of Atmospheric Pollutants
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This work presents an analytical solution of the two-dimensional advection–diffusion equation of fractional order, in the sense of Caputo and applied it to the dispersion of atmospheric pollutants. The solution is obtained using Laplace decomposition and homotopy perturbation methods, and it considers the vertical eddy diffusivity dependency on the longitudinal distance of the source with fractional exponents of the same order of the fractional derivative (K\(\propto\)xα). For validation of the model, simulations were compared with data from Copenhagen experiments considering moderately unstable conditions. The best results were obtained with α = 0.98, considering wind measured at 10 m, and α = 0.94 with wind measured at a height of 115 m.
KeywordsDecomposition method advection–diffusion equation planetary boundary layer pollutant dispersion
We thank the financial support of CNPq and FAPESB.
- Crank, J. (1979). The mathematics of diffusion (p. 414). Oxford: Oxford University Press.Google Scholar
- Jumarie, G. (2008). Fourier’s transformation of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivatives. Journal of Applied Mathematics & Informatics, 26, 1101–1121.Google Scholar
- Pimentel, L. C. G., Perez-Grerrero, J. S., Ulke, A. G., Duda, F. P., & Heilbron Filho, P. F. L. (2014). Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470, 20140021.CrossRefGoogle Scholar
- Podlubny, I. (1999). Fractional differential equations (p. 340). Cambridge: Academic.Google Scholar
- Wang, Y.X., Si, H.Y. & Mo, L.F. (2008). Homotopy perturbation method for solving reaction-diffusion equations. Mathematical Problems in Engineering Article ID 795838.Google Scholar