A Model Using Fractional Derivatives with Vertical Eddy Diffusivity Depending on the Source Distance Applied to the Dispersion of Atmospheric Pollutants PAULO HENRIQUE FARIAS XAVIER, 1 ERICK GIOVANI SPERANDIO NASCIMENTO, 1 and DAVIDSON MARTINS MOREIRA 1 Abstract—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 / x a ). For validation of the model, simulations were compared with data from Copenhagen experiments considering moderately unstable conditions. The best results were obtained with a = 0.98, considering wind measured at 10 m, and a = 0.94 with wind measured at a height of 115 m. Key words: Decomposition method, advection–diffusion equation, planetary boundary layer, pollutant dispersion. 1. Introduction The process of pollutant dispersion is usually modelled using the advection–diffusion equation; thus, the scientific community has a great interest in obtaining numerical and analytical solutions of this equation. In this sense, there are significant scientific advances in relation to obtaining analytical/semi-an- alytical solutions of the advection–diffusion equation with traditional derivatives (integer order) using dif- ferent methodologies (Rounds 1955; Yeh and Huang 1975; Moreira et al. 2005, 2009, 2014; Sharan and Modani 2006; Essa et al. 2007; Tirabassi et al. 2008; Pimentel et al. 2014). In addition, in recent years there has been a strong motivation in the different areas of knowledge with non-integer order equations, that is, fractional derivatives (Debnath 2003), where very recently, solutions were proposed for the advection–diffusion equation of fractional order in atmospheric pollutant dispersion. The first work was given by Goulart et al. (2017), with a solution obtained through the method of separation of vari- ables. The second work, by Moreira and Moret (2018), used the technique of integral transformations. However, the literature still has a gap in obtaining analytical solutions of the advection–diffusion equa- tion considering integer and fractional order derivatives in atmospheric dispersion problems, par- ticularly using Laplace’s decomposition and homotopy perturbation methods. The homotopy per- turbation method was developed by the Chinese mathematician He (1999) and has been employed to solve a wide variety of linear and non-linear prob- lems in several areas (Ganji and Rafei 2006; Wang et al. 2008; Yildirim and Kocak 2009; among others). This method introduces a homotopy parameter that varies between 0 and 1 and has a striking feature that only a few disturbance terms are sufficient to obtain an accurate solution. Thus, the novelty of the present work is the use of the homotopy perturbation method coupled with the Laplace transformation to solve the fractional order advection–diffusion equation with eddy diffusivity that is dependent on the longitudinal distance of the source with exponent of the same order of the fractional derivative (K / x a ). This methodology is an elegant combination and can provide an exact (or approximate) solution to a given equation in a simpler way than the traditional meth- ods found in the literature. It is noted that the memory effect, which is important in regions close to high and low sources, has usually been taken into account by the vertical eddy diffusivity, which depends on the distance from 1 SENAI CIMATEC, Av. Orlando Gomes, 1845-Piata ˜, Sal- vador, Bahia, Brazil. E-mail: davidson.moreira@gmail.com Pure Appl. Geophys. Ó 2018 Springer Nature Switzerland AG https://doi.org/10.1007/s00024-018-1977-8 Pure and Applied Geophysics