Uma nova perspectiva na solução da equação de advecção-difusão fracionária usando o método GILTT e derivadas conformáveis

Abstract

The dispersion of pollutants in the atmosphere is a perpetual source of challenging issues due to its intrinsic physical complexity, one of which is the description of anomalous dif fusion caused by atmospheric turbulence. This matter calls for new paradigms, and it is within this context that a growing interest in fractional partial differential equations arises. Such modeling provides a fresh perspective to analyze and comprehend turbulence com plexity, bridging the gap left by traditional differential equations, which fail to adequately describe turbulent diffusion in the atmosphere. This advancement brings forth new chal lenges in seeking solutions, as most existing approaches are based on techniques applied to integer-order differential equations. Given this scenario, the primary objective of this study is to explore the potential of fractional derivatives in the mathematical modeling of atmospheric pollutant dispersion. To achieve this goal, this thesis proposes the resolu tion of two distinct models of the fractional advection-diffusion equation. The first model addresses the transient two-dimensional fractional advection-diffusion equation, while the second deals with the stationary two-dimensional fractional advection-diffusion equation. To this end, the GILTT (Generalized Integral Laplace Transform Technique) method and the conformable derivative are combined. In the first model, fractional parameters are introduced in all terms of the equation, whereas in the second model, fractional para meters are considered in the longitudinal advective and diffusive terms. This procedure allows for the consideration of anomalous behavior in the dispersion process, resulting in a new methodology called α-GILTT. Simulations were compared with ground-level integrated concentration data from the Copenhagen and Prairie Grass experiments. In the case of the transient model, statistical results of pollutant concentration at ground level showed little influence of fractional parameters under low fractional conditions, as the Copenhagen experiment is considered moderately unstable. However, sensitivity tests with fractional parameters allow the conclusion that they exert an effective influence on controlling pollutant diffusion and advection in the atmosphere, suggesting dependence on atmospheric stability. Regarding the stationary model, statistical results indicate that fractional parameters have little influence in the Copenhagen experiments (moderately unstable) with the parameterizations used under low fractional conditions. However, for the Prairie Grass experiment (strongly convective), results showed greater dependence on fractional parameters (integer order: NMSE = 0.90, COR = 0.81, FAT2 = 0.63; non integer order: NMSE = 0.56, COR = 0.89, FAT2 = 0.84), suggesting that fractional parameters are dependent on atmospheric stability and can be used, along with good parameterization of traditional physical parameters (wind profiles and diffusion coeffici ents), to improve the atmospheric pollutant dispersion process. Lastly, a short-duration source was added to the transient model. Sensitivity tests demonstrated that fractional parameters influence the pollutant dispersion behavior for points located near the source.