Hybrid Approach for the Time-Dependent Fractional Advection–Diffusion Equation Using Conformable Derivatives

Abstract

Nowadays, several applications in engineering and science are considering fractional partial differential equations. However, this type of equation presents new challenges to obtaining analytical solutions, since most existing techniques have been developed for integer order differential equations. In this sense, this work aims to investigate the potential of fractional derivatives in the mathematical modeling of the dispersion of atmospheric pollutants by obtaining a semi-analytical solution of the time-dependent fractional, two-dimensional advection–diffusion equation. To reach this goal, the GILTT (Generalized Integral Laplace Transform Technique) and conformal derivative methods were combined, taking fractional parameters in the transient and longitudinal advective terms. This procedure allows the anomalous behavior in the dispersion process to be considered, resulting in a new methodology called α-GILTT. A statistical comparison between the traditional Copenhagen experiment dataset (moderately unstable) with the simulations from the model showed little influence on the fractional parameters under lower fractionality conditions. However, the sensitivity tests with the fractional parameters allow us to conclude that they effectively influence the dispersion of pollutants in the atmosphere, suggesting dependence on atmospheric stability.