Prediction of Air Pollution Levels Through Time-Fractional Advection-Diffusion Equation

Abstract

Traditional models are often not able to capture the (spatial) persistence phenomenon found in air transport so that it is still a challenge predicting the spread of large-scale contamination. In this paper, these long-term effects are accounted for in a one-dimensional time-fractional advection-diffusion equation (Caputo definition) based model. Uniform and non-uniform Dirichlet boundary conditions are applied to test the model. The Eigen function expansion is employed to obtain an analytic solution as it balances mathematical utility and physical understanding. More importantly, it is rigorously shown that the solution is well-posed by utilizing the necessary basic assumptions on the Lipschitz condition with regard to its proofs for the existence, uniqueness, and continuous dependence on the initial input values for the solution. This shows how the fractional calculus could be more precise with regard to the characteristics related to the dispersion of the pollutant in the real world than being confined to the completion of an equation for this task.