Analytical solution of two-dimensional fractional advection-diffusion equation with instantaneous source and time-varying coefficients

Abstract

This article presents an analytical study of the two-dimensional Advection-Diffusion Equation (ADE) incorporating the Caputo-Fabrizio (CF) time-fractional derivative. This work introduces a formulation with time-varying diffusion and advection, featuring an instantaneous point injection at the boundary. Analytical solutions in a half-plane are derived using the Laplace transform with respect to time and the Fourier transform with respect to spatial coordinates. The resulting solution, expressed in terms of special functions and integral representations, encompasses both the Fractional Advection-Diffusion Equation (FADE) and the classical ADE. The analysis shows that, for each fractional order, the concentration profile follows a bel-shaped curve where it begins at a low value, rises to a peak, and then decays with increasing distance from the source. Moreover, temporal variations in diffusion and velocity coefficients significantly affect transport dynamics, underscoring the importance of accounting for time-dependent factors to accurately model real-world fractional transport processes.