Analytical Solution of the Advection-Diffusion Equation with Reaction and Spatially Dependent Coefficient for Exponentially Decay Source Injection

Abstract

This study presents an analytical investigation of the one-dimensional advection-diffusion equation (ADE) involving reaction and source term with spatially dependent coefficient for exponential decay boundary conditions. The ADE is introduced and subsequently solved using the Laplace transformation approach. A change of spatial variables is first introduced to reduce the governing equation to one with constant coefficients. The Laplace transform is then applied, and the inverse transformation is evaluated using the Gaver–Stehfest algorithm. The results demonstrate that the temporal evolution of the solution is strongly governed by diffusion coefficient, inhomogeneity and the velocity indicating their critical roles in characterizing the transport dynamics. Higher diffusion coefficients produce broader concentration distributions. Similarly for higher flow velocity with particularly away from the source, the difference between the velocities becomes clearer. Meanwhile, medium inhomogeneity affects the spatial distribution of concentration as the pollutant moves further from the source, whereas higher flow velocity extends the transport distance and allows the contaminant to persist over a larger region.