A note on the solution to a 1D advection–diffusion equation with exponentially decaying inlet boundary condition

Abstract

An analytical solution is presented for a one-dimensional advection–diffusion equation (ADE) with an exponentially decaying inlet boundary condition and a non-zero gradient at the outlet. The solution is derived using the Laplace transform method, and a numerical solution is obtained through an explicit finite difference scheme for comparison. The numerical results show good agreement with the analytical solution. Additionally, this work corrects an error in a previously published analytical solution (van Genuchten and Alves, 1982), which applied a zero-gradient condition at the outlet.