An Integral Solution for the Diffusion‐Advection Equation

Abstract

A new integral formulation, based on Green's second identity, is used to solve the unsteady transport (diffusion‐advection) equation which governs the storage and movement of pollutants in porous media. It uses the fundamental solution for the terms of the differential equation with the highest derivatives (the order of the equation) and, by applying Green's second identity, those terms are cast into a boundary integral while the remaining terms with lower derivatives are weighted with that fundamental solution and integrated over the solution domain. The resulting integral representation is no longer a boundary integral but comprises both boundary and domain integral portions. Because the terms with the highest derivatives constitute the Laplacian, the fundamental solution employed in the formulation is the logarithmic function. The boundary integral portion of the formulation is similar to that encountered when the boundary element method is applied to elliptic equations but the domain portion, which contains the temporal derivative and advection terms, is evaluated by discretizing the solution domain into elements, as is frequently done in finite element formulations. The classic one‐dimensional semi‐infinite and a two‐dimensional semi‐infinite diffusion‐advection problems are solved for a wide range of local Peclet numbers in order to verify and demonstrate the usefulness of the present formulation. The results are satisfactory.