Exact Analytical Solution to the One-Dimensional Advective-Dispersive Equation with a Decaying Source Term

This paper presents an exact analytical solution of one-dimensional transport of a contaminant undergoing advection, dispersion, sorption, and first-order decay, subject to unique boundary conditions, a first-order decaying contaminant concentration in time at the source with a constant concentration at infinity. This exact solution provides a method to study a new area of physical processes and to evaluate approximate methods and models that have been developed to analyze these problems. Recently two fields have developed interest in modeling physical processes that are best conceptualized using a decaying source boundary condition. These two fields are long-term modeling of recalcitrant non-aqueous phase liquid (NAPL) spills and radioactive waste disposal. Both of which can be approximated using a decaying source term. This paper briefly introduces some general field transport problems that can conceptually be described using a decaying source boundary condition. The paper then describes the governing equations, boundary and initial conditions, and the solution techniques used to develop the analytical solution. Next the analytic solution is provided and discussed along with some example cases. The results of the exact analytical solution are compared to another similar solution from the literature.