Analytical Solutions for Advection and Advection-Diffusion Equations with Spatially Variable Coefficients

Analytical solutions are provided for the one-dimensional transport of a pollutant in an open channel with steady unpolluted lateral inflow uniformly distributed over its whole length. This practical problem can be described approximately by spatially variable coefficient advection and advection-diffusion equations with the velocity proportional to distance, and the diffusion coefficient proportional to the square of the velocity. Using a simple transformation, the governing equations can be transformed into constant coefficient problems that have known analytical solutions for general initial and boundary conditions. Analytical solutions to the spatially variable coefficient advection and advection-diffusion equations, written in conservative and nonconservative forms, are presented. The analytical solutions are simple to evaluate and can be used to validate models for solving the advection and advection-diffusion equations with spatially variable coefficients. The analytical solutions show that nonconservative forms of the equations can yield exact solutions that are not consistent with the physical problem.