Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions

Analytical solutions of the advection–dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originally established for diffusion type problems, to the case of advective–dispersive transport subject to transient (time-dependent) boundary conditions. Generalized analytical formulas are established which relate the exact solutions to corresponding time-independent auxiliary solutions. Explicit analytical expressions were developed for the instantaneous pulse problem formulated from the generalized Dirac delta function for situations with first-type or third-type inlet boundary conditions of both finite and semi-infinite domains. The developed generalized equations were evaluated computationally against other specific solutions available from the literature. Results showed the consistency of our expressions.

• We extend the Duhamel theorem to the case of advective–dispersive solute transport.
• Analytical formulas relate exact solutions to time-independent auxiliary solutions.
• Explicit analytical expressions are developed for selected particular cases.
• Results are compared with other specific solutions from the literature.