Analytical Solutions of ADE with Temporal Coefficients for Continuous Source in Infinite and Semi-Infinite Media

Abstract

In the present technical note, two aspects commonly used to obtain the analytical solution of advection–diffusion equation (ADE) with time-dependent coefficients describing solute transport due to a continuous source in infinite and semi-infinite porous media, respectively, have been addressed. One is regarding describing a continuous source in an infinite medium and the other is in the context of the analytical solution of the ADE with time-dependent coefficients in a semi-infinite medium. Primarily, this note establishes that in an infinite medium, the correct concentration attenuation pattern from a continuous or instantaneous source may be obtained through the solution of the ADE only if the pollutant’s source is defined by a nonhomogeneous term of the ADE in the form of the Dirac delta function, whereas the solution of the ADE with a first-order decay term when the continuous source is expressed by means of an initial condition in the form of the Heaviside function does not exhibit the expected attenuation pattern. In the second part, it is shown that the analytical solutions of the ADE with temporally dependent coefficients in a semi-infinite medium may be obtained, contrary to it being held in the literature that it could not be obtained.