Instantaneous and exponential decay point source effects on two-dimensional heavy metal transport with adsorption and desorption in soil

In the mathematical modeling of the advection-diffusion equation (ADE) governing the transport of heavy metals in soil, it is essential to consider scenarios where the pollution source is not localized at the boundary but rather introduced as a source term within the equation, reflecting realistic situations where abrupt contamination events may occur within the soil matrix. Considering also adsorption and desorption, the ADE incorporate additional source term. This source term representing as a point source that satisfies instantaneous release and exponential decay release which can be used to simulate different types of pollution events in soil. Utilizing the Laplace transform method to obtain solutions, the dynamics of pollutant behavior in both cases are investigated. The results show that regardless of release type, the concentration reaches a peak near the source point area and remains higher while stabilizing over time. However, when the retardation factor varied, both shows similar behavior on the first day only but not at the fourth day. Additionally, the amount of particles release and exponential decay coefficient follows or aligns with its equations.