# Mathematical model for pollutant and virus dispersion: An analytical solution for pulse type input source

### Abstract

The mathematical model is a tool employed in the current study to forecast the spread of pollutants/viruses. The effect and treatment of infectious viruses/pollutants are described using the transport equation, advection-dispersion equation (ADE). The analytical solution of the one-dimensional advection-dispersion equation yields a description of the propagation of pollutant/viruses along the flow domain from a point source. Using the Laplace transform approach, an analytical solution is found. The spatial-temporal dependent functions are used to calculate the ADE coefficient. The adsorption (retardation) coefficient is also considered to be inversely proportional to flow velocity with temporal dependency and zero order production coefficients are taken as inversely proportional to adsorption parameter. The environment also contains a small amount of infectious viruses and pollutants in a variety of forms. In this case, the initial concentration is also taken into account as the sum of the initial pollutant/virus concentration and the ratio of the zero order production to the flow rate. There are two boundary conditions, one is a pulse-type input source boundary, and the other is an infinite-range flux source barrier. The ADE variable is reduced to a constant coefficient upon the addition of new space and time coordinates. Examples are provided in the section on results and discussions.