Mathematical model for expanding suitable areas for fish survival in a polluted river by using the solution of the two coupled pollution and aeration equations

Laplace transformation was used to derive an analytical solution for a one-dimensional advection–diffusion equation with exponential pollution source terms. Additionally, a numerical solution was obtained for two coupled advection–diffusion equations for the concentrations of pollutants and dissolved oxygen using an explicit finite difference scheme. In this study, it is assumed that the river has a uniform solute concentration prior to the injection of pollutants. At the origin, the input pollutant concentration source may increase over time for various reasons and the assumption is made that the concentration gradient approaches zero at x → ∞ . The parameters that have a role in reducing the concentration of pollutants along the river have been studied in detail with the help of figures. This simple model facilitates an understanding of the pollution-aeration process and its relationship to injected pollution along a river. The study discusses a straightforward procedure aimed at regulating farming, industrial, and urban practices, and imposing restrictions if necessary. With accuracy, this study determines the intervals within the river where fish can survive at a given time, along the river with the maximum permissible water velocity and the minimum permissible oxygen transfer from air to water. This study mathematically proves the fact that high pollutant concentrations can be reduced by increasing the concentration of dissolved oxygen.