Application of Mathematical Model in Biological Passive Defense: Simulation of Pollution Flow in Groundwater

Pollution of water resources poses a significant and fundamental challenge to human life and the well-being of all living organisms. The presence of management gaps in this critical area creates vulnerabilities for potential bioterrorist attacks and contamination of water sources. An essential and practical approach to managing groundwater resources, particularly in identifying the time and location of pollutant releases into water, involves the utilization of mathematical models    governing pollutant flow. Compared to other practical methods, mathematical modeling offers a reliable and cost-effective solution that is computationally feasible. The primary objective of this research is to simulate the initial pollutant intensity function (at t=0) through a mathematical backward in time inverse problem. This issue stands as one of the foremost challenges in water resources management and engineering, garnering significant attention from researchers. In this study, initially, a mathematical model of pollutant flow in groundwater is presented, where the  initial pollutant intensity function is unknown. Then, a new numerical method based on the      pseudo-solution method is employed to calculate an approximation for this function using a linear combination of Bernstein polynomials. The mathematical model is numerically analyzed and   scrutinized through a specific example. The numerical results indicate that the initial pollutant intensity function can be precisely simulated by utilizing the pollutant intensity function at any time t>0. For an error level of δ=0.01 in the input data, the approximate solution obtained through the introduced numerical method exhibits a maximum error of 4%.