Identify the source of pollution with an Inverse-time analytical solution to the pollution transport equation

Despite the existence of the regulations that penalize wastewater discharging into water sources, repeated violations in terms of discharging has been occurred for many years. Therefore, identifying the pollutant source is of high significance. Pollution introduced into the river is transmitted to the downstream via the advection-dispersion process. Direct advection-dispersion equation (Direct ADE) or positive time step can be solved using traditional numerical and analytical methods. In the true solution, the upstream Pollutant concentration time pattern is already specified, and the aim is to find the temporal and spatial distributions of pollutant concentration at downstream.In contrast, in inverse solution, a totally different approach is adopted. Inverse solution of ADE in the form of negative reverse time step falls into the category of the partial differential equations (III-posed), and the answers to this equation do not achieve convergence. Therefore, the quasi-reversibility method is used. The quasi-reversibility method is basically built upon the addition of the stability term, which includes the fourth-order term and the stabilization coefficient and plays a major role in the convergence of the inverse answers to ADE. In the next step, ADE and stability term are solved by Fourier transform (FT). This solution leads to a highly oscillatory integral using FT. By solving this integral, Pollutant sources time pattern (time series) can be identified. To resolve the integral, two hypothetical examples (simple and intricate) were used, and the validation of the numerical results indicates the high capability of the analytical model.