Analytical solution for river pollution modelling in determining the level of dissolved oxygen concentrations using laplace transforms

In the present study, the advection-diffusion equation is solved to determine the level of dissolved oxygen (DO) concentration. For problems related to river pollution, DO is the most important element to predict the water quality level other than Biochemical Oxygen Demand (BOD) or pollutant concentration. The problems are considered and solved either with dispersion oxygen or without dispersion for unsteady state condition which are developed to solve the problems in order to have a realistic model. Then, two cases are considered for each problem; (i) presence of half saturated oxygen for DO and (ii) absence half saturated oxygen for DO. For simplified cases, the linear model is analytically solved by using Laplace and inverse Laplace transform method. But, for the nonlinear model we solved numerically by finite difference method. Then, the results are presented graphically with the dependency on various respective parameters. Finally, the comparison between both steady and unsteady states is made. Such a mathematical model with analytical solutions can help the management of water quality and generically able to solve river pollution problems with slight modifications for other rivers of interest