Analytical solution of the pollution transport equation with variable coefficients in river using the Laplace Transform

Rivers are one of the most important natural water resources in the world. Pollution transport modeling in rivers is performed by the partial advection-dispersion-reaction equation (ADRE). In the present study, using the Laplace transform, which is a powerful and useful tool in solving differential equations, the analytical solution of the ADRE equation was obtained in a finite domain with variable coefficients for the upstream and downstream Dirichlet boundary conditions and the initial zero condition in the river. To use the analytical solution in this study, three examples are presented, each of which, the river are divided into two, four, and eight parts, which, while the parameters of flow, pollution, and river geometry are variable in all three examples, for each of the examples, the accuracy of the analytical solution available when the segmentation of the intervals increases as compared to the numerical solution. By specifying the matrices of velocity, dispersion coefficient, cross-section, etc. as input to the problem, the diffusion matrix is calculated and, consequently, a complex system of equations is created that doubles the complexity of the work. The amount of pollutant concentration is calculated by solving the system of the above equations. The numerical solution is used to validate the existing analytical solution, the results showed that the greater the number of river divisions, the higher the accuracy of the solution, and the two analytical and numerical solutions will be well compatible with each other. Given the ability and performance of the existing analytical solution, it can be acknowledged that the analytical solution in this study can be used as a tool to validate and verification numerical solutions and other analytical solutions for the coefficients of the equation.