Analytical solution of mass transport equation in river network

In this study, by Laplace transform method, the analytical solution of the pollution transport equation in the limited domain for the river network to the upstream and downstream Dirichlet boundary conditions, and the initial condition of zero was extracted, and simulation was performed for two branch and loop networks with fixed and variable boundary conditions. After naming the nodes, by forming matrices of how to connect, flow characteristics and geometry of the river for each network as input to the problem, the diffusion matrix is created based on a function of the Laplace variable, which The value of the concentration in each node is calculated by solving the complex device created and using the inverse Laplace transform. Then, using the analytical solution extracted in a branch of the river for the pollution transfer equation, the analytical solution can calculate the value of pollution concentration at any desired location and time along with the river network. Finally, for validation, the analytical solution was compared with the numerical solution, and then the statistical error indices were calculated. The results indicate the optimal performance and high ability of analytical solution in modeling the two networks and its good adaptation to the numerical solution, which can replace numerical solution due to high accuracy and speed of calculations. Generally, due to common errors in numerical solutions such as numerical dispersion error, Round-off error, Truncation error of Taylor expansion mathematical sentences, analytical solutions, if any, for the river network are recommended over numerical solutions. Also, in the performed simulations, due to the change of the inlet flow and the flow cross-section to each, changes in the pollution concentration occur in the areas where the branches connect to each other, increasing or decreasing. The proposed analytical solution for river networks can model more complex river networks and can be considered a criterion for the validation of numerical solutions. Also, the existing analytical solution can be used as a tool to validate other analytical solutions in the river network.