# Analytical Solution of Pollution Transport Equation with Arbitrary Time Pattern of Multiple Point Sources using Green’s Function Method

Document Type : Research Article

Authors

1 MSc. Student, Water Structures Department, Tarbiat Modares University

2 Professor, Water Structures Department, Tarbiat Modares University

3 . Assistant Professor, Water Structures Department, Tarbiat Modares University

Abstract

Application of mathematical models of pollution transport in rivers is very important. It is necessary to utilize analytical solutions for verification of numerical methods.The purpose of this study is to determine 1-D analytical solution of the pollution transport equation (ADRE[1]) with constant velocity and dispersion coefficient for arbitrarily time patterns of multiple point sources using Green's function method (GFM). General solution of ADRE equation was determined in semi-infinite domain. Final explicit solution depends on the existence of Green’s function related to the original problem. In order to find the Green’s function of each problem, a powerful tool called “Adjoint Operator” was employed. By locating the Green’s function in the general solution associated with the main boundary value problem, the final solution of ADRE equation was specified. Verification of the proposed solution was achieved by comparing the present results to the ones of Van Genuchten and Alves (1982) for the same conditions of flow and time step pattern of entrance for the pollution loading. The results obtained from both solutions were completely consistent. To generalize the proposed solution, the concentration resulting from two point sources with irregular time pattern was determined using GFM. Due to the lack of analytical solution in these cases, the result were compared with the results obtained from MIKE11 model. The Final graphs and statistical analysis show good agreement between the results of MIKE11 and the proposed solution. The main innovation aspect of this research is determining the analytical solution of ADRE equation for multiple active point sources with irregular and arbitrary time pattern.

Keywords

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### History

• Receive Date: 22 October 2014
• Accept Date: 22 October 2014
• First Publish Date: 22 October 2014