Asymptotic Solution for a Water Quality Model in a Uniform Stream

We employ approximate analytical method, namely, Optimal Homotopy Asymptotic Method (OHAM), to investigate a one-dimensional steady advection-diffusion-reaction equation with variable inputs arises in the mathematical modeling of dispersion of pollutants in water is proposed. Numerical values are obtained via Runge-Kutta-Fehlberg fourth-fifth order method for comparison purpose. It was found that OHAM solution agrees well with the numerical solution. An example is included to demonstrate the efficiency, accuracy, and simplicity of the proposed method. 1. Introduction Differential equations have been the focus of many studies due to their frequent appearance in various applications in physics, fluid mechanics, biology, and engineering. Consequently, considerable attention has been given to the solutions of higher order ordinary differential equations, integral equations, and fractional order partial differential equations of physical interest. Number of literatures concerning the application of higher order differential equations in nonlinear dynamics has grown rapidly in the recent years [1–5]. Several numerical and semianalytical methods have been developed for solving high order boundary value problems [6–9]. A mathematical model for the dispersion of pollutants in a river is presented. The optimal homotopy asymptotic method for assessment of the chemical oxygen demand (COD) concentration in a river is considered. Pochai and Tangmanee [10] have provided a mathematical model of water pollution with the help of numerical method. Furthermore, Pochai and coworkers [11–14] have used numerical methods for the solution of hydrodynamic model with constant coefficients in the uniform reservoir and stream. The optimal homotopy asymptotic method is an approximate analytical tool that is simple and straightforward and does not require the existence of any small or large parameter as does traditional perturbation method. Optimal Homotopy Asymptotic Method (OHAM) has been successfully applied to a number of nonlinear problems arising in fluid mechanics and heat transfer by various researchers [15–19]. This paper is organized as follows. First in Section 2, advection-diffusion-reaction equation is presented. In Section 3 we described the basic principles of OHAM. The OHAM solution of the problem is given in Section 4. Section 5 is devoted for the concluding remarks. 2. Dispersion in a Stream The dispersion of chemical oxygen demand (COD) is described by the advection-diffusion-reaction equation (ADRE) [11] in the domain : where is the concentration of COD at the