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Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation

Received: 23 November 2015     Accepted: 6 December 2015     Published: 8 January 2016
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Abstract

In the present study we have applied diffusion – reaction equation to describe the dynamics of river pollution and drawn numerical solution through simulation study. The diffusion-reaction equation is turn to be a partial differential equation since the independent variables are more than one that include spatial and temporal coordinates. The diffusion-reaction equation is widely applied to environmental studies in general and to river pollution studies in particular. River pollution models are special cases and are included in the broad area known as environmental studies. The diffusion – reaction equation is characterized by the reaction term. When the reaction term depends on the concentration of the contaminants then the original single diffusion-reaction equation will evolve to be a system of equations and this lead to analytical problems. The diffusion-reaction equations are difficult to solve analytically and hence we consider numerical solutions. For this purpose we first separate diffusion and reaction terms from the diffusion-reaction equation using splitting method and then apply numerical techniques such as Crank – Nicolson and Runge – Kutta of order four. These numerical methods are preferred because the systems of equations are solved accurately and efficiently. Detailed discussion of the results and their interpretations are included.

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)
DOI 10.11648/j.ajam.20150306.24
Page(s) 335-340
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

River Pollution, Dissolved Oxygen, Biological Oxygen Demand, Diffusion-Reaction Equation, Splitting Method, Simulation Study, Crank – Nicolson Method, Runge – Kutta Method

References
[1] David J. and Logan (2006). Applied Mathematics, John Wiley and Sons, Interscience.
[2] Tsegaye Simon (2013). Numerical Simulation of Diffusion – Reaction Equations: Application from River Pollution Model, Hawassa University, Hawassa, Ethiopia (Unpublished M. Sc. Thesis).
[3] Won Y., Wenwu C., Tae – Sang C. and John M. (2005). Applied Numerical Methods Using MATLAB.
[4] Aly – Khan K. (2003). Solving Reaction – Diffusion Equations 10 Times Faster.
[5] Sanderson A. R., Meyer M. D., Kirby R. M. and Johnson C. R. (2007). A Frame works for Exploring Numerical Solutions of Advection-Reaction-Diffusion Equations Using a GPU-Based Approach, Computer visual Sci. Vol. (10), PP. 1-16.
[6] Holzbecker E. (2007), Environmental Modeling: Using MATLAB.
[7] Nas S. S., Bayram A., Nas E. and Bulut V. N. (2008). Effects of Some Water Quality Parameters on the Dissolved Oxygen Balance of Streams, Polish J. of Environ. Stud. Vol.17, PP. 531-538.
[8] Craster R. V. and Sassi R. (2006). Spectral Algorithms for Reaction – Diffusion Equations.
[9] Gerischa A. and Chaplain M. A. J. (2004). Robust Numerical Methods for Taxis – Diffusion – Reaction systems: Applications to Biomedical Problems. Mathematical and Computer Modeling 43 (2006), Pp. 4975.
[10] Hamdi A. (2006). Identification of Point Sources in Two Dimensional Advection – Diffusion – Reaction Equation: Application to Pollution Sources in a River. Stationary Case. Inverse Problems in Science and Engineering, 15, 8, 885–870.
[11] Scott A. S. (2012). A Local Radial Basis Function Method for Advection-Diffusion-Reaction Equations on Complexly Shaped Domains.
[12] Sportisse B. (2007). A Review of Current Issues in Air Pollution Modeling and Simulation, Computer Geo Sci. Vol. 11, Pp. 159-181.
[13] Verwer J., Hundsdorfer G., Willem H. and Joke G. (2002). Numerical Time Integration for Air Pollution Models, Surv. Math. Ind. Vol. 10, Pp. 107–174.
[14] Brain J. McCartin Sydney B. and Forrester Jr. (2002). A Fractional Step – Exponentially. Fitted Hopscotch Scheme for the Streeter – Phelps Equations of River Self – purification, Engineering Computations. Vol. 19(2), and Pp. 177–189.
[15] Mesterton – Gibbons M. (2007). A Concrete Approach to Mathematical Modelling, John Wiley and Sons.
[16] Kiely G. (1997). Environmental Engineering, McGraw-Hill.
[17] Mihelcic J. R. (1999). Fundamentals of Environmental Engineering, Wiley.
[18] Schnoor J. (1996). Environmental Modeling: Fate and Transport of Pollutants in Water, Air, and Soil, Wiley – Interscience.
[19] Evans G., Blackdge J. and Yardley. (2000). Numerical Methods for Partial Differential Equations. Springer – Verlag, London.
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[22] Shahraiyni Taheri H. and Ataie B. (2009). Comparison of Finite Difference Schemes for Water Flow in Unsaturated Soils, International Journal of Aerospace and Mechanical Engineering. Vol. 3(1), Pp. 1–5.
[23] Huangsdorfer W. (1996). Numerical Solution of Advection – Diffusion – Reaction Equations, Lecture Notes for a PhD Course, CWI Netherlands.
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  • APA Style

    Tsegaye Simon, Purnachandra Rao Koya. (2016). Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation. American Journal of Applied Mathematics, 3(6), 335-340. https://doi.org/10.11648/j.ajam.20150306.24

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    ACS Style

    Tsegaye Simon; Purnachandra Rao Koya. Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation. Am. J. Appl. Math. 2016, 3(6), 335-340. doi: 10.11648/j.ajam.20150306.24

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    AMA Style

    Tsegaye Simon, Purnachandra Rao Koya. Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation. Am J Appl Math. 2016;3(6):335-340. doi: 10.11648/j.ajam.20150306.24

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  • @article{10.11648/j.ajam.20150306.24,
      author = {Tsegaye Simon and Purnachandra Rao Koya},
      title = {Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {6},
      pages = {335-340},
      doi = {10.11648/j.ajam.20150306.24},
      url = {https://doi.org/10.11648/j.ajam.20150306.24},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.24},
      abstract = {In the present study we have applied diffusion – reaction equation to describe the dynamics of river pollution and drawn numerical solution through simulation study. The diffusion-reaction equation is turn to be a partial differential equation since the independent variables are more than one that include spatial and temporal coordinates. The diffusion-reaction equation is widely applied to environmental studies in general and to river pollution studies in particular. River pollution models are special cases and are included in the broad area known as environmental studies. The diffusion – reaction equation is characterized by the reaction term. When the reaction term depends on the concentration of the contaminants then the original single diffusion-reaction equation will evolve to be a system of equations and this lead to analytical problems. The diffusion-reaction equations are difficult to solve analytically and hence we consider numerical solutions. For this purpose we first separate diffusion and reaction terms from the diffusion-reaction equation using splitting method and then apply numerical techniques such as Crank – Nicolson and Runge – Kutta of order four. These numerical methods are preferred because the systems of equations are solved accurately and efficiently. Detailed discussion of the results and their interpretations are included.},
     year = {2016}
    }
    

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  • TY  - JOUR
    T1  - Modeling and Numerical Simulation of River Pollution Using Diffusion-Reaction Equation
    AU  - Tsegaye Simon
    AU  - Purnachandra Rao Koya
    Y1  - 2016/01/08
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ajam.20150306.24
    DO  - 10.11648/j.ajam.20150306.24
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 335
    EP  - 340
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20150306.24
    AB  - In the present study we have applied diffusion – reaction equation to describe the dynamics of river pollution and drawn numerical solution through simulation study. The diffusion-reaction equation is turn to be a partial differential equation since the independent variables are more than one that include spatial and temporal coordinates. The diffusion-reaction equation is widely applied to environmental studies in general and to river pollution studies in particular. River pollution models are special cases and are included in the broad area known as environmental studies. The diffusion – reaction equation is characterized by the reaction term. When the reaction term depends on the concentration of the contaminants then the original single diffusion-reaction equation will evolve to be a system of equations and this lead to analytical problems. The diffusion-reaction equations are difficult to solve analytically and hence we consider numerical solutions. For this purpose we first separate diffusion and reaction terms from the diffusion-reaction equation using splitting method and then apply numerical techniques such as Crank – Nicolson and Runge – Kutta of order four. These numerical methods are preferred because the systems of equations are solved accurately and efficiently. Detailed discussion of the results and their interpretations are included.
    VL  - 3
    IS  - 6
    ER  - 

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Author Information
  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

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