Applied Mathematics and Physics
ISSN (Print): 2333-4878 ISSN (Online): 2333-4886 Website: https://www.sciepub.com/journal/amp Editor-in-chief: Vishwa Nath Maurya
Open Access
Journal Browser
Go
Applied Mathematics and Physics. 2022, 10(1), 24-31
DOI: 10.12691/amp-10-1-2
Open AccessArticle

Application of Advection Diffusion Equation for Determination of Contaminants in Aqueous Solution: A Mathematical Analysis

Muhammad Masudur Rahaman1, Humaira Takia2, , Md. Kamrul Hasan3, Md. Bellal Hossain1, Shamim Mia4 and Khokon Hossen2

1Department of Mathematics, Patuakhali Science and Technology University, Dumki, Patuakhali-8602, Bangladesh

2Department of Physics and Mechanical Engineering, Patuakhali Science and Technology University, Dumki, Patuakhali-8602, Bangladesh

3Department of Civil Engineering, Uttara University, House No. 4 & 6, Road No. 15, Uttara, Dhaka-1230, Bangladesh

4Department of Agronomy, Patuakhali Science and Technology University, Dumki, Patuakhali-8602, Bangladesh

Pub. Date: March 29, 2022

Cite this paper:
Muhammad Masudur Rahaman, Humaira Takia, Md. Kamrul Hasan, Md. Bellal Hossain, Shamim Mia and Khokon Hossen. Application of Advection Diffusion Equation for Determination of Contaminants in Aqueous Solution: A Mathematical Analysis. Applied Mathematics and Physics. 2022; 10(1):24-31. doi: 10.12691/amp-10-1-2

Abstract

Water pollution is one of the leading environmental issues faced especially at developing countries all over the World. However, the study of pollution movement is a necessary basis for solving water quality problems. The mathematical model for soluble and insoluble water pollutants has been formulated in light of partial differential equations. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection diffusion equation. Therefore, the study explored finite difference method for the numerical solutions of advection diffusion equation (ADE). Moreover, the study considered the ADE as an initial boundary value problem (IBVP) for the estimation of water pollution. By implementing a finite difference scheme for the IBVP, we examined as well as analyzed the extent of water pollution at different times and different points in an 1-dimensional spatial domain. We estimated relative error of the schemes in comparison with an exact solution of ADE and the numerical features of the rate of convergence are presented graphically. Finally, we compared our numerical solution of the ADE with solution obtained by finite element method. Numerical analysis indicates that water treatment is an essential tool for obtaining quality water for human consumption.

Keywords:
advection diffusion equation finite difference schemes water pollutants mathematical modeling feasibility of solution stability analysis

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Bartram, Jamie, and Richard Ballance, eds,“Water quality monitoring: a practical guide to the design and implementation of freshwater quality studies and monitoring programmes,” CRC Press, 1996.
 
[2]  Jackson, C., “Problems, Perceptions and Perfection - The Role of The Drinking Water Inspectorate in Water Quality Incidents and Emergencies,” In Internatinal Conference on Water Contamination Emergencies: Can We Cope? (pp. 38-43).2003.
 
[3]  Gane, Sharon, “Solution of the advection equation using finite difference schemes and the method of characteristics,” 2000.
 
[4]  Wang, Q., Li, S., Jia, P., Qi, C., & Ding, F. A Review of Surface Water Quality Models, 2013.
 
[5]  Khandoker Nasrin Ismet Ara, Md. Masudur Rahaman And Md. Sabbir Alam, “Numerical Solution Of Advection Diffusion Equation Using Semi-Discretization Scheme,” Scientific Research Publishing, Applied Mathematics, 12, 1236-1247, 2021.
 
[6]  Bhar, E. and Gurarslan, G., “Numerical Solution of Advection diffusion Equation Using Operator Splitting Method,” International Journal of Engineering & Applied Sciences, vol-9, 76-88, 2017.
 
[7]  Ahmed, S., A. “Numerical Algorithm for solving Advection Diffusion Equation with constant and variable coefficients,” The Open Numerical Methods Journal, vol-4, 1-7, 2012.
 
[8]  F. B. Agusto and O. M. Bamigbola, “Numerical Treatment of the Mathematical Models for Water Pollution,” Research Journal of Applied Sciences 2(5): 548-556, 2007.
 
[9]  Changjun Zhu and Shuwen Li,“Numerical Simulation of River Water Pollution Using Grey Differential Model,” Journal of computers, Vol. No.9, 2010.
 
[10]  A. Kumar, D. Kumar, Jaiswal and N. Kumar, “Analytical solution of one dimensional advection diffusion equation with variable coefficients in a finite domain,” J. Earth Syst. Sci. Vol. 118, No.5, pp. 539-549, 2009.
 
[11]  Dilip Kumar Jaiswal, Atul Kumar, Raja Ram Yadav, “Analytical solution of the advection diffusion equation with temporally dependent Coefficients,” Journal of water Resource and Protection, 3, 76-84,2011.
 
[12]  A. EI. Badia, T Ha-Duong and A. Hamdi, “Identification of a point source in a linear advection-dispersion-reaction equation: application to a pollution source problem,” Institute of Physics Publishing, Inverse Problems 21, 1-17, 2005.
 
[13]  Young San Park, Jong Jin Baik, “Analytical solution of the advection diffusion equation for a ground level finite area source,” Atmospheric Environment. 42, 9063-9069, 2008.
 
[14]  M. Thongmoon and R. Mckibbin, “A comparison of some numerical methods for the advection diffusion equation,” Inf. Math. Sci., Vol.10, pp49-52, 2006.
 
[15]  Ahsan, M., “Numerical Solution of The Advection Diffusion Equation Using Laplace Transform Finite Analytical Method,” River Basin Management, 10(June), 177-188, 2012.
 
[16]  Scott A. Socoloofsky Gerhard H. Jirka, “Advection Diffusion Equation”, 2004.
 
[17]  R. Szymkiewicz, “Solution of the Advection diffusion equation using the spline function and finite elements method,” Communications in numerical methods in engineering, vol. 9, 197-206, 1993.