Hostname: page-component-f554764f5-rvxtl Total loading time: 0 Render date: 2025-04-20T13:03:40.816Z Has data issue: false hasContentIssue false
  • English
  • Français

ON MODELLING WATER QUALITY WITH STOCHASTIC DIFFERENTIAL EQUATIONS

Part of: Stochastic analysis Equations and systems with randomness

Published online by Cambridge University Press:  09 January 2024

MAHMOUD B. A. MANSOUR Open the ORCID record for MAHMOUD B. A. MANSOUR [Opens in a new window]
MAHMOUD B. A. MANSOUR*
Affiliation:
Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on biochemical kinetics, a stochastic model to characterize wastewater treatment plants and dynamics of river water quality under the influence of random fluctuations is proposed in this paper. This model describes the interaction between dissolved oxygen (DO) and biochemical oxygen demand (BOD), and is in the form of stochastic differential equations driven by multiplicative Gaussian noises. The stochastic persistence problem for the model of the system is analysed. Further, a numerical simulation of the stationary probability distributions of BOD and OD by approximations of the stochastic process solution is presented. These results have implications for the prediction and control of pollutants.

Keywords

stochastic differential equationswater quality dynamicspersistence

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 34F05: Equations and systems with randomness 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 65 , Issue 3 , July 2023 , pp. 273 - 284
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Australian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Boano, F., Revelli, R. and Ridolfi, L., “Stochastic modelling of DO and BOD components in a stream with random inputs”, Adv. Water Resour. 29 (2006) 1341–1350; doi:10.1016/j.advwaters.2005.10.007.CrossRefGoogle Scholar
Chang, T. S. and Chen, C. T., “On the Routh–Hurwitz criterion”, IEEE Transact. Autom. Control 19 (1974) 250–251; doi:10.1109/TAC.1974.1100537.CrossRefGoogle Scholar
Cunha, A. C., Coneglian, C. M. R. and Poletti, E. C. C., “Sewage discharge and water self-decay: Streeter and Phelps model application”, Comput. Appl. Math. 37 (2018) 3514–3524; doi:10.1007/s40314-017-0526-x.CrossRefGoogle Scholar
Curi, W. E., Unny, T. E. and Kay, J. J., “A stochastic physical system approach to modeling river water quality”, Stoch. Hydrol. Hydraul. 9(1995) 117–132; doi:10.1007/BF01585602.CrossRefGoogle Scholar
Gray, A., Greenhalgh, D., Hu, L., Mao, X. and Pan, J., “A stochastic differential equation SIS epidemic model”, SIAM J. Appl. Math. 71 (2011) 876–902; doi:10.1137/10081856X.CrossRefGoogle Scholar
Guaca, D. C. and Poletti, E. C. C., “Modelling and numerical simulation of dissolved oxygen and biochemical oxygen demand concentrations with Holling type III kinetic relationships”, Appl. Math. Comput. 415 (2022) Article ID 126690; doi:10.1016/j.amc.2021126690.Google Scholar
Higham, D., “An algorithmic introduction to numerical simulation of stochastic differential equations”, SIAM Rev. 43 (2001) 525–546; doi:10.1137/S0036144500378302.CrossRefGoogle Scholar
Kallianpur, G. and Xiong, J., “Stochastic models of environmental pollution”, Adv. Appl. Probab. 26 (1994) 377–403; doi:10.2307/1427442.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations (Springer, Berlin–Heidelberg, 2010); doi:10.1007/978-3-662-12616-5.Google Scholar
Øksendal, B., Stochastic differential equations (Springer, Berlin–Heidelberg, 2003); doi:10.1007/978-3-642-14394-6.CrossRefGoogle Scholar
Papadopoulos, A. S. and Tiwari, R. C., “Bayesian approach for BOD and DO when the discharged pollutants are random”, Ecol. Modell. 71 (1994) 245–257; doi:10.1016/0304-3800(94)90136-8.CrossRefGoogle Scholar
Pimpunchat, B., “A mathematical model for pollution in a river and its remediation by aeration”, Appl. Math. Lett. 22 (2009) 304–308; doi:10.1016/j.aml.2008.03.026.CrossRefGoogle Scholar
Revelli, R. and Ridolfi, L., “Stochastic dynamics of BOD components in a stream with random inputs”, Adv. Water Resour. 27 (2004) 943–952; doi:10.1016/j.advwaters.2004.05.009.CrossRefGoogle Scholar
Stijnen, J. W., Heemink, A. W. and Ponnambalam, K., “Numerical treatment of stochastic river quality models driven by coloured noise”, Water Resour. Res. 39 (2003) 1053–1062; doi:10.1029/2001WR001054.CrossRefGoogle Scholar