A Lyapunov-Sylvester numerical method for solving a reverse osmosis model

Saloua Helali; Anouar Ben Mabrouk; Mohamed Rashad; Nizar Bel Hadj Ali; Munirah A. Ȧlanazi; Marwah A. Alsharif; Elham M. Al-Ali; Lubna A. Alharbi; Manahil S. Mustafa; Saloua Helali; Anouar Ben Mabrouk; Mohamed Rashad; Nizar Bel Hadj Ali; Munirah A. Ȧlanazi; Marwah A. Alsharif; Elham M. Al-Ali; Lubna A. Alharbi; Manahil S. Mustafa

doi:10.3934/math.2024852

AIMS Mathematics

2024, Volume 9, Issue 7: 17531-17554. doi: 10.3934/math.2024852

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A Lyapunov-Sylvester numerical method for solving a reverse osmosis model

1.
Department of Physics, Faculty of Science, University of Tabuk, King Faisal road, Tabuk 47512, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, King Faisal road, Tabuk 47512, Saudi Arabia
3.
National School of Engineers, University of Gabes, Street Omar Elkhattab, Zrig 6029, Gabes, Tunisia
4.
Department of Computer Science, Faculty of Computers and Information Technology, University of Tabuk, King Faisal road, Tabuk 47512, Saudi Arabia
5.
Department of Statistic, Faculty of Sciences, University of Tabuk, King Faisal road, Tabuk 47512, Saudi Arabia

Received: 15 December 2023 Revised: 30 April 2024 Accepted: 09 May 2024 Published: 21 May 2024

Clean water is a necessity for many organisms, especially human life. Due to many factors, there is a significant shortage of potable water. This has led to efforts involving recovering water from wastewater or the sea through different technologies. Recently, the desalination of seawater via the reverse osmosis system has shown to be a promising method for drinking water treatment and recovery. Such a technique relies on mathematical models based on many parameters, resulting in special PDEs to model the reverse osmosis system. This paper develops a numerical method to solve a reverse osmosis model. The governing PDE is converted into a Sylvester equation that is proved to be uniquely solvable, stable, consistent, and convergent. The numerical scheme developed is validated with experimental data from the literature, and some numerical simulations.
- reverse osmosis,
- advection diffusion,
- numerical methods,
- Sylvester equation
Citation: Saloua Helali, Anouar Ben Mabrouk, Mohamed Rashad, Nizar Bel Hadj Ali, Munirah A. Ȧlanazi, Marwah A. Alsharif, Elham M. Al-Ali, Lubna A. Alharbi, Manahil S. Mustafa. A Lyapunov-Sylvester numerical method for solving a reverse osmosis model[J]. AIMS Mathematics, 2024, 9(7): 17531-17554. doi: 10.3934/math.2024852

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Abstract

Clean water is a necessity for many organisms, especially human life. Due to many factors, there is a significant shortage of potable water. This has led to efforts involving recovering water from wastewater or the sea through different technologies. Recently, the desalination of seawater via the reverse osmosis system has shown to be a promising method for drinking water treatment and recovery. Such a technique relies on mathematical models based on many parameters, resulting in special PDEs to model the reverse osmosis system. This paper develops a numerical method to solve a reverse osmosis model. The governing PDE is converted into a Sylvester equation that is proved to be uniquely solvable, stable, consistent, and convergent. The numerical scheme developed is validated with experimental data from the literature, and some numerical simulations.

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