Optimal control of water pollutant transmission by utilizing a combined Jacobi collocation method and mountain Gazelle algorithm

Articles in Press

Document Type : Research Article

Authors

• A. Ebrahimzadeh ¹
• R. ‎Khanduzi ²
• R. Mirabbasi ³
• E. Hashemizadeh ⁴

¹ Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran.

² Department of Mathematics and Statistics, Gonbad Kavous University, P.O. Box 49717- 99151, Gonbad Kavous, Golestan, Iran.

³ Department of Water Engineering, Faculty of Agriculture, Shahrekord University, Shahrekord, Iran.

⁴ Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

10.22067/ijnao.2024.89204.1491

Abstract

Water pollution can have many adverse effects on the environment and human health. The study of the transmission of water pollutants over a finite lifespan is carried out using an optimal control problem (OCP), with the system governed by ordinary differential equations. By utilizing the collocation approach, the OCP is transmuted to a nonlinear programming problem, and then the mountain Gazelle algorithm is applied to determine the optimal control and state solutions. A practical study demonstrates the effect of treatment on reducing water pollutants during a finite time.

Keywords

• Optimal control
• Jacobi polynomials
• Transmission of water pollutant
• Collocation method
• Mountain Gazelle algorithm

Main Subjects

• Computational Mathematics

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References

[1] Abdollahzadeh, B., Gharehchopogh, F.S., Khodadadi, N., and Mirjalili, S. Mountain gazelle optimizer: a new nature-inspired metaheuristic al-gorithm for global optimization problems, Adv. Eng. Softw. 174 (2022), 103282.

[2] Aribowo, W., Abualigah, L., Oliva, D., and Prapanca, A. A novel mod-ified mountain gazelle optimizer for tuning parameter proportional in-tegral derivative of DC motor, Bull. Electr. Eng. Inform. 13(2) (2024) 745–752.
[3] Bonyaha, E., Agbekpornub, P., and Unlud, C. Mathematical modeling of transmission of water pollution, J. Prime Res. Math., 17(2) (2021), 20–38.
[4] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., Spectral methods: fundamentals in single domains, Springer Science and Business Media, 2007.
[5] Cizmas, L., Sharma, V.K., Gray, C.M., and McDonald, T.J., Pharma-ceuticals and personal care products in waters: occurrence, toxicity, and risk, Environ. Chem. Lett. 13 (2015) 381–394.
[6] Doha, E.H., Bhrawy, A.H., and Ezz-Eldien, S.S., A new Jacobi opera-tional matrix: an application for solving fractional differential equations, Appl. Math. Model. 36(10) (2012) 4931–4943.
[7] Doha, E.H., On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A: Math. Gen. 37(3) (2004) 657.
[8] Drag, P., Styczeń, K., Kwiatkowska, M., and Szczurek, A., A review on the direct and indirect methods for solving optimal control problems with differential-algebraic constraints. In Recent Advances in Computational Optimization: Results of the Workshop on Computational Optimization WCO 2014 (pp. 91–105). Springer International Publishing, 2016.
[9] Gong, Q., Ross, I.M., Kang, W., and Fahroo, F., On the pseudospectral covector mapping theorem for nonlinear optimal control, In Proceedings of the 45th IEEE Conference on Decision and Control (pp. 2679–2686) (2006) IEEE.

[10] Ebrahimzadeh, A., Hashemizadeh, E., Mirabbasi, R., A numerical so-lution of the mathematical models for water pollution by shifted Jacobi polynomials, TWMS J. Appl. Eng. Math. 14 (2023) 1–15.
[11] Gael, P.K. Water pollution: Causes, effects and control, New Age Inter-national, 2006.
[12] Guo, G., and Cheng, G. Mathematical modelling and application for simulation of water pollution accidents, Process Saf. Environ. Prot. 127 (2019), 189–196.
[13] Khodadadi, N., El-Kenawy, E.S.M., De Caso, F., Alharbi, A.H., Khafaga, D.S., and Nanni, A. The mountain Gazelle optimizer for truss structures optimization, Appl. Comput. Inform. 3(2) (2023), 116–144.
[14] Hashemizadeh, E., and Ebadi, M.A. A numerical solution by alternative Legendre polynomials on a model for novel coronavirus (COVID-19), Adv. Differ. Equ. 2020(1) (2020), 223.
[15] Hussaini, M.Y., and Zang, T.A. Spectral methods in fluid dynamics, Annu. Rev. Fluid Mech. , 19(1) (1987), 339–367.
[16] Kroeze, C., Gabbert, S., Hofstra, N., Koelmans, A.A., Li, A., Löhr, A., Ludwig, F., Strokal, M., Verburg, C., Vermeulen, L. and van Vliet, M.T. Global modelling of surface water quality: a multi-pollutant approach, Curr. Opim. Environ. Sustain. 23 (2016), 35–45.
[17] Luke, Y. The special functions and their approximations, vol. 2, Aca-demic Press, New York, 1969.
[18] Maleknejad, K., and Ebrahimzadeh, A. An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems, Math. Commun., 19(2) (2014) 417–435.
[19] Magdy, F. E. Z., Hasanien, H. M., Sabry, W., Ullah, Z., Alkuhayli, A., and Yakout, A. H. Mountain gazelle algorithm-based optimal control strategy for improving LVRT capability of grid-tied wind power stations. IEEE Access, 11 (2023) 129479-129492.

[20] Maashi, M., Alabdulkreem, E., Maray, M., Shankar, K., Darem, A.A., Alzahrani, A., and Yaseen, I. Elevating survivability in next-gen IoT-Fog-Cloud networks: Scheduling optimization with the metaheuristic mountain gazelle algorithm, IEEE Transactions on Consumer Electron-ics, 2024.
[21] Parsaie, A., and Haghiabi, A.H. Computational modeling of pollution transmission in rivers, Appl. Water Sci. 7 (2017), 1213-1222.
[22] Rao, A. V. A survey of numerical methods for optimal control, Adv. Astronaut. Sci. 135(1) (2009) 497–528.
[23] Ren, H., Lin, F., Tao, Y., Wei, T., Kang, B., Li, Y., and Li, X. Research on the Optimal Regulation of Sudden Water Pollution, Toxics, 11(2) (2023), 149.
[24] Sadjadi, S.J., and Ponnambalam, K. Advances in trust region algorithms for constrained optimization, Appl. Numer. Math. 29(3) (1999), 423–443.
[25] Shah, N.H., Patel, S.N., Satia, M.H., and Thakkar, F.A. Optimal control for transmission of water pollutants, Int. J.Math. Eng. Manag. Sci. 3(4) (2018), 381–391.
[26] Tang, W., Pei, Y., Zheng, H., Zhao, Y., Shu, L., and Zhang, H. Twenty years of China’s water pollution control: Experiences and challenges, Chemosphere, 295 (2022) 133875.
[27] Utama, D.M., Sanjaya, B.D., and Nugraha, A. A modified mountain gazelle optimizer for minimizing energy consumption on no-wait per-mutation flow shop scheduling problem, Int. J. Intell. Eng. Syst. 17(2) (2024) 609–620.
[28] Weng, C.H. Water environment and recent advances in pollution control technologies, Environ. Sci. Pollut. Res. 29(9) (2022), 12462–12464.
[29] Yuan, Y.X. A review of trust region algorithms for optimization, In Iciam, 99(1) (2000), 271–282.

[30] Zellagui, M., Belbachir, N., and El-Sehiemy, R.A. Solving the optimal power flow problem in power systems using the mountain gazelle algo-rithm, Eng. Proc. 56(1) (2023), 176.