Research Article
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Göl kirliliği probleminin çözümü için hızlı ve güvenilir bir sayısal yaklaşım

Year 2025, Volume: 27 Issue: 2, 736 - 749, 15.07.2025

Sevin Gümgüm

https://doi.org/10.25092/baunfbed.1676679

Abstract

Su, hava gibi çevrenin temel bileşenlerinden biridir ve su kaynaklarının bozulması tüm canlı organizmaları tehdit etmektedir. Bu nedenle su kirliliği sorununu araştırmak büyük önem taşımaktadır. Göller, su kaynaklarının büyük bir bölümünü oluşturmaktadır. Bu çalışmanın amacı, üç adet birbiriyle bağlantılı gölden oluşan bir sistemdeki kirlilik dinamiklerini Gegenbauer dalgacık yöntemi ile analiz etmektir. Problem, her bir göldeki kirlilik oranının zamana göre değişimini temsil eden üç doğrusal diferansiyel denklem sistemi ile modellenmiştir. Zaman türevlerine, kesikli Gegenbauer dalgacık serisiyle yaklaşılmış ve diferansiyel denklem sistemi, cebirsel denklem sistemine dönüştürülmüştür. Elde edilen sayısal sonuçlar, literatürde mevcut diğer sayısal sonuçlarla karşılaştırılarak önerilen tekniğin güvenilir ve hızlı olduğu gösterilmiştir. Ayrıca, yöntemin yüksek doğruluk sağladığı ve bu nedenle diğer ekolojik olayların çözümünde de kullanılabileceği ortaya konmuştur.

Keywords

Göl kirlinmesi Su kirliliği Gegenbauer dalgacıkları

References

  • Biazar, J., Farrokhi, L., Islam, M.R. Modeling the pollution of a system of lakes, Applied Mathematics and Computation, 178 (2006), 423–430. Doi:10.1016/j.amc.2005.11.05
  • Prakasha, D.G., Veeresha, P. Analysis of Lakes pollution model with Mittag-Leffler kernel, Journal of Ocean Engineering and Science, 5 (2020), 310–322. Doi: 10.1016/j.joes.2020.01.004.
  • Sokhanvar, E., Yousefi, S.A. The Bernoulli Ritz-collocation method to the solution of modelling the pollution of a system of lakes, Caspian Journal of Mathematical Sciences, 3(2) (2014), 253–265.
  • Merdan, M. Homotopy perturbation method for solving modelling pollution of a system of lakes, SDU Journal of Science, 4(1) (2009), 99–111.
  • Yüzbaşı, Ş., Şahin, N., Sezer, M. A collocation approach to solving the model of pollution for a system of lakes, Mathematical and Computer Modelling, 55 (2012), 330–341.
  • Sabermahani, S., Ordokhani, Y. An analytical method for solving the model of pollution for a system of lakes of lakes, First international conference on combinatorics, cryptography and computation, 35-40 (2016). Doi: 10.5281/zenodo.2652404.
  • Yıldırım, G., Yüzbaşı, Ş., Boubaker Collocation Method for Approximate Solutions of the Model of Pollution for a System of Lakes, Turkish Journal of Mathematics and Computer Science, 16(1) (2024), 240–254. Doi: 10.47000/tjmcs.1167568.
  • Benhammouda, B., Vazquez-Leal, H., Hernandez-Martinez, L. Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 645726, 12 pages. Doi:10.1155/2014/645726.
  • Ghosh, I., Chowdhury, M.S.H., Mt Aznam, S., Rashid, M.M. Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method, Journal of Applied Mathematics, 2021 (2021), Article ID 6664307, 16 pages. Doi:10.1155/2021/6664307.
  • Dogan, N. Series Solution of Modelling the Pollution, Turkish Journal of Mathematics and Computer Science, 13(1) (2021), 81–88. Doi:10.47000/tjmcs.879627Merdan, M. Homotopy perturbation method for solving modelling pollution of a system of lakes, SDU Journal of Science, 4(1) (2009), 99–111.
  • Yönet, N., Gürbüz, B. and Gökçe, A. An alternative numerical approach for an improved ecological model of interconnected lakes with a fixed pollutant, Computational and Applied Mathematics , 42:56 (2023). Doi:10.1007/s40314-023-02191-3.
  • Biazar, J., Shahbala, M. and Ebrahimi, H. VIM for Solving the Pollution Problem of a System of Lakes, Journal of Control Science and Engineering, 2010 (2010), Article ID 829152, 6 pages. Doi:10.1155/2010/
  • Beler, A., Özaltun Simşek, G., Gümgüm, S. Numerical Solutions of the HIV Infection Model of 〖CD4〗^((+)) Cells by Laguerre Wavelets, Mathematics and Computers in Simulation, 209 (2023), 205–219. Doi:10.1016/j.matcom.2023.02.016.
  • Çelik, ̇İbrahim Generalization of Gegenbauer Wavelet Collocation Method to the Generalized Kuramoto–Sivashinsky Equation, International Journal of Applied and Computational Mathematics , 4 (2018. Doi:10.1007/s40819-018-0546-2.
  • Ersoy Özdek, D. Laguerre Wavelet Solution of Bratu And Duffing Equations, TWMS Journal Of Applied And Engineering Mathematics, 11(1) (2021), 66–77.
  • Gümgüm, S., Özdek, D.E., Özaltun, G., Bildik, N. Legendre wavelet solution of neutral differential equations with proportional delays, Journal of Applied Mathematics and Computing, 1(1-2) (2019), 389–404. Doi:10.1007/s12190-019-01256-z.
  • Gümgüm, S., Ersoy Özdek, D., Özaltun, G. Legendre wavelet solution of high order nonlinear ordinary delay differential equations, Turkish Journal of Mathematics, 43(3) (2019), 1339–1352. Doi:10.3906/mat-1901-109.
  • Özaltun Simsek, G., Gümgüm, S. Numerical solutions of Troesch and Duffing equations by Taylor wavelets, Hacettepe Journal of Mathematics and Statistics, 52(2) (2023), 292–302. Doi:10.15672/hujms.1063791
  • Özaltun Şimşek, G., Beler, A., Gümgüm, S. Gegenbauer Wavelet Solutions of the SIR and the SITR Systems of the COVID-19 Disease, International Journal of Biomathematics, (2024). Doi: 10.1142/S1793524524501432
  • Özaltun, G., Konuralp, A., Gümgüm, S. Gegenbauer wavelet solutions of fractional integro-differential equations, Journal of Computational and Applied Mathematics, 420(09) (2023). Doi:10.1016/j.cam.2022.114830.

A fast and reliable numerical approach for solving lake pollution problem

Year 2025, Volume: 27 Issue: 2, 736 - 749, 15.07.2025

Sevin Gümgüm

https://doi.org/10.25092/baunfbed.1676679

Abstract

Just like air, water is an essential component of the environment and deterioration of water sources threatens all living organisms. Hence, investigating the water pollution problem is highly important. Lakes constitutes a big portion of water sources. The aim of this study is to analyze the dynamics of pollution in a system of three interconnected lakes using the Gegenbauer wavelet method. The problem is modeled by three ordinary linear differential equations representing the rate of pollution in each lake with respect to time. Time derivates are approximated by the truncated Gegenbauer wavelet series and the system of differential equations are converted into a system of algebraic equations. We show that the proposed technique is reliable and fast by comparing the numerical results with other numerical results available in the literature. We also show that the method is highly accurate and hence can also be used to solve other ecological phenomena as well.

Keywords

Lake contamination Water Pollution Gegenbauer wavelets

References

  • Biazar, J., Farrokhi, L., Islam, M.R. Modeling the pollution of a system of lakes, Applied Mathematics and Computation, 178 (2006), 423–430. Doi:10.1016/j.amc.2005.11.05
  • Prakasha, D.G., Veeresha, P. Analysis of Lakes pollution model with Mittag-Leffler kernel, Journal of Ocean Engineering and Science, 5 (2020), 310–322. Doi: 10.1016/j.joes.2020.01.004.
  • Sokhanvar, E., Yousefi, S.A. The Bernoulli Ritz-collocation method to the solution of modelling the pollution of a system of lakes, Caspian Journal of Mathematical Sciences, 3(2) (2014), 253–265.
  • Merdan, M. Homotopy perturbation method for solving modelling pollution of a system of lakes, SDU Journal of Science, 4(1) (2009), 99–111.
  • Yüzbaşı, Ş., Şahin, N., Sezer, M. A collocation approach to solving the model of pollution for a system of lakes, Mathematical and Computer Modelling, 55 (2012), 330–341.
  • Sabermahani, S., Ordokhani, Y. An analytical method for solving the model of pollution for a system of lakes of lakes, First international conference on combinatorics, cryptography and computation, 35-40 (2016). Doi: 10.5281/zenodo.2652404.
  • Yıldırım, G., Yüzbaşı, Ş., Boubaker Collocation Method for Approximate Solutions of the Model of Pollution for a System of Lakes, Turkish Journal of Mathematics and Computer Science, 16(1) (2024), 240–254. Doi: 10.47000/tjmcs.1167568.
  • Benhammouda, B., Vazquez-Leal, H., Hernandez-Martinez, L. Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 645726, 12 pages. Doi:10.1155/2014/645726.
  • Ghosh, I., Chowdhury, M.S.H., Mt Aznam, S., Rashid, M.M. Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method, Journal of Applied Mathematics, 2021 (2021), Article ID 6664307, 16 pages. Doi:10.1155/2021/6664307.
  • Dogan, N. Series Solution of Modelling the Pollution, Turkish Journal of Mathematics and Computer Science, 13(1) (2021), 81–88. Doi:10.47000/tjmcs.879627Merdan, M. Homotopy perturbation method for solving modelling pollution of a system of lakes, SDU Journal of Science, 4(1) (2009), 99–111.
  • Yönet, N., Gürbüz, B. and Gökçe, A. An alternative numerical approach for an improved ecological model of interconnected lakes with a fixed pollutant, Computational and Applied Mathematics , 42:56 (2023). Doi:10.1007/s40314-023-02191-3.
  • Biazar, J., Shahbala, M. and Ebrahimi, H. VIM for Solving the Pollution Problem of a System of Lakes, Journal of Control Science and Engineering, 2010 (2010), Article ID 829152, 6 pages. Doi:10.1155/2010/
  • Beler, A., Özaltun Simşek, G., Gümgüm, S. Numerical Solutions of the HIV Infection Model of 〖CD4〗^((+)) Cells by Laguerre Wavelets, Mathematics and Computers in Simulation, 209 (2023), 205–219. Doi:10.1016/j.matcom.2023.02.016.
  • Çelik, ̇İbrahim Generalization of Gegenbauer Wavelet Collocation Method to the Generalized Kuramoto–Sivashinsky Equation, International Journal of Applied and Computational Mathematics , 4 (2018. Doi:10.1007/s40819-018-0546-2.
  • Ersoy Özdek, D. Laguerre Wavelet Solution of Bratu And Duffing Equations, TWMS Journal Of Applied And Engineering Mathematics, 11(1) (2021), 66–77.
  • Gümgüm, S., Özdek, D.E., Özaltun, G., Bildik, N. Legendre wavelet solution of neutral differential equations with proportional delays, Journal of Applied Mathematics and Computing, 1(1-2) (2019), 389–404. Doi:10.1007/s12190-019-01256-z.
  • Gümgüm, S., Ersoy Özdek, D., Özaltun, G. Legendre wavelet solution of high order nonlinear ordinary delay differential equations, Turkish Journal of Mathematics, 43(3) (2019), 1339–1352. Doi:10.3906/mat-1901-109.
  • Özaltun Simsek, G., Gümgüm, S. Numerical solutions of Troesch and Duffing equations by Taylor wavelets, Hacettepe Journal of Mathematics and Statistics, 52(2) (2023), 292–302. Doi:10.15672/hujms.1063791
  • Özaltun Şimşek, G., Beler, A., Gümgüm, S. Gegenbauer Wavelet Solutions of the SIR and the SITR Systems of the COVID-19 Disease, International Journal of Biomathematics, (2024). Doi: 10.1142/S1793524524501432
  • Özaltun, G., Konuralp, A., Gümgüm, S. Gegenbauer wavelet solutions of fractional integro-differential equations, Journal of Computational and Applied Mathematics, 420(09) (2023). Doi:10.1016/j.cam.2022.114830.
There are 20 citations in total.

Details

Primary Language English
Subjects Ordinary Differential Equations, Difference Equations and Dynamical Systems
Journal Section Research Articles
Authors

Sevin Gümgüm 0000-0002-0594-2377

Early Pub Date July 11, 2025
Publication Date July 15, 2025
Submission Date April 15, 2025
Acceptance Date May 29, 2025
Published in Issue Year 2025 Volume: 27 Issue: 2

Cite

APA Gümgüm, S. (2025). A fast and reliable numerical approach for solving lake pollution problem. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27(2), 736-749. https://doi.org/10.25092/baunfbed.1676679
AMA Gümgüm S. A fast and reliable numerical approach for solving lake pollution problem. BAUN Fen. Bil. Enst. Dergisi. July 2025;27(2):736-749. doi:10.25092/baunfbed.1676679
Chicago Gümgüm, Sevin. “A Fast and Reliable Numerical Approach for Solving Lake Pollution Problem”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27, no. 2 (July 2025): 736-49. https://doi.org/10.25092/baunfbed.1676679.
EndNote Gümgüm S (July 1, 2025) A fast and reliable numerical approach for solving lake pollution problem. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27 2 736–749.
IEEE S. Gümgüm, “A fast and reliable numerical approach for solving lake pollution problem”, BAUN Fen. Bil. Enst. Dergisi, vol. 27, no. 2, pp. 736–749, 2025, doi: 10.25092/baunfbed.1676679.
ISNAD Gümgüm, Sevin. “A Fast and Reliable Numerical Approach for Solving Lake Pollution Problem”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27/2 (July 2025), 736-749. https://doi.org/10.25092/baunfbed.1676679.
JAMA Gümgüm S. A fast and reliable numerical approach for solving lake pollution problem. BAUN Fen. Bil. Enst. Dergisi. 2025;27:736–749.
MLA Gümgüm, Sevin. “A Fast and Reliable Numerical Approach for Solving Lake Pollution Problem”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 27, no. 2, 2025, pp. 736-49, doi:10.25092/baunfbed.1676679.
Vancouver Gümgüm S. A fast and reliable numerical approach for solving lake pollution problem. BAUN Fen. Bil. Enst. Dergisi. 2025;27(2):736-49.