Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach
This paper aims to develop a fractional order mathematical model addressing water pollution dynamics. The model is designed to elucidate the effect of pollutants and propose effective strategies for mitigating their spread in various water bodies such as rivers, lakes, oceans, or streams. Firstly, w...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-06-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818125000889 |
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| author | Pasquini Fotsing Soh Mathew Kinyanjui David Malonza Roy Kiogora |
| author_facet | Pasquini Fotsing Soh Mathew Kinyanjui David Malonza Roy Kiogora |
| author_sort | Pasquini Fotsing Soh |
| collection | DOAJ |
| description | This paper aims to develop a fractional order mathematical model addressing water pollution dynamics. The model is designed to elucidate the effect of pollutants and propose effective strategies for mitigating their spread in various water bodies such as rivers, lakes, oceans, or streams. Firstly, we formulate and analyze a nonlinear ordinary differential equations model that integrates a fractional derivative to capture the memory effect of pollutants in water. We initiate the analysis by establishing the existence of a unique positive and bounded solution. We then compute the basic reproduction number, which dictates the global dynamics of the model. Furthermore, we rigorously demonstrate the existence of a unique pollution-free equilibrium and the endemic equilibrium, and prove their global stability under appropriate assumptions on the basic reproduction number. Additionally, we conduct a global sensitivity analysis of the basic reproduction number to assess the variability in model predictions. Secondly, we enrich this initial model by extending it to a fractional partial differential system, incorporating spatial variables and diffusion terms to elucidate the transmission dynamics of pollutants in a spatially uniform environment. We establish the existence of a unique positive and bounded solution, along with the global stability of both pollution-free and endemic equilibria. To complement our theoretical findings, we perform numerical simulations using finite difference techniques and implemented via MATLAB. |
| format | Article |
| id | doaj-art-c565983920f742e6a794d7bfec4548bb |
| institution | OA Journals |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-c565983920f742e6a794d7bfec4548bb2025-08-20T02:29:43ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-06-011410116110.1016/j.padiff.2025.101161Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approachPasquini Fotsing Soh0Mathew Kinyanjui1David Malonza2Roy Kiogora3Department of Mathematics, Institute for Basic Sciences, Technology and Innovation, Pan African University, Juja, Nairobi, Kenya; Corresponding author.Department of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Juja, Nairobi, KenyaDepartment of Mathematics and Actuarial Science, South Eastern Kenya University, Kitui, Nairobi, KenyaDepartment of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Juja, Nairobi, KenyaThis paper aims to develop a fractional order mathematical model addressing water pollution dynamics. The model is designed to elucidate the effect of pollutants and propose effective strategies for mitigating their spread in various water bodies such as rivers, lakes, oceans, or streams. Firstly, we formulate and analyze a nonlinear ordinary differential equations model that integrates a fractional derivative to capture the memory effect of pollutants in water. We initiate the analysis by establishing the existence of a unique positive and bounded solution. We then compute the basic reproduction number, which dictates the global dynamics of the model. Furthermore, we rigorously demonstrate the existence of a unique pollution-free equilibrium and the endemic equilibrium, and prove their global stability under appropriate assumptions on the basic reproduction number. Additionally, we conduct a global sensitivity analysis of the basic reproduction number to assess the variability in model predictions. Secondly, we enrich this initial model by extending it to a fractional partial differential system, incorporating spatial variables and diffusion terms to elucidate the transmission dynamics of pollutants in a spatially uniform environment. We establish the existence of a unique positive and bounded solution, along with the global stability of both pollution-free and endemic equilibria. To complement our theoretical findings, we perform numerical simulations using finite difference techniques and implemented via MATLAB.http://www.sciencedirect.com/science/article/pii/S2666818125000889Reaction–diffusion systemCaputo derivativeExistence and uniqueness resultStability analysisFinite difference |
| spellingShingle | Pasquini Fotsing Soh Mathew Kinyanjui David Malonza Roy Kiogora Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach Partial Differential Equations in Applied Mathematics Reaction–diffusion system Caputo derivative Existence and uniqueness result Stability analysis Finite difference |
| title | Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach |
| title_full | Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach |
| title_fullStr | Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach |
| title_full_unstemmed | Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach |
| title_short | Non-local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science: A spatio-temporal approach |
| title_sort | non local operator as a mathematical tool to improve the modeling of water pollution phenomena in environmental science a spatio temporal approach |
| topic | Reaction–diffusion system Caputo derivative Existence and uniqueness result Stability analysis Finite difference |
| url | http://www.sciencedirect.com/science/article/pii/S2666818125000889 |
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