LINEAR-THETA METHOD FOR THE DISCRETIZATION AND NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH MULTIPLE RETARDATIONS
DOI:
https://doi.org/10.33003/fjs-2024-0806-2865Keywords:
Multiple delay, DiscretizationAbstract
This study presents a special case of proximal point algorithm for solving linear programming problem (LPP). This method, also known as the Alternating Direction Method of multipliers (ADMM), was deployed because of its strong convergence properties of the method of multipliers, the decomposability property of dual ascent and the potential to solve large- scale structured optimization problems. The update formulas for the LPP were derived from the associated augmented Lagrangian with the primal and dual residuals also derived for the convergence of the algorithm. The Game theory was re-structured into a LPP amenable to the ADMM. Prisoner’s Dilemma in Game theory was tested with the ADMM provided the matrix operator is invertible to guarantee its convergence. Other Numerical examples were also tested and it was discovered that the developed algorithm performs faster than the conventional simplex method.
References
Abah, M. A., Otitoju, O., Okoli, E. C., Ozioma, P. E., Bando, D. C. & Zephaniah, H. S. (2021). Determination of selected pesticide residues in leafy vegetables (Amaranthus spinosus) consumed in Donga, Taraba State. International Journal of Biochemistry, Bioinformatics and Biotechnology Studies, 6(2), 9-16.
Adeola, F. O. (2004). Boon or bane? The environmental and health impacts of persistent organic pollutants (POPs). Human Ecology Review, 11(1), 27-34.
American Public Health Association (APHA). (2017). Standard methods for the examination of water and wastewater (23rd ed.). American Public Health Association.
Bhatia, S. C. (2006). Environmental chemistry. CBS Publishers.
Botwe, B. O., Ntow, W. J., Kelderman, P. & Drechsel, P. (2011). Pesticides residues contamination of vegetables and their public health implication in Ghana. Journal of Environmental Issues and Agriculture in Developing Countries, 3(1), 11-13.
Bruton, M. N. (1979). The food and feeding behaviour of Clarias gariepinus (Pisces: Clariidae) in Lake Sibaya, South Africa, with its emphasis on its role as a predator of cichlids. Transactions of the Zoological Society of London, 35(1), 47-114.
Burton, F. L., Tchobanoglous, G. & Stensel, H. D. (2003). Wastewater engineering: Treatment, disposal, and reuse (4th ed.). McGraw-Hill. Pp. 56.
Burton, G. A., DeZwart, D., Diamond, J., Dyer, S., Joubert, L., Kidd, K., .. & Wentsel, R. (2003). The role of pollution in the global decline of amphibians. Environmental Science & Technology, 37(7), 199-208.
Chen, L., Zhao, Y., Li, F. & Wang, Q. (2016). Assessment of organic pollutants in water and sediment. Journal of Environmental Sciences, 45, 123-130.
Cheng, Y., Li, M., Liu, J. & Wang, Z. (2022). Analysis of water contaminants using gas chromatography-mass spectrometry. Environmental Science and Technology, 56(4), 2342-2351.
Clay, D. (1979). Population biology, growth and feeding of African catfish (Clarias gariepinus) with particular reference to juveniles and their importance in fish culture. Archive für Hydrobiologie, 87, 453-482.
Damià, B. (2005). Pesticides and the environment. Science of the Total Environment, 347(1-3), 1-3.
EPA. (2017). Report on the environment: Pesticides. Environmental Protection Agency. Retrieved from: https://www.epa.gov/roe/pesticides. 24.03.2024
European Commission. (2016). Directive 2016/122/EC of the European Parliament and of the Council. Official Journal of the European Union, 49, 372-374.
FDA. (2010). Food additives permitted in feed and drinking water of animals; fatty acid esters of mixed polyols. Federal Register, 75(49), 11744-11745.
Hecht, T., Uys, W., & Britz, P. J. (1996). Culture of sharp-tooth catfish, Clarias gariepinus, in Southern Africa. Southern African Journal of Aquatic Sciences, 12(1), 13-24.
HHS. (2015). Report on carcinogens, eleventh edition. U.S. Department of Health and Human Services. Retrieved from https://ntp.niehs.nih.gov/ntp/roc/content/profiles/ethyleneoxide.pdf 24.06.2024
IARC. (2012). Chemical agents and related occupations. IARC Monographs on the Evaluation of Carcinogenic Risks to Humans, 100F, 9-562.
International Organization for Standardization (ISO). (2017). Water quality — Sampling — Part 12: Guidance on sampling of bottom sediments (ISO 5667-12:2017).
Kim, K. H., Kabir, E. & Jahan, S. A. (2017). Exposure to pesticides and the associated human health effects. Science of the Total Environment, 575, 525-535.
Kronimus, A., Schwarzbauer, J., Dsikowitzky, L., Heim, S. & Littke, R. (2004). Anthropogenic organic contaminants in sediments of the Lippe River, Germany. Water Research, 38, 3473–3484.
Lee, J. S., Choi, H. J. & Hwang, S. J. (2019). Seasonal variation of agricultural runoff in a stream of Korea. Water Research, 161, 315-326.
Li, J., Song, Y. & Cai, Z. (2018). Human exposure to persistent organic pollutants via food consumption in China. Food Chemistry, 239, 387-394.
National Research Council (NRC). (2003). Oil in the sea III: Inputs, fates, and effects. The National Academies Press.
Nnorom, I. C., Igwe, J. C. & Oji-Nnorom, C. G. (2021). Determination of organic pollutants in water bodies in Southeastern Nigeria. Journal of Environmental Chemistry and Ecotoxicology, 13(2), 27-36.
Okoli, E. C., Otitoju, O., Abah, M. A., Ozioma, P. E., Bando, D. C. & Zephaniah, H. S. (2021). Ecological risk assessment of pesticide residues in fish samples from River Donga in Donga, Taraba State, Nigeria. International Journal of Biochemistry, Bioinformatics and Biotechnology Studies, 6(2), 1-8.
Olajire, A. A., Oluwole, F. A. & Akinloye, A. (2014). Distribution and sources of polycyclic aromatic hydrocarbons in water and sediments of Asejire Lake, Nigeria. Journal of Environmental Protection, 5(11), 1113-1125.
Olawale, M., Adekunle, S. & Yusuf, A. (2021). Seasonal variation of pollutants in Nigerian lakes. Environmental Monitoring and Assessment, 193(6), 384.
Pal, A., Gin, K. Y. H., Lin, A. Y. C. & Reinhard, M. (2010). Impacts of emerging organic contaminants on fresh water resources: Review of recent occurrences, sources, fate, and effects. Science of the Total Environment, 408, 6062–6069.
Skelton, P. (2001). A complete guide to the freshwater fishes of southern Africa. Struik Publishers.
Smith, J. M. (2020). Introduction to Organic Chemistry. 5th ed. New York: Wiley. P.45.
Smith, J., Doe, A. & Brown, L. (2022). Industrial runoff impacts on sediment quality in urban waterways. Environmental Pollution, 271, 116303.
Smith, K. & Jones, R. (2018). Water sampling and analysis techniques. Environmental Science and Technology, 52(3), 134-145.
Tang, Z., Li, L., Pan, D. & Zhang, Z. (2021). Spatial distribution and risk assessment of persistent organic pollutants in surface sediments of Bohai Bay, China. Marine Pollution Bulletin, 163, 111919.
Teugels, G. G. (1986). A systematic revision of the African species of the genus Clarias (Pisces; Clariidae). Annales du Musée Royal de l'Afrique Centrale, Sciences Zoologiques, 247, 1-199.
UNEP. (2019). Global Chemicals Outlook II: From legacies to innovative solutions. United Nations Environment Programme. Retrieved from https://wedocs.unep.org/bitstream/handle/20.500.11822/28113/GCOII.pdf. 06.04.2024
United States Environmental Protection Agency (USEPA). (1996). Method 3510C: Separatory funnel liquid-liquid extraction.
United States Environmental Protection Agency.
United States Environmental Protection Agency (USEPA). (2016). Field sampling guidance document. United States Environmental Protection Agency.
Usman, L. U. (2016). Effects of anthropogenic activities on the seasonal variation of some physico-chemical parameters in Ajiwa Lake, Katsina State, Nigeria. Annals of Experimental Biology, 4(4), 11-16.
Vittoli, A. L., Trivisano, C., Gessa, C., Gherard, I. M., Simoni, A. & Vienello, G. (2010). Quality of municipal wastewater compared to surface waters of the river and artificial canal network in different areas of the eastern Po Valley (Italy). Water Quality Exportation Health, 2(1), 1-13.
Wang, J., Li, X., Zhang, Y. & Chen, D. (2018). Quantification of persistent organic pollutants in environmental matrices using GC-MS. Analytical Chemistry, 90(5), 2730-2736.
Wang, S., Chen, H. & Yang, Y. (2021). Impact of agricultural runoff on water quality in rural areas. Agricultural Water Management, 246, 106683.
Wang, X., Zhang, Y., & Ma, Z. (2020). Assessment of persistent organic pollutants in fish from the Yangtze River. Environmental Science and Pollution Research, 27(30), 37421-37430.
Yang, X., Zhou, S. & Wang, H. (2019). Sediment sampling protocols for organic pollutants analysis. Environmental Pollution, 252, 1005-1012.
Zhang, Q., Wang, X. & Zhao, L. (2023). Industrial pollutants in aquatic environments: A comprehensive study. Journal of Environmental Management, 303, 1145-1158.
Published
How to Cite
Issue
Section
FUDMA Journal of Sciences
