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Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative

Author

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  • Rubayyi T. Alqahtani

    (Department of Mathematics and Statistics, College of Science Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia)

  • Abdullahi Yusuf

    (Department of Computer Engineering, Biruni University, 34010 Istanbul, Turkey
    Department of Mathematics, Federal University Dutse, Jigawa 7156, Nigeria)

  • Ravi P. Agarwal

    (Department of Mathematics, Texas A M University-Kingsville, Kingsville, TX 78363, USA)

Abstract

In this paper, the wastewater treatment model is investigated by means of one of the most robust fractional derivatives, namely, the Caputo fractional derivative. The growth rate is assumed to obey the Contois model, which is often used to model the growth of biomass in wastewaters. The characteristics of the model under consideration are derived and evaluated, such as equilibrium, stability analysis, and steady-state solutions. Further, important characteristics of the fractional wastewater model allow us to understand the dynamics of the model in detail. To this end, we discuss several important analyses of the fractional variant of the model under consideration. To observe the efficiency of the non-local fractional differential operator of Caputo over its counter-classical version, we perform numerical simulations.

Suggested Citation

  • Rubayyi T. Alqahtani & Abdullahi Yusuf & Ravi P. Agarwal, 2021. "Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative," Mathematics, MDPI, vol. 9(6), pages 1-19, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:675-:d:521642
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    References listed on IDEAS

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    1. Dubey, Ved Prakash & Kumar, Rajnesh & Kumar, Devendra, 2019. "Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 762-772.
    2. Kirtphaiboon, Sarinya & Humphries, Usa & Khan, Amir & Yusuf, Abdullahi, 2021. "Model of rice blast disease under tropical climate conditions," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    3. Bhattacharya, S.C. & Khai, Pham Quang, 1987. "Kinetics of anaerobic cowdung digestion," Energy, Elsevier, vol. 12(6), pages 497-500.
    4. Acay, Bahar & Inc, Mustafa, 2021. "Fractional modeling of temperature dynamics of a building with singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Abdulhameed, M. & Muhammad, M.M. & Gital, A.Y. & Yakubu, D.G. & Khan, I., 2019. "Effect of fractional derivatives on transient MHD flow and radiative heat transfer in a micro-parallel channel at high zeta potentials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 519(C), pages 42-71.
    6. Qureshi, Sania & Yusuf, Abdullahi, 2019. "Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 111-118.
    7. Yusuf, Abdullahi & Acay, Bahar & Mustapha, Umar Tasiu & Inc, Mustafa & Baleanu, Dumitru, 2021. "Mathematical modeling of pine wilt disease with Caputo fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    8. Jajarmi, Amin & Yusuf, Abdullahi & Baleanu, Dumitru & Inc, Mustafa, 2020. "A new fractional HRSV model and its optimal control: A non-singular operator approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    9. Qureshi, Sania & Yusuf, Abdullahi, 2019. "Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 32-40.
    10. Goulart, A.G. & Lazo, M.J. & Suarez, J.M.S., 2019. "A new parameterization for the concentration flux using the fractional calculus to model the dispersion of contaminants in the Planetary Boundary Layer," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 518(C), pages 38-49.
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