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Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

Author

Listed:
  • Rubayyi T. Alqahtani

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia)

  • Shabir Ahmad

    (Department of Mathematics, University of Malakand, Lower Dir 18300, Khyber Pakhtunkhwa, Pakistan)

  • Ali Akgül

    (Department of Mathematics, Art and Science Faculty, Siirt University, Siirt TR-56100, Turkey)

Abstract

To lower the concentration of organic pollutants in the effluent stream, wastewater must be treated before being discharged into the environment. The question of whether wastewater treatment facilities can successfully reduce the concentration of micropollutants found in their influent streams is becoming increasingly pressing. The removal of micropollutants in treatment plants is investigated using a model that incorporates biodegradation and sorption as the key processes of micropollutant removal. This article provides the mathematical analysis of the wastewater model that describes the removal of micropollutant in treatment plants under a non-local operator in Caputo sense. The positivity of the solution is presented for the Caputo fractional model. The steady state’s solution of model and their stability is presented. The fixed point theorems of Leray–Schauder and Banach are used to deduce results regarding the existence of the solution of the model. Ulam–Hyers (UH) types of stabilities are presented via functional analysis. The fractional Euler method is used to find the numerical results of the proposed model. The numerical results are illustrated via graphs to show the effects of recycle ratio and the impact of fractional order on the evolution of the model.

Suggested Citation

  • Rubayyi T. Alqahtani & Shabir Ahmad & Ali Akgül, 2021. "Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense," Mathematics, MDPI, vol. 9(21), pages 1-21, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2787-:d:671557
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    References listed on IDEAS

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    1. Ahmad, Shabir & Ullah, Aman & Arfan, Muhammad & Shah, Kamal, 2020. "On analysis of the fractional mathematical model of rotavirus epidemic with the effects of breastfeeding and vaccination under Atangana-Baleanu (AB) derivative," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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    Cited by:

    1. Xu, Changjin & Liu, Zixin & Pang, Yicheng & Akgül, Ali & Baleanu, Dumitru, 2022. "Dynamics of HIV-TB coinfection model using classical and Caputo piecewise operator: A dynamic approach with real data from South-East Asia, European and American regions," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    2. Xu, Changjin & Liu, Zixin & Pang, Yicheng & Saifullah, Sayed & Inc, Mustafa, 2022. "Oscillatory, crossover behavior and chaos analysis of HIV-1 infection model using piece-wise Atangana–Baleanu fractional operator: Real data approach," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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