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Mathematical study of Algae as a bio-fertilizer using fractal–fractional dynamic model

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  • Mahmood, Tariq
  • ur Rahman, Mati
  • Arfan, Muhammad
  • Kayani, Sadaf-Ilyas
  • Sun, Mei

Abstract

Algae plays a vital role as bio-fertilizer and soil stabilizer in the field of agriculture. This paper considers a fractal–fractional dynamics model in sense of Caputo derivative that study the reuse of detritus generated by the dead algae as fertilizer for crop. In particular, we study dynamics model in which algae recovers nitrogen and phosphorus (waste nutrients) from water and is reused to enhance agricultural production. The theoretical results are established for the considered model with the aid of fixed point theory, whereas Ulam–Hyers approach is used for the stability of system. The numerical simulations are computed by using the well-known technique of fractional Adams–Bashforth. For simulation of the problem we consider different values for fractional order and fractal dimension by using some pre-existing data. Several graphical representation is given to understand the system at different fractional order and fractal dimension.

Suggested Citation

  • Mahmood, Tariq & ur Rahman, Mati & Arfan, Muhammad & Kayani, Sadaf-Ilyas & Sun, Mei, 2023. "Mathematical study of Algae as a bio-fertilizer using fractal–fractional dynamic model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 207-222.
    • Handle: RePEc:eee:matcom:v:203:y:2023:i:c:p:207-222
    DOI: 10.1016/j.matcom.2022.06.028
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    References listed on IDEAS

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    1. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    2. Rahman, Mati ur & Arfan, Muhammad & Shah, Kamal & Gómez-Aguilar, J.F., 2020. "Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
    4. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    5. J. Michael Beman & Kevin R. Arrigo & Pamela A. Matson, 2005. "Agricultural runoff fuels large phytoplankton blooms in vulnerable areas of the ocean," Nature, Nature, vol. 434(7030), pages 211-214, March.
    6. 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. Stefania Tomasiello & Jorge E. Macías-Díaz, 2023. "A Mini-Review on Recent Fractional Models for Agri-Food Problems," Mathematics, MDPI, vol. 11(10), pages 1-12, May.

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