A NUMERICAL STUDY OF COMPLEX DYNAMICS OF A CHEMOSTAT MODEL UNDER FRACTAL-FRACTIONAL DERIVATIVE

Zareen A. Khan (), Kamal Shah, Bahaaeldin Abdalla () and Thabet Abdeljawad
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Zareen A. Khan: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
Kamal Shah: ��Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia‡Department of Mathematics, University of Malakand Chakdara Dir (L), 18000, Khyber Pakhtunkhwa, Pakistan
Bahaaeldin Abdalla: ��Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia
Thabet Abdeljawad: ��Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia§Department of Medical Research, China Medical University, Taichung 40402, Taiwan¶Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea∥Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako, Makgatho Health Sciences University, Ga-Rankuwa, South Africa

FRACTALS (fractals), 2023, vol. 31, issue 08, 1-14

Abstract: In this paper, we study the existence of numerical solution and stability of a chemostat model under fractal-fractional order derivative. First, we investigate the positivity and roundedness of the solution of the considered system. Second, we find the existence of a solution of the considered system by employing the Banach and Schauder fixed-point theorems. Furthermore, we obtain a sufficient condition that allows the existence of the stabling of solutions by using the numerical-functional analysis. We find that the proposed system exists as a unique positive solution that obeys the criteria of Ulam–Hyers (U-H) and generalized U-H stability. We also establish a numerical analysis for the proposed system by using a numerical scheme based on the Lagrange interpolation procedure. Finally, we provide two numerical examples to verify the correctness of the theoretical results. We remark that the structure described by the considered model is also sometimes called side capacity or cross-flow model. The structure considered here can be also seen as a limiting case of the pattern chemostats in parallel with diffusion connection. Moreover, the said model forms in natural and engineered systems and can significantly affect the hydrodynamics in porous media. Fractal calculus is an excellent tool to discuss fractal characteristics of porous media and the characteristic method of the porous media.

Keywords: Chemostat Model; Fractal-fractional Order Derivative; Fixed-point Theorem; Numerical Solution; Stability (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:31:y:2023:i:08:n:s0218348x23401813

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DOI: 10.1142/S0218348X23401813

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