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

Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia

E-mail Address: zakhan@pnu.edu.sa

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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

E-mail Address: kamalshah408@gmail.com

E-mail Address: kshah@psu.edu.sa

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BAHAAELDIN ABDALLA

Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia

E-mail Address: babdallah@psu.edu.sa

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, and

THABET ABDELJAWAD

https://orcid.org/0000-0002-8889-3768

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

E-mail Address: tabdeljawad@psu.edu.sa

Corresponding author.

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https://doi.org/10.1142/S0218348X23401813Cited by:29 (Source: Crossref)

This article is part of the issue:

Special Issue on Analysis and Modeling of Heat and Mass Transfer in Fractal Porous Media: Part I
Guest Editors: Boqi Xiao, Ali Akgül, Dahua Shou and Gongbo Long

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:

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Vol. 31, No. 08

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History

Received 14 March 2023

Accepted 15 April 2023

Published: August 23, 2023

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This is an Open Access article in the “Special Issue on Analysis and Modeling of Heat and Mass Transfer in Fractal Porous Media”, edited by Boqi Xiao (Wuhan Institute of Technology, China), Ali Akgül (Siirt University, Turkey), Dahua Shou (The Hong Kong Polytechnic University, Hong Kong) & Gongbo Long (Wuhan Institute of Technology, China) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC) License which permits use, distribution and reproduction in any medium, provided that the original work is properly cited and is used for non-commercial purposes.

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

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