Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

H. M. Srivastava; H. I. Abdel-Gawad; Khaled Mohammed Saad

doi:10.3934/dcdss.2020433

Article Contents

2021, Volume 14, Issue 10: 3785-3801. Doi: 10.3934/dcdss.2020433

This issue Previous Article A robust computational framework for analyzing fractional dynamical systems Next Article Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function

Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

1.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
2.
Department of Medical Research, China Medical University Hospital, Taichung 40204, Taiwan
3.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
4.
Department of Mathematics, Faculty of Science, Cairo University, Al Orman, Giza 12613, Egypt
5.
Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia
6.
Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

^* Corresponding author: H. M. Srivastava
^* Corresponding author: H. M. Srivastava

Received: October 31, 2019

Revised: May 31, 2020

Early access: November 2020

Published: October 2021

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Abstract

The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.

When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.

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Mathematics Subject Classification: Primary 34A08; Secondary 35A20, 35A22.

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Figure 1. The concentration of the reactant $ \alpha_{1} $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ t = 100 $, respectively

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Figure 2. The concentration of the autocatalyst $ \beta $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ t = 100 $, respectively

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Figure 3. The concentration of the reactant $ \alpha_{2} $ displayed against $ x $ for $ \gamma = 0.1 $, $ k = 0.01 $ and $ L = 100 $ in (a), (b) and (c) for $ t = 30 $, and $ 100 $, respectively

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Figure 4. The concentration of $ \alpha_{1} $, $ \beta $ and $ \alpha_{2} $ displayed against $ x $ for $ \gamma = 0.6 $, $ k = 0.09 $ and $ L = 100 $. $ (\cdots) $ for $ t = 0 $, $ (–) $ for $ t = 30 $ and $ (-) $ for $ t = 100 $, respectively

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Figure 5. Contour plots of $ \alpha_{1} $, $ \beta $ and $ \alpha_{2} $, respectively, for $ \gamma = 0.09 $, $ k = 0.01 $ and $ L = 100 $

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Figure 6. Approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $ for $ t = 50 $, $ k = 0.09 $, $ \gamma = 0.03 $ and $ L = 10 $. $ (-) $ for approximate analytical solutions; $ (–) $ for numerical solutions. The error estimate is of order $ ( $$ 10^{-8}) $

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Figure 7. Approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $ for $ t = 50 $, $ k = 0.01 $, $ \gamma = 0.001 $ and $ L = 100 $. $ (-) $ for approximate analytical solutions; $ (–) $ for numerical solutions.Th error estimate is of order $ (10^{-4}) $

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Figure 8. The absolute error between the approximate analytical and numerical solutions of $ \alpha_{i}(x,t) $ and $ \beta(x,t) $ displayed against $ x $, $ \alpha_{1}(x,t) $ (solid line), $ \beta(x,t) $ (doted-line) and $ \alpha_{2}(x,t) $ (dashed line) for $ k = 0.09 $, $ \gamma = 0.1 $ and $ L = 100 $. (a) $ t = 30 $, (b) $ t = 50 $ and (c) $ t = 100 $. The error estimate is of order $ (10^{-3} $)

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