Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic

Alqahtani, Aisha M.

doi:10.1007/s10910-018-0859-8

Cited by 22 publications

(18 citation statements)

References 29 publications

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“…Therefore, every trajectory z uvw (t) crosses the plane Σ (2) at some t uvw ∈ (0.34, 0.4) and by the implicit function theorem it follows that the function t uvw is continuous in (u, v, w) ∈ S (1) .…”

Section: Constructing a Map On The First Segmentmentioning

confidence: 96%

“…where • is the Euclidean norm of tensor quantities of type (1, 0), (1, 1), (1,2), and (1, 3), respectively [10].…”

Section: Preliminariesmentioning

confidence: 99%

“…(1) of the parallelepiped S (1) onto the plane Σ (2) , which relates each point (u, v, w) ∈ S (1) to a point z uvw (t uvw ) where z uvw (t) intersects the plane Σ (2) . We denote the set of these points by S (2) . The continuity of the map Φ (2) (1) follows from the theorem on the continuous dependence of solutions on the initial point.…”

Section: Constructing a Map On The First Segmentmentioning

confidence: 99%

“…The mathematical model of such reactions is also called Brusselator and serves as an important tool for their study. In fact, the Brusselator is adequately described by a system of two parabolic equations (diffusion and transfer equations) with respect to the concentrations of substances involved in the reaction that occurs in a given region in R 3 (see [20, system (7.11)] and also [1,2,30,34]. In practice, one mainly considers models in the form of a system of ordinary differential equations for averaged concentrations of substances A, B, X, Y, D, and E involved in the reaction dA dt = −k 1 A, dB dt = −k 2 BX, dX dt = k 1 A − k 2 BX + k 3 X 2 Y − k 4 X, dY dt = k 2 BX − k 3 X 2 Y, dD dt = k 2 BX, dE dt = k 4 X, (1.2) where the parameters k i = const characterize the reaction rate constants of reaction (1.1).…”

Section: Introductionmentioning

confidence: 99%

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Four–dimensional Brusselator Model With Periodical Solution

Akhmedov

Азамов

Ibragimov

2020

UMJ

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In the paper, a four-dimensional model of cyclic reactions of the type Prigogine's Brusselator is considered. It is shown that the corresponding dynamical system does not have a closed trajectory in the positive orthant that will make it inadequate with the main property of chemical reactions of Brusselator type. Therefore, a new modified Brusselator model is proposed in the form of a four-dimensional dynamic system. Also, the existence of a closed trajectory is proved by the DN-tracking method for a certain value of the parameter which expresses the rate of addition one of the reagents to the reaction from an external source.

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Section: Constructing a Map On The First Segmentmentioning

confidence: 96%

“…where • is the Euclidean norm of tensor quantities of type (1, 0), (1, 1), (1,2), and (1, 3), respectively [10].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Constructing a Map On The First Segmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Four–dimensional Brusselator Model With Periodical Solution

Akhmedov

Азамов

Ibragimov

2020

UMJ

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“…In paper [19], authors discussed the asymptotic behavior of the solutions of the Brusselator model numerically and focused on the stable pattern formation. In paper [20], author studied various types of pattern formation of the Brusselator model arising in chemical reactions with the numerical investigation. We are interested to establish the existence of PTW solutions of the Brusselator model numerically as a function of wave speed and a bifurcation parameter b throughout the periodic pattern formation.…”

Section: Introductionmentioning

confidence: 99%

Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

Nazimuddin¹,

Ali²

2019

IJMSC

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In this paper, we analyze the pattern formation in a chemical reaction-diffusion Brusselator model. Twocomponent Brusselator model in two spatial dimensions is studied numerically through direct partial differential equation simulation and we find a periodic pattern. In order to understand the periodic pattern, it is important to investigate our model in one-dimensional space. However, direct partial differential equation simulation in one dimension of the model is performed and we get periodic traveling wave solutions of the model. Then, the local dynamics of the model is investigated to show the existence of the limit cycle solutions. After that, we establish the existence of periodic traveling wave solutions of the model through the continuation method and finally, we get a good consistency among the results.

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Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

Hussain

2022

Math Methods in App Sciences

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This article presents an effective and simple approximation scheme to analytically approximate solution behavior of a multi‐dimensional reaction–diffusion Brusselator system. This system has applications in cooperative processes of chemical kinetics and biochemical reaction processes. The proposed residual power series method is based on multiple power series expansion and residual error function formulation. The convergence properties and error bounds of the proposed scheme are theoretically established and numerically verified. Two benchmark problems in two‐ and three‐dimensional space are considered to validate the efficiency and accuracy of the proposed scheme. The obtained results verified reliability, accuracy, and simplicity of the proposed scheme against the available results in literature.

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Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic

Cited by 22 publications

References 29 publications

Four–dimensional Brusselator Model With Periodical Solution

Four–dimensional Brusselator Model With Periodical Solution

Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

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