The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution

Ang, W.T.

doi:10.1016/s0955-7997(03)00059-6

Cited by 46 publications

(37 citation statements)

References 10 publications

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“…In our calculations we have used 89 nodes for DRBEM-FEM and 16 nodes for DQM-FEM to catch the behavior of the solution. The number of collocation points used in DQM solution is considerably small comparing to the number of points used in [16] and DRBEM procedure used in this study.…”

Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 97%

“…It is reported in [15,16] that the solution (u, v) of the system (53) tends to (1, 1/2) for increasing t. For the time discretization M is taken as 4 in each time block. The time step Dt = 0.1 is found to be suitable as is the case in the first problem.…”

Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 99%

“…The time step Dt = 0.1 is found to be suitable as is the case in the first problem. In [16], they use DRBEM with a predictor-corrector approach in time for solving the same problem, and they used at least N=113 nodes in order to catch the behavior of the solution. In our calculations we have used 89 nodes for DRBEM-FEM and 16 nodes for DQM-FEM to catch the behavior of the solution.…”

Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 99%

“…A class of a system of second order nonlinear reaction-diffusion equations in twospace dimensions known as Brusselator system has been solved in [15] using a linear combination of first-order finite difference schemes leading to a second-order approximations for the components of the solution. In the study of Ang [16], the two-dimensional reaction-diffusion Brusselator system is solved by using DRBEM for space and FDM for time direction which includes a linerization procedure. In [17], the one-dimensional systems of linear and nonlinear PDE's, and the reaction-diffusion Brusselator model have been handled by Adomain decomposition method.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

Meral

Tezer‐Sezgin

2011

Communications in Nonlinear Science and Numerical Simulation

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Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 97%

Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 99%

Section: Nonlinear Reaction-diffusion Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

Meral

Tezer‐Sezgin

2011

Communications in Nonlinear Science and Numerical Simulation

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“…Author of [55] presented a solution for the reaction-diffusion Brusselator equation using the decomposition method of Adomian. The main aim of [56] is to apply the dual-reciprocity boundary element method for the numerical solution of a class of twodimensional reaction-diffusion Brusselator system which arises in the modeling of certain chemical reaction-diffusion processes. Authors of [57] introduced the local extrapolation of first order locally one-dimensional exponential time differencing scheme for the numerical solution of multi-dimensional nonlinear reaction-diffusion systems.…”

Section: The Brusselator Modelmentioning

confidence: 99%

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

Dehghan

Abbaszadeh

2016

Computers & Mathematics with Applications

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Available online xxxx Keywords: Element free Galerkin (EFG) and movingKriging interpolation Fourth-order Runge-Kutta method Mathematical biology and Turing system Gray-Scott model and Allen-Cahn equation Ginzburg-Landau model and Brusselator equation Predator-prey model with additional food supply to predator a b s t r a c tIn this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge-Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on threedimensional PDEs, we select some equations and system of PDEs such as Allen-Cahn, Gray-Scott, Ginzburg-Landau, Brusselator models, predator-prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

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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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The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution

Cited by 46 publications

References 10 publications

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

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

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