Nonlinear Galerkin finite element method applied to the system of reaction–diffusion equations in one space dimension

Mach, Jan; Bene, Michal; Strachota, Pavel

doi:10.1016/j.camwa.2017.02.032

Cited by 7 publications

(8 citation statements)

References 29 publications

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“…Again the complexity is higher than in DQM. Mach et al (2017) developed finite-element nonlinear Galerkin method for coupled reaction-diffusion problems but converting its theory into coding is more tough than in DQM.…”

Section: Introductionmentioning

confidence: 99%

A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

Jiwari

Tomasiello

Tornabene

2018

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Purpose This paper aims to capture the effective behaviour of nonlinear coupled advection-diffusion-reaction systems and develop a new computational scheme based on differential quadrature method. Design/methodology/approach The developed scheme converts the coupled system into a system of ordinary differential equations. Finally, the obtained system is solved by a four-stage RK4 scheme. Findings The developed scheme helped to capture the different types of patterns of nonlinear time-dependent coupled advection-diffusion-reaction systems such as Brusselator model, Chemo-taxis model and linear model which are similar to the existing patterns of the models. Originality/value The originality lies in the fact that the developed scheme is new for coupled advection-diffusion-reaction systems such as Brusselator model, Chemo-taxis model and linear models. Second, the captured pattern is similar to the existing patterns of the models.

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Section: Introductionmentioning

confidence: 99%

A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

Jiwari

Tomasiello

Tornabene

2018

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“…For the discretization of problem ( 1)-( 3) the finite difference method (FDM) was selected due to its suitability for solving the reaction-diffusion equations. As shown in [11], [16] and [20] the FDM performs better than e.g. the general Galerkin method or the finite-element method in its standard form.…”

Section: Using the Uniform Boundedness Ofmentioning

confidence: 98%

“…the general Galerkin method or the finite-element method in its standard form. A comparison of such methods is presented in [16].…”

Section: Using the Uniform Boundedness Ofmentioning

confidence: 99%

Mathematical model of signal propagation in excitable media

Kantner¹,

Beneš²

2021

Discrete &Amp; Continuous Dynamical Systems - S

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This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.

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“…The model has also been explored by Kai et al [24] using an innovative technique known as the second-order explicit implicit methodology. In recent years, the nonlinear Galerkin finite element approach has become increasingly prevalent as a means to investigate the model [25,26]. Mach [27] has performed an in-depth examination of the quantitative evaluation of the model's numerical solution.…”

Section: Introductionmentioning

confidence: 99%

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

Ali,

Trisha,

Islam

2023

GLM

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The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L∞ norm.

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Nonlinear Galerkin finite element method applied to the system of reaction–diffusion equations in one space dimension

Cited by 7 publications

References 29 publications

A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

Mathematical model of signal propagation in excitable media

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

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