scholarly journals Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dongfang Li ◽  
Chao Tong ◽  
Jinming Wen
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Variable Delay ◽  
Discrete Energy ◽  
Continuous Problems ◽  
Rate Of Decay ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

An Improved Error Estimate for a Numerical Method for a System of Coupled Singularly Perturbed Reaction-diffusion Equations

2003 ◽  
Vol 3 (3) ◽  
pp. 417-423 ◽  
Author(s):  
Torsten Linss ◽  
Niall Madden
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Perturbation Parameter ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Central Difference ◽  
Central Difference Scheme ◽  
Coupled Reaction ◽  
Theoretical Results

AbstractWe consider a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations. We show that the scheme is almost second-order convergent, uniformly in the perturbation parameter. We present the results of numerical experiments to confirm our theoretical results.

scholarly journals Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1650135 ◽  
Author(s):  
C. A. Cardoso ◽  
J. A. Langa ◽  
R. Obaya
Keyword(s):  
Chaotic Dynamics ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Minimal Set ◽  
Full Characterization ◽  
The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

scholarly journals On the Issue of Radiation-Induced Instability in Binary Solid Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Santosh Dubey ◽  
S. K. Joshi ◽  
B. S. Tewari
Keyword(s):  
Solid Solution ◽  
Stability Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Control Parameters ◽  
Binary Solid Solution ◽  
Nonlinear Reaction ◽  
Radiation Induced ◽  
The Stability

The stability of a binary solid solution under irradiation has been studied. This has been done by performing linear stability analysis of a set of nonlinear reaction-diffusion equations under uniform irradiation. Owing to the complexity of the resulting system of eigenvalue equations, a numerical solution has been attempted to calculate the dispersion relations. The set of reaction-diffusion equations represent the coupled dynamics of vacancies, dumbbell-type interstitials, and lattice atoms. For a miscible system (Cu-Au) under uniform irradiation, the initiation and growth of the instability have been studied as a function of various control parameters.

scholarly journals Numerical treatment of singularly perturbed delay reaction-diffusion equations

2020 ◽  
Vol 12 (1) ◽  
pp. 15-24
Author(s):  
Gashu Gadisa Kiltu ◽  
Gemechis File Duressa ◽  
Tesfaye Aga Bullo
Keyword(s):  
Time Delay ◽  
Present Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Numerical Treatment ◽  
The Stability

This paper presents a uniform convergent numerical method for solving singularly perturbed delay reaction-diffusion equations. The stability and convergence analysis are investigated. Numerical results are tabulated and the effect of the layer on the solution is examined. In a nutshell, the present method improves the findings of some existing numerical methods reported in the literature. Keywords: Singularly perturbed, Time delay, Reaction-diffusion equation, Layer

A Revisit of the Semi-Adaptive Method for Singular Degenerate Reaction-Diffusion Equations

2012 ◽  
Vol 2 (3) ◽  
pp. 185-203 ◽  
Author(s):  
Qin Sheng ◽  
A. Q. M. Khaliq
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Adaptive Method ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Key Characteristics

AbstractThis article discusses key characteristics of a semi-adaptive finite difference method for solving singular degenerate reaction-diffusion equations. Numerical stability, monotonicity, and convergence are investigated. Numerical experiments illustrate the discussion. The study reconfirms and improves several of our earlier results.

A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

2019 ◽  
Vol 84 (4) ◽  
pp. 669-678
Author(s):  
Lennon Ó Náraigh ◽  
Khang Ee Pang
Keyword(s):  
Steady State ◽  
A Priori ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Mathematical Framework ◽  
Source Terms ◽  
The Stability

Abstract We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction–diffusion equations with periodic source terms in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch’s theorem and Poincaré’s inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

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