scholarly journals A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics

2009 ◽  
Vol 23 (3) ◽  
pp. 1041-1060
Author(s):  
Maria do Carmo Pacheco de Toledo ◽  
◽  
Sergio Muniz Oliva ◽  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Discretization Scheme ◽  
One Dimensional ◽  
Equation With Delay

scholarly journals A reaction-diffusion equation on a thin L-shaped domain

1995 ◽  
Vol 125 (2) ◽  
pp. 283-327 ◽  
Author(s):  
Jack K. Hale ◽  
Geneviève Raugel
Keyword(s):  
Diffusion Equation ◽  
Lower Semicontinuity ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Reaction Diffusion Equation ◽  
Limit Equation ◽  
One Dimensional ◽  
Shaped Domain

We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.

scholarly journals Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Fengyan Wu ◽  
Qiong Wang ◽  
Xiujun Cheng ◽  
Xiaoli Chen
Keyword(s):  
Diffusion Equation ◽  
Necessary Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Sufficient And Necessary Conditions ◽  
Equation Solvability ◽  
Equation With Delay ◽  
Theoretical Results

This paper is concerned with the analysis of the linear θ-method and compact θ-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When θ∈[0,1/2), sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When θ∈[1/2,1], the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.

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