Exponential Decay and Stability of Volterra Diffusion Equations

2000 ◽  
pp. 317-326
Keyword(s):  
Exponential Decay ◽  
Diffusion Equations

scholarly journals Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2473-2504 ◽  
Author(s):  
Claire Chainais-Hillairet ◽  
Maxime Herda
Keyword(s):  
Finite Volume ◽  
Exponential Decay ◽  
Large Time ◽  
Nonlinear Models ◽  
Numerical Test ◽  
Diffusion Equations ◽  
Time Behaviour ◽  
Finite Volume Schemes ◽  
Large Time Behaviour ◽  
Convection Diffusion

Abstract We are interested in the large-time behaviour of solutions to finite volume discretizations of convection–diffusion equations or systems endowed with nonhomogeneous Dirichlet- and Neumann-type boundary conditions. Our results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations or drift–diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centred convections or Scharfetter–Gummel discretizations. We illustrate our theoretical results on several numerical test cases.

Gradient-like parabolic semiflows on BUC(ℝN)

1998 ◽  
Vol 128 (6) ◽  
pp. 1281-1291 ◽  
Author(s):  
Daniel Daners ◽  
Sandro Merino
Keyword(s):  
Banach Space ◽  
Exponential Decay ◽  
Decay Estimate ◽  
Reaction Diffusion ◽  
Continuous Functions ◽  
Stationary Solutions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Uniformly Continuous

We prove that a class of weighted semilinear reaction diffusion equations on RN generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on RN. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.

Computer Simulation of Amperometric Biosensor Response to Mixtures of Compounds

2002 ◽  
Vol 7 (2) ◽  
pp. 3-14 ◽  
Author(s):  
R. Baronas ◽  
J. Christensen ◽  
F. Ivanauskas ◽  
J. Kulys
Keyword(s):  
Computer Simulation ◽  
Enzymatic Reaction ◽  
Diffusion Equations ◽  
Response Data ◽  
Finite Difference Technique ◽  
Stationary Diffusion ◽  
Biosensor Array ◽  
Menten Kinetic ◽  
Biosensor Response

A mathematical model of amperometric biosensors has been developed. The model bases on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetic of the enzymatic reaction. The model describes the biosensor response to mixtures of multiple compounds in two regimes of analysis: batch and flow injection. Using computer simulation, large amount of biosensor response data were synthesised for calibration of a biosensor array to be used for characterization of wastewater. The computer simulation was carried out using the finite difference technique.

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