The initial boundary value problem for a nonlocal singularly perturbed reaction-diffusion equation

2012 ◽  
Vol 52 (6) ◽  
pp. 926-931 ◽  
Author(s):  
N. N. Nefedov ◽  
A. G. Nikitin
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value

On a singular initial-boundary-value problem for a reaction-diffusion equation arising from a simple model of isothermal chemical autocatalysis

1992 ◽  
Vol 437 (1901) ◽  
pp. 657-671 ◽  
Keyword(s):  
Simple Model ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Rate Constant ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value

In this paper we examine the evolution that occurs when a localized input of an autocatalyst B is introduced into an expanse of a reactant A. The reaction is autocatalytic of order p,so A -> B at rate k [A] [B] p with rate constant k . We examine the case when 0 < p < 1, with p>/ 1 having been examined by Needham & Merkin (Phil. Trans. R. Soc. Lond. A 337, 261—274 (1991)). In particular, we show that the fully reacted state is not achieved (as t-> oo) via the propagation of a travelling wavefront (as for p>/ 1) but is approached uniformly in space as t-00.

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. I. Continuous diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 335-360 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

We examine the effects of a concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics. The diffusivity is taken as a continuous monotone, a decreasing function of concentration that has compact support, of the form that arises in polymerization processes. We consider piecewise-classical solutions to an initial-boundary value problem. The existence of a family of permanent form travelling wave solutions is established, and the development of the solution of the initial-boundary value problem to the travelling wave of minimum propagation speed is considered. It is shown that an interface will always form in finite time, with its initial propagation speed being unbounded. The interface represents the surface of the expanding polymer matrix.

The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations

1994 ◽  
Vol 348 (1687) ◽  
pp. 229-260 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Travelling Waves ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

In this paper we examine the effects of concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics and ecology. We consider piecewise classical solutions to an initial boundary-value problem. The existence of a family of permanent form travelling wave solutions is established and the development of the solution of the initial boundary-value problem to the travelling wave of minimum propagation speed is considered. For certain types of initial data, ‘waiting time’ phenomena are encountered.

Sign in / Sign up

Export Citation Format

Share Document