scholarly journals Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Petr Stehlík ◽  
Jonáš Volek
Keyword(s):  
Maximum Principle ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Maximum Principles ◽  
Time Step ◽  
Reaction Diffusion Model ◽  
Continuous Reaction ◽  
Initial Boundary Value

We study reaction-diffusion equations with a general reaction functionfon one-dimensional lattices with continuous or discrete timeux′  (or  Δtux)=k(ux-1-2ux+ux+1)+f(ux),x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time) exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

scholarly journals THE REACTION-DIFFUSION MODEL OF BACTERIA COMMUNICATION UNDER VARIATION IN THE LAW OF POPULATION GROWTH

2021 ◽  
pp. 14-23
Author(s):  
Shuai Yixuan ◽  
◽  
Alexey Pavlovich Khmelev ◽  
Anna Gennadievna Maslovskaya ◽  
◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Communication Process ◽  
Reaction Diffusion Equation ◽  
Bacterial Communication ◽  
Reaction Diffusion Model ◽  
Implicit Finite Difference Scheme ◽  
The Law ◽  
Initial Boundary Value

The paper proposes a modification of the mathematical model of the bacterial communication process based on using the law of multiphase dynamics of changes in the microorganisms number. The model is described as an initial-boundary value problem for an evolutionary «reaction – diffusion» equation. An implicit finite-difference scheme is derived for the numerical solution of an applied problem. The results of computational experiments are discussed.

Positivity for elliptic and parabolic systems

1978 ◽  
Vol 79 (3-4) ◽  
pp. 183-191 ◽  
Author(s):  
J. F. G. Auchmuty
Keyword(s):  
Initial Conditions ◽  
Elliptic Systems ◽  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Biological Phenomena ◽  
Weakly Coupled ◽  
Coupling Terms ◽  
Initial Boundary Value

SynopsisThe positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

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.

scholarly journals Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Vladislav V. Kravchenko ◽  
Josafath A. Otero ◽  
Sergii M. Torba
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Cauchy Data ◽  
Variable Coefficients ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Analytic Approximation ◽  
Transmutation Operator ◽  
Initial Boundary Value

A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

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.

scholarly journals Identifying an Unknown Coefficient in the Reaction-Diffusion Equation Using He’s VIM

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
F. Parzlivand ◽  
A. M. Shahrezaee
Keyword(s):  
Unknown Parameter ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Main Property ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equation ◽  
Unknown Coefficient ◽  
Initial Boundary Value ◽  
Inverse Heat Problem

An inverse heat problem of finding an unknown parameter p(t) in the parabolic initial-boundary value problem is solved with variational iteration method (VIM). For solving the discussed inverse problem, at first we transform it into a nonlinear direct problem and then use the proposed method. Also an error analysis is presented for the method and prior and posterior error bounds of the approximate solution are estimated. The main property of the method is in its flexibility and ability to solve nonlinear equation accurately and conveniently. Some examples are given to illustrate the effectiveness and convenience of the method.

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