scholarly journals Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1035
Author(s):  
Christopher N. Angstmann ◽  
Bruce I. Henry
Keyword(s):  
Diffusion Equation ◽  
Reaction Kinetics ◽  
Rate Equation ◽  
Reaction Rate ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Mass Action ◽  
Diffusion Equations ◽  
Mixed Systems ◽  
Reaction Rate Equation

A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

scholarly journals An extension of the principle of spatial averaging for inertial manifolds

1999 ◽  
Vol 66 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Hyukjin Kwean
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Specific Class ◽  
Spatial Averaging ◽  
The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

The Model of the Pearl River Water Pollution Based on Reaction Diffusion Equation

2013 ◽  
Vol 464 ◽  
pp. 199-202
Author(s):  
Guan Jia Huang ◽  
Shao Mei Fang ◽  
Jian Xian ◽  
Jie Mei Jiang
Keyword(s):  
Water Pollution ◽  
Diffusion Equation ◽  
River Water ◽  
Reaction Diffusion ◽  
Pollution Sources ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Pearl River ◽  
The Pearl River

In this paper, we study the Pearl River cadmium spill. Based on the reaction diffusion equations, we can monitor the concentration of pollutants any time. We find that in 20 kilometers distance from pollution sources, the concentration of cadmium is high. At last, we improve the model.

Uniqueness, multiplicity and stability for positive solutions of a pair of reaction–diffusion equations

1996 ◽  
Vol 126 (4) ◽  
pp. 777-809 ◽  
Author(s):  
Yihong Du
Keyword(s):  
Diffusion Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Complete Understanding ◽  
Satisfactory Description

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.

scholarly journals On the maximization problem for solutions of reaction–diffusion equations with respect to their initial data

2020 ◽  
Vol 15 ◽  
pp. 71
Author(s):  
Grégoire Nadin ◽  
Ana Isis Toledo Marrero
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Adjoint Problem ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Maximization Problem ◽  
Biological Conservation ◽  
Open Problems

We consider in this paper the maximization problem for the quantity ∫ Ωu(t, x)dx with respect to u0 =: u(0, ⋅), where u is the solution of a given reaction diffusion equation. This problem is motivated by biological conservation questions. We show the existence of a maximizer and derive optimality conditions through an adjoint problem. We have to face regularity issues since non-smooth initial data could give a better result than smooth ones. We then derive an algorithm enabling to approximate the maximizer and discuss some open problems.

scholarly journals Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

2015 ◽  
Vol 16 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Bao Quoc Tang
Keyword(s):  
Diffusion Equation ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Random Attractors ◽  
External Forces ◽  
Stochastic Reaction

The existence of a unique random attractors in [Formula: see text] for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544–1583] instead of the usual one. The asymptotic compactness of solutions in [Formula: see text] is established by combining “tail estimate” technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.

scholarly journals Adaptive Extraction of Oil Painting Texture Features Based on Reaction Diffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Qicai Huang
Keyword(s):  
Diffusion Equation ◽  
Total Variation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Support Vector ◽  
Oil Painting ◽  
New Model ◽  
Oil Paintings

The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.

Entire solutions of reaction—diffusion equations with balanced bistable nonlinearities

2006 ◽  
Vol 136 (6) ◽  
pp. 1207-1237 ◽  
Author(s):  
Xinfu Chen ◽  
Jong-Shenq Guo ◽  
Hirokazu Ninomiya
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Entire Solutions ◽  
Stable Equilibria ◽  
Travelling Fronts

This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

Sign in / Sign up

Export Citation Format

Share Document