scholarly journals Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Xiaoquan Ding ◽  
Jifa Jiang
Keyword(s):  
Diffusion Equation ◽  
Stochastic Equation ◽  
Dimensional Space ◽  
Reaction Diffusion ◽  
Random Attractor ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Random Dynamical System ◽  
Large Space ◽  
Uniform Estimates

This paper is devoted to a stochastic retarded reaction-diffusion equation on alld-dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

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.

scholarly journals An efficient convergent method for the delay reaction-diffusion equations

2021 ◽  
Author(s):  
Marzieh Heidari ◽  
Mehdi Ghovatmand ◽  
Mohammad Hadi Noori Skandari
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
System Of Equations ◽  
Equivalent System ◽  
Numerical Examples ◽  
Delay Function ◽  
Mild Conditions ◽  
Convergent Method

In this manuscript, we consider the delay reaction-diffusion equation and implement an efficient spectral collocation method to approximate the solution of this equation. We first replace the delay function in the delay reaction-diffusion equation and achieve an equivalent system of equations. We then utilize the Legendre-Gauss-Lobatto and two-dimensional interpolating polynomial to approximate the solution of obtained system. Moreover, we prove the convergent of method under some mild conditions. Finally, the capability and efficiency of the method is illustrated by providing several numerical examples and comparing them with others

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 Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

2019 ◽  
Vol 17 (1) ◽  
pp. 1411-1434
Author(s):  
Zongfei Han ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Multiplicative Noise ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Random Dynamical System ◽  
Exponential Attractor

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

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.

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