scholarly journals Distributed Parameter State Estimation for the Gray–Scott Reaction-Diffusion Model

Systems ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 71
Author(s):  
Petro Feketa ◽  
Alexander Schaum ◽  
Thomas Meurer
Keyword(s):  
Finite Number ◽  
Exponential Convergence ◽  
Reaction Diffusion ◽  
Distributed Parameter ◽  
System State ◽  
State Information ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Complete State ◽  
The One

A constructive approach is provided for the reconstruction of stationary and non-stationary patterns in the one-dimensional Gray-Scott model, utilizing measurements of the system state at a finite number of locations. Relations between the parameters of the model and the density of the sensor locations are derived that ensure the exponential convergence of the estimated state to the original one. The designed observer is capable of tracking a variety of complex spatiotemporal behaviors and self-replicating patterns. The theoretical findings are illustrated in particular numerical case studies. The results of the paper can be used for the synchronization analysis of the master–slave configuration of two identical Gray–Scott models coupled via a finite number of spatial points and can also be exploited for the purposes of feedback control applications in which the complete state information is required.

On a reaction–diffusion model for sterile insect release method on a bounded domain

2014 ◽  
Vol 07 (03) ◽  
pp. 1450030 ◽  
Author(s):  
Weihua Jiang ◽  
Xin Li ◽  
Xingfu Zou
Keyword(s):  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Whole Space ◽  
The One ◽  
Sterile Insect Release ◽  
Release Method ◽  
Dimensional Domain ◽  
Sterile Insect Release Method

We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method (SIRM). Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci.5 (1992) 221–247] where the habitat is assumed to be the one-dimensional whole space ℝ, we consider this system in a bounded one-dimensional domain (interval). Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that: (i) when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; (ii) when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

Pyrolysis behaviors dominated by the reaction–diffusion mechanism in the fluorine-free metal–organic decomposition process

2020 ◽  
Vol 8 (48) ◽  
pp. 17417-17428
Author(s):  
Jiangtao Shi ◽  
Yue Zhao ◽  
Yue Wu ◽  
Jingyuan Chu ◽  
Xiao Tang ◽  
...  
Keyword(s):  
Diffusion Model ◽  
Diffusion Mechanism ◽  
Reaction Diffusion ◽  
Decomposition Process ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Metal Organic ◽  
Free Metal ◽  
Organic Decomposition

In this work, pyrolysis behaviors dominated by the reaction–diffusion mechanism were investigated. And one-dimensional reaction–diffusion model is proposed.

A Set of Plane Measure Zero Containing all Finite Polygonal Arcs

1970 ◽  
Vol 22 (4) ◽  
pp. 815-821 ◽  
Author(s):  
D. J. Ward
Keyword(s):  
Finite Number ◽  
Measure Zero ◽  
One Dimensional ◽  
The One ◽  
The Given ◽  
Plane Measure

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.

scholarly journals Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

1983 ◽  
Vol 24 (4) ◽  
pp. 392-416 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Nonlinear Elliptic ◽  
Special Cases ◽  
Exothermic Chemical Reaction ◽  
Dimensional Shape ◽  
The One ◽  
Special Case

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

scholarly journals Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Kolade M. Owolabi ◽  
Kailash C. Patidar
Keyword(s):  
Patch Size ◽  
Reaction Diffusion ◽  
Mathematical Community ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Critical Patch Size ◽  
The One ◽  
Exponential Time Differencing Method ◽  
Theoretical Results

We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted asLmust be greater than the critical patch sizeLc. It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function.

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