Reaction-diffusion equation , model system

The model we employ in the first study involves a general system of reaction-diffusion equations. This system includes the diffusional-thermal model [1], in which the effect of a given fluid dynamical field is employed in the transport (reaction-diffusion) equations describing the evolution of the temperature field and the concentrations of the chemical species participating in the chemical reactions. Thus the qualitative effects of thermal expansion are considered to be weak, and... 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

A disadvantage of ODE models is that they assume spatially homogeneous systems, an assumption that sometimes may lead to wrong predictions. Although in many cases spatial effects can be incorporated in the function I , there are situations where one may need to take into account diffusion and transport of proteins from one compartment to another. For the purpose, reaction-diffusion equations (RDEs) of the form... 

Basic features of the light-sensitive BZ reaction are reproduced by the Oregonator model given as a system of two coupled reaction-diffusion equations ... 

The computer simulations of chemical kinetics in a straight tube reactor [1065] were based on an equation combining diffusion, convection, and reaction terms. The sample dispersion without chemical reactions gave very similar results to that of Vanderslice [1061], yet the value of that paper is that it expanded the study to computation of FIA response curves for fast and slower chemical reactions. The numerically evaluated equation was similar to that of Vanderslice [1061], however with inclusion of a term for reaction rate. Two model systems were chosen and spectro-photometrically monitored in a FIA system with appropriately con-... 

It should be pointed out that although //is not consumed by the reaction, its value is not locally constant because diffusion during PEB and evaporation (as well as unwanted side reactions involving the acid, i.e., all acid loss mechanisms) may induce local variations in the concentration of H. This condition necessitates the use of reaction-diffusion equations to accurately model this system. However, the assumption that H is constant is not without merit, for it is valid under certain conditions. Besides, it helps to simplify the problem. 

Peskin et al [1993] have proposed the Brownian ratchet theory to describe the active force production. The main component of that theory was the interaction between a rigid protein and a diffusing object in front of it. If the object undergoes a Brownian motion, and the fiber undergoes polymerization, there are rates at which the polymer can push the object and overcome the external resistance. The problem was formulated in terms of a system of reaction-diffusion equations for the probabilities of the polymer to have certain number of monomers. Two limiting cases, fast diffusion and fast polymerization, were treated analytically that resulted in explicit force/velocity relationships. This theory was subsequently extended to elastic objects and to the transient attachment of the filament to the object. The correspondence of these models to recent experimental data is discussed in the article by Mogilner and Oster [2003]. 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

Linear diffusion satisfactorily describes the transport mechanism for a single population. For interacting populations, linear diffusion terms imply that the populations are able to mix completely, with the movement of one cell type unaffected by the presence of cells of the other type. The reality is exactly the opposite. Cell movement is typically halted by contact with another cell. This phenomenon is known as contact inhibition and is very well documented for many types of cells. Sherratt introduced a phenomenological model to account for contact inhibition [402]. Consider the interaction between normal and tumor cells with concentrations pm(xj) and Pt(x, t), respectively. The overall cell flux of both populations is given by x(Pn + Pt)- a fraction Pn/(Pn + Pt) of this flux corresponds to normal cells, so that the flux of normal cells is - [pn/(Pn + Pr)] x(Pn + Pr)> a similar expression for the flux of tumor cells. These expressions indicate that the movement of one population is inhibited by the presence of the other. The system of dimensionless reaction-diffusion equations reads [402]... 

Vanag and Epstein have formulated a four-variable model to understand pattern formation in the BZ-AOT system [449,453]. Their model builds on the Oregonator, see Sect. 1.4.8. It assumes that the chemistry within the water core of the droplets is well described by the two-variable Oregonator rate equations (1.131). It further assumes that the species in the oil phase are inert, since they lack reaction partners, the key reactants all being confined to the aqueous core of the droplets. Consequently, only transfer reactions occur for the activator B1O2 and inhibitor Br2 in the oil phase. The rate terms for the two transfer reactions are added to the rate terms of the two-variable Oregonator model. The reaction-diffusion equations of the four-variable model of the BZ-AOT system are given in nondimensionalized form by... 

Development-controlling prepattern mechanisms have been modelled in reaction-diffusion context. In the celebrated paper of Turing (1952) a model was presented in terms of reaction-diffusion equations to show how spatially inhomogeneous arrangements of material might be generated and maintained in a system in which the initial state is a homogeneous distribution. Two components were involved in the model, and the reactions were described by linear differential equations. The model in one spatial dimension s is ... 

Adaptive computations of nonlinear systems of reaction-diffusion equations play an increasingly important role in dynamical process simulation. The efficient adaptation of the spatial and temporal discretization is often the only way to get relevant solutions of the underlying mathematical models. The corresponding methods are essentially based on a posteriori estimates of the discretization errors. Once these errors have been computed, we are able to control time and space grids with respect to required tolerances and necessary computational work. Furthermore, the permanent assessment of the solution process allows us to clearly distinguish between numerical and modelling errors - a fact which becomes more and more important. 

The inlet conditions for the numerical simulations are based on the experimental conditions. The simulations are performed with the three different models for internal diffusion as given in Section 2.3 to analyze the effect of internal mass transfer limitations on the system. The thickness (100 pm), mean pore diameter, tortuosity (t = 3), and porosity ( = 60%) of the washcoat are the parameters that are used in the effectiveness factor approach and the reaction-diffusion equations. The values for these parameters are derived from the characterization of the catalyst. The mean pore diameter, which is assumed to be 10 nm, hes in the mesapore range given in the ht-erature (Hayes et al., 2000 Zapf et al., 2003). CO is chosen as the rate-limiting species for the rj-approach simulations, rj-approach simulations are also performed with considering O2 as the rate-hmiting species. 

The choice of theoretical tools used to analyze the behavior of these systems depends on the scales on which one wishes to model the dynamics and the questions being asked. If the phenomena of interest are truly macroscopic, for example, the chemical waves in the BZ reaction, then an appropriate level of description is through reaction-diffusion equations. This macroscopic description focuses on chemical concentrations in small (but still macroscopic) system volumes and considers how these local concentrations change as a result of reaction and diffusion. In such an approach, one bypasses the molecular level of description completely and focuses directly on phenomena occurring on macroscopic scales. 

In order to model pattern formation in chemical systems at the macroscopic level, we must be able to solve the reaction-diffusion equation [1]. The simplest numerical method that one can use to solve this equation is an Euler scheme, where space is divided into a regular grid of points with separation Ax. In Figure 2, we show such a grid for a two-dimensional system. Each grid point is labeled by r = (f, ). Time is also divided into small segments At. 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

In order to construct mesoscopic models, we again begin by partitioning the system into cells located at the nodes of a regular lattice, but now the cells are assumed to contain some small number of molecules. We cannot use a continuum description of the dynamics in a cell as we did for the reaction-diffusion equation. Instead, we describe the reactions and motions of molecules using stochastic rules that mimic the dynamics of these processes on meso-scales. The stochastic element arises because we do not take into account the detailed motions of all solvent species or the dynamics on microscopic scales. Nevertheless, because the number of molecules in a cell may be small, we must account for the fact that this number can change by random reactive events and random motions of molecules that take them into and out of a... 

Pattern Formation and Periodic Structures in Systems Modeled by Reaction-Diffusion Equations. 

Solution of the reaction-diffusion equations that describe the spatial behavior of a chemically reacting system is an extremely demanding computational task. For this reason, it is a great advantage to have a simple model with the smallest possible number of variables. The availability of such a model, the Oregonator [69], is one of the reasons for the popularity of the Belousov-Zhabotinsky reaction for theoretical studies of spatial phenomena. 

To calculate the critical perturbation necessary for wave initiation, we choose a modified Oregonator model [18-20] for an excitable and an oscillatory Belousov-Zhabotinsky reaction and numerically solve the deterministic reaction-diffusion equations for the system in one spatial dimension. We determine both the critical radius and the critical concentration change necessary for trigger wave propagation to proceed. We review the model in this section before proceeding to the results of the critical perturbations necessary for wave initiation. 

Cellular automata can be constructed as simplified models of reaction-diffusion systems. Often the main features of an apparently very complex dynamics can be captured in a simple rule. The Greenbeig-Hastings rules [13] are an example of simple cellular automaton rules that model excitable media. We term such cellular automaton models classical cellular automata since they are constructed in the spirit of the original cellular automaton models of Von Neumann. There is a laige literature [14] on this topic that deals with the mathematical properties of different abstract cellular automaton rules [15], as well as studies that attempt to model rather detailed features of specific reaction-diffusion equations [16,17]. 

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