Reaction diffusion equation

The mass- and energy-balance equations for our new system, allowing for diffusion and conduction along one spatial dimension r, can be written as 

In all cases we will assume that the initial distribution of the precursor P is uniform, p(r, t = 0) — p0 for all 0 r a0. There is no mechanism by which a concentration gradient in P can then appear spontaneously, so the concentration of the precursor is then determined by the ordinary differential  

This equation can be integrated to give the exponential decay  

This form can be substituted into eqn (10.2), so we have equations two coupled 

The temperature and the concentration of the intermediate A are thus functions of both time and position, as will be the local value of the reaction r te constant kx given by the Arrhenius law 

The classical and simplest model for spatial spread or dispersal is the diffusion equation or Pick s second law, which in spatially one-dimensional systems reads 

If the particles or individuals react or interact according to some rate law F p) and at the same time undergo diffusion, it is legitimate to combine the diffusion equation and the rate equation p = F p) [178], The result is the well-known reaction-dijfusion (RD) equation  

Mendez et al., Reaction-Transport Systems, Springer Series in Synergetics, DOI 10.1007/978-3-642-11443-4 2, Springer-Verlag Berlin Heidelberg 2010 

Besides the simple mathematical approach of combining the rate equation and the diffusion equation, two fundamental approaches exist to derive the reaction-diffusion equation (2.3), namely a phenomenological approach based on the law of conservation and a mesoscopic approach based on a description of the underlying random motion. While it is fairly straightforward to show that the standard reaction-diffusion equation preserves positivity, the problem is much harder, not to say intractable, for other reaction-transport equations. In this context, a mesoscopic approach has definite merit. If that approach is done correctly and accounts for all reaction and transport events that particles can undergo, then by construction the resulting evolution equation preserves positivity and represents a valid reaction-transport equation. For this reason, we prefer equations based on a solid mesoscopic foundation, see Chap. 3. 

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Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

In order to investigate such front instabilities quantitatively one may derive an equation for the profile ( i(y, t) of the front directly from the reaction-diffusion equation. This Kuramoto-Sivashinsky equation 1691... 

In a moving co-ordinate system, the traveling wave equations typically reduce to a system of parameterized nonlinear ordinary differential equations. The solutions of this system corresponding to pulses and fronts for the original reaction-diffusion equation are called homoclinic and heteroclinic orbits, correspondingly, or just connecting orbits. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

If nonreactive MPC collisions maintain an instantaneous Poissonian distribution of particles in the cells, it is easy to verify that reactive MPC dynamics yields the reaction-diffusion equation,... 

Starting from an initial state where half the system has species A and the other half B, a reaction front will develop as the autocatalyst B consumes the fuel A in the reaction. The front will move with velocity c. The reaction-diffusion equation can be solved in a moving frame, z = x — ct, to determine the front profile and front speed,... 

Rayleigh parameters, two-pathway excitation, coherence spectroscopy, 155-159 Reaction-diffusion equations, multiparticle collisions, reactive dynamics, 108-111 Reaction rates ... 

The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

When applying a mechanistic model, nearly all of the computational effort resides in step (3).109 In most mechanistic models, step (3) is modeled by one-dimensional reaction-diffusion equations of the form... 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

Mass transfer One more difficulty arises from the fact that there are two phases in the reactor (i) hydrocarbon and (ii) acid. The reaction occurs in the acid phase while reactants are feed in hydrocarbon phase. This implies that, in order to reaction occurs, there is mass transfer from hydrocarbon to acid phase. The mass transfer is a very complex phenomenon which can involve the reaction-diffusion equation. However, such a phenomenon is beyond of the goal of this chapter. Both isobutane and propilene are feed in hydrocarbon phase. Solubility of propylene in acid phase is very... 

In population genetics there is experimental evidence that many mutations are neutral, which is consistent with Kimura s theory of neutral evolution [19]. Kimura s theory is based on a neutrality condition, that is, on the assumption that the natality and mortality functions as well as the transport (migration) coefficients are the same for the main population as well as for the mutants. For neutral mutations the nonlinear reaction-diffusion equations for the spreading of a mutation within a growing population which is expanding in space have a... 

Let us go back to the Brusselator (1). In a one-dimensional medium of unit length the reaction-diffusion equations (12) become... 

This shows that the usual ideas associated with propagating waves in electromagnetism or fluid dynamics do not describe the behaviors found here. These differences could be expected because of the mathematical structure of reaction-diffusion equations, which owing to their parabolic character propagate information with infinite velocity. On the contrary, in the case of classical wave equations or hyperbolic equations there is a well-defined domain of influence and a characteristic velocity of propagation of information. ... 

The reaction-diffusion equation describes how the local concentration of the reactant A varies within some infinitesimal volume element at some point in the reaction zone. For the above reaction kinetics, this equation will have the following form ... 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Now that we have two governing reaction-diffusion equations and two independent concentrations we can hope for a more varied range of local stabilities and perhaps also for sustained oscillatory solutions. The latter may then be manifest as solutions which are distributed in both space and time. 

Brown, K. J. and Eilbeck, J. C. (1982). Bifurcation, stability diagrams and varying diffusion coefficient in reaction-diffusion equations. Bull. Math. Biol., 44, 87-102. 

Burnell, J. G., Lacey, A. A., and Wake, G. C. (1983). Steady-states of the reaction-diffusion equations, part 1 questions of existence and continuity of solution branches. J. Aust. Math. Soc., B24, 374-91. 

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