Travelling front solution

Reaction-transport equations allow us to model the spread of invading populations. Traveling front solutions describe the invasive process, and their velocity is a quantity that can be obtained from observational data. In this chapter we review some recent reaction-transport models and compare the theoretical predictions with the observed values for the rate of population invasions. 

It is not difficult to see that this possesses a travelling front solution of the form... 

Distance Traveled by Solvent Front fa Distance Traveled by Solute (b) r... 

Stationary, traveling wave solutions are expected to exist in a reference frame attached to the combustion front. In such a frame, the time derivatives in the set of equations disappear. Instead, convective terms appear for transport of the solid fuel, containing the unknown front velocity, us. The solutions of the transformed set of equations exist as spatial profiles for the temperature, porosity and mass fraction of oxygen for a given gas velocity. In addition, the front velocity (which can be regarded as an eigenvalue of the set of equations) is a result from the calculation. The front velocity and the gas velocity can be used to calculate the solid mass flux and gas mass flux into the reaction zone, i.e., msu = ps(l — e)us and... 

Ye, Q.-X. and Wang, M.-X. (1987). Traveling wave front solutions of Noyes-Field system for Belousov-Zhabotinskii reaction. Nonlinear Anal. Theor. Methods. Appl., 11, 1289-302. 

Thus, it is clear that the front behavior of (4.16) is completely dominated by the linear behavior close to the unstable state Cu. The full solution is only close to Cu in the leading edge of the travelling front, so that we can say that the full front is pulled by the linear... 

Distance travelled by solute Distance travelled by solvent front... 

We observe that the solutions we seek are not traveling wave solutions in the strict sense of the word. For a traveling wave solution u, we must have u = u x), where x = x + ct for some constant speed c. It can be also written d x = x — ip t) with (p t) = —ct. Here, ip t) is the coordinate of the front (or any other characteristic point of the solution). We seek solutions with (fi = ip y,t). That is, we look for solutions u of the problem (3.87) having the form... 

Note that the reaction-telegraph equation (2.19) differs from the ad hoc HRDE (2.15) by the additional term —xF p) dp/dt) on the left-hand side. It can be shown that solutions of (2.19) converge to solutions of the reaction-diffusion equation (2.3) as T 0 [494]. Traveling wave front solutions for the reaction-telegraph equation have been investigated by several authors [201, 176, 282, 291, 285, 136, 288, 137, 114, 116, 115, 117]. 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

Pojman and his co-workers demonstrated the feasibility of traveling fronts in solutions of thermal free-radical initiators in a variety of neat monomers at ambient pressure using liquid monomers with high boiling points (5-7) and with a solid monomer, acrylamide (8,9), Fronts in solution have also been developed (10). The macrokinetics and dynamics of frontal polymerization have been examined in detail (//). A patented process has been developed for producing functionally-gradient materials (12,13). 

Direct autocatalysis occurs in biological polymerizations, such as DNA and RNA replication. In normal biological processes, RNA is produced from DNA. The RNA acts as the carrier of genetic information in peptide synthesis. However, RNA has been found to be able to replicate itself, for example, with the assistance of the Q/li replicase enzyme. Bauer and McCaskill have created traveling fronts in populations of short self-replicating RNA variants (Bauer ct ah, 1989 McCaskill and Bauer, 1993). If a solution of monomers of triphosphorylated adenine, gua-... 

The spatial differentials included in (5) are approximated by a nearest neighbor differential scheme. The scheme is shown in Figure 9, for the sake of simplicity merely in two dimensions, d= 2. Prom the current source node i,j) there are 2 = 4 possible directions for the front to propagate. The minimum of the travel-time solutions indicates where the front passes first, and hence the neighbor in this direction is selected as the next source node. 

TLC measurements used for the identification of unknown solutes depends on two basic pai ameters. Firstly, the distance traveled by the solvent front, measured from the sampling point or sampling boundary, and secondly, on the distance traveled by the spot from the sampling point or sampling boundary. These are the sole... 

The primary parameter used in TLC is the (Rf) factor which is a simple ratio of the distance traveled by the solute to the distance traveled by the solvent front. The (Rf)... 

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 all TLC work, identification is confirmed by measurement of the distance travelled along the plate by the analyte compared to the solvent front, the Rf value, and by reference to standard solutions run on the same plate. Further confirmation can be obtained by using a reference sample and measuring this under the same conditions as the sample. This allows measurement of the Rx value. This is illustrated in Figure 4.1. 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

This partial derivative is the velocity of the concentration front hi the bed. The constant pattern assumption presupposes that this velocity is constant, or in other words, is independent of the solution concentration. This means that all points on the breakthrough curve are traveling in the bed under the same velocity, and thus a constant shape of this curve is established (Wevers, 1959). According to the above equation, this could happen only if (Perry and Green, 1984)... 

A useful technique which can be applied in cases where constant velocity solutions arise is that of changing from a fixed coordinate system (the present x coordinate has an origin whose position is fixed in space) to travelling-wave coordinates. With the latter, the origin moves from left to right with the front at the same constant velocity c we are constantly adjusting our frame of reference so that within the frame the reaction front appears stationary. This new coordinate z is defined in terms of x and t by... 

A self-similar solution is found in which the pressure II at the shock wave front propagating in the gas decreases as a power function of the distance traveled, X II X n, where 1 < n < 2. In the general form, for any adiabatic index of the gas, it is proved that 1 < n < 2 for 7 = 7/5 the numerical value of n is 1.333. The law found yields the greatest possible rate of the plane shock wave decay under any circumstances. 

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