Traveling Wave Solution

In the same manner as an ordinary distribution line, the following traveling wave solutions for the voltage and the current are easily obtained [45]  

---

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... 

One may obtain traveling wave solutions with other kinds of boundary conditions. This is, for example, the case when the reaction medium can be visualized as a closed curve in a two-dimensional space, or a closed surface in three-dimensional space (periodic boundary conditions).2... 

Another type of shape-preserving behaviour, the one we shall be preoccupied with in the rest of this section, is characteristic for travelling wave solutions of the form... 

Note that a travelling wave solution is related to a similarity solution via the following known transformation ... 

Observe that a monotonic travelling wave solution to (3.2.2a) with boundary conditions... 

P. S. Hagan, Travelling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal., 18 (1982), pp. 717-738. 

The evaluation of stability for travelling-wave solutions is by no means a simple process and will not even be attempted here. The result that systems governed by quadratic Fisher equations tend to pick up their minimum permitted velocity will be used later. 

Note that we have, by definition, yr = c2y " andyl = c2y, showing that the wave equation is satisfied for all traveling wave shapes yr and vy. However, the derivation of the wave equation itself assumes the string slope is much less than 1 at all times and positions [Morse, 1981]. The traveling-wave solution of the wave equation was first published by d Alembert in 1747 [Lindsay, 1973]. 

Substituting into the traveling-wave solution of the wave equation gives... 

It can be checked that, for small displacements, the following modified traveling wave solution satisfies the lossy wave equation ... 

Again the discrete-time simulation of the decaying traveling-wave solution is an exact implementation of the continuous-time solution at the sampling positions and... 

Fig. 4.7. Numerical results for the block velocities against time for each block for a different choice of the parameter set values N = 250, K = 100, V = 0.05 a = 2.5), where earthquake-like events are absent and travelling wave solutions dominate (Ananthkrishnan and Ramachandan 1994).

Glasser, B.J., Kevrekidis, I.G. and Sundaresan, S. (1996), One- and two-dimensional traveling wave solutions in gas-fluidized beds, J. Fluid Mech., 306, 183-221. 

We can show that the function of Eq. (8.100) is a traveling wave by showing that a node in the wave moves along the string. When t = 0, there is a node at X = 0. At a later time, this node is located at a value of x such that k(x — ct) is still equal to zero. At a time r, x = ct at the node, so that the speed of the wave is equal to c. The traveling wave solution in Eq. (8.100) is not a solution in which the variables are separated. However, using Eq. (14) of Appendix B, we can show that... 

We begin with a discussion of a one-dimensional traveling wave solution which is a mathematical representation of the uniformly propagating reaction wave that we are discussing. This discussion is a bit technical but gives a good insight into the combustion theory. 

Since the system (3.22), (3.23) is already written in a moving coordinate system, the traveling wave solution is a stationary solution of this problem,... 

In order to present here some basic results of the weakly stability analysis, we consider below a general reaction-diffusion system of equations. We assume that the problem has a one-dimensional traveling wave solution that loses stability in the same way as the gasless combustion wave as discussed in greater detail below. A study of a general reaction-diffusion system rather than a specific model is useful, because it allows us to focus on general properties of the solution, independent of a particular model. 

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... 

A. J. Bernoff, R. Kuske, B. J. Matkowsky, and V. Nofpfkt,Mean-field effects for counterpropagating traveling-wave solutions of reaction-diffusion systems, SIAM J. Appl. Math., 55 (1995), pp. 485-519. 

A. I. VOLPERT, V. A. VOLPERT, AND V. A. YohFEKI, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, RI, 1994. 

It was introduced in 1937 in the seminal contributions of R. A. Fisher [132] and A. N. Kolmogorov, together with 1. G. Petrovskii and N. S. Piskunov [232] as a model for the spreading of an advantageous gene. Consequently, we will refer to (4.1) also as the FKPP equation. It is the simplest and most well-known equation that has traveling wave solutions. 

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... 

In this model, one may determine travelling wave solutions, by solving -vd T=DdlT+Q (3)... 

Kopell and Howard have written several nice expository papers on travelling wave solutions of reaction-diffusion equations (e.g. Kopell and Howard, 1974, 1975) One result which is particularly worth mertioning is that one can construct axisymmetric periodic travelling waves which are completely regular at the origin (see also Greenberg, 1975). That is. 

---