Singular perturbation analysis

Singular perturbation analysis was employed to study the velocity of pulled fronts, and it was shown that the solvability integrals diverge [103, 104, 448]. Therefore we will use this method only for non-KPP kinetics. We assume 5 = 0(e), weak heterogeneities, i.e., S = as in (6.51) with a = 0(1). Equation (6.51), together with the corresponding boundary conditions, becomes 

To study (6.53) we carry out a nonrigorous asymptotic analysis. We assume that the domain is divided into two regions, an interior or boundary layer region, whose width is 0(e), where p varies rapidly, and an outer region where p is almost constant. In other words, either p = 0(e ) or /o = 1 + 0(e ), where i and 2 are positive real numbers. To solve (6.53) in the outo region we expand p as follows  

By substituting (6.54) into (6.53) and equating terms of equal power in e, we obtain 

To study the dynamics in the interior of the boundary layer we transform (6.53) to the reference frame of the front, i.e., we define the new variable z = [x - S(t)]/s, where S t) represents the position of the front. The derivatives in (6.51) transform according to 

Since we assumed o = o(z) in (6.59), the first equation (6.63) is equivalent to the homogeneous ( = 0) reaction-diffusion equation transformed to the front reference frame, z = x—SqI, which travels with constant velocity So. We set Sq = c, and So = ct with S(0) = 0. Since j has a zero eigenvalue, id o/dz = 0, it cannot be inverted to obtain the solutions of (6.64) and (6.65). Instead those equations have 

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Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

Fox, R. O. (1989). Steady-state IEM model Singular perturbation analysis near perfectmicromixing limit. Chemical Engineering Science 44, 2831-2842. 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

A more cogent mathematical treatment of this problem was given in the 1970s by several mathematical biologists. For details see books by Lin and Segel [130] and Murray [146], Here we provide a brief account of this approach. The approach uses the somewhat advanced mathematical method of singular perturbation analysis, but does provides a deep appreciation of the Michaelis-Menten enzyme kinetics. 

Notice that when r - oo,v - S0/(Km + S0). This is exactly the value of v that we arrived at for r = 0. Thus as r -> oc (on the fast timescale), v approaches the derived initial condition for the slow timescale (r = 0). Hence, the entire transient for Michaelis-Menten kinetics can be represented by combining the short timescale result, Equation (4.29), with the long timescale result, the solution to Equation (4.25). The two results match seamlessly at r = oo and x = 0. This is known as asymptotic matching in singular perturbation analysis [110]. 

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

These methods have some limitations. Singular perturbation analysis is an effective tool if the solution is known to the leading order and if the reaction term is not given by KPP kinetics. The solution to the lowest order can be found for some particular non-KPP kinetic terms, but it is not known in general. This method requires, of course, that a small parameter is present in the model. It is necessary to assume that the spatial heterogeneities in the system introduce a small variation in the reaction... 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

Fig. 6.8 Comparison of the temporal evolution of the velocity of fronts (in dimensionless units) between the singular perturbation analysis result given by (6.73) (solid lines), the local velocity approach (6.76) (dashed lines), and numerical results (symbols) for different values of s. Here = X and / = p ( — p). Reprinted with permission from [290]. Copyright 2003 by the American Physical Society...

Singular perturbation analysis does not provide a fully analytical result for the very important case of KPP kinetics. It is not possible to go beyond the first order in 5, because the exact unperturbed solution is not known and the integrals in the solvability condition diverge. Proceeding as in Sect. 4.2.1 for (6.50) with KPP kinetics, we obtain to leading order the following equation for the action functional (e = 0) ... 

For the singular perturbation analysis of some other chemical kinetics problems in the critical case see, for example, [35]. 

The singular perturbation analysis developed below is a heuristic version of the rigurous results derived by Fife [105]. According to this work, the original stationary problem (Equation (9)) can in principle be reduced to a more tractable equation. For the specific case of piece-wise linear slow manifolds, this reduced system can be solved analytically [62,104]. 

L0vas, T., Mastorakos, E., Goussis, D.A. Reduction of the RACM scheme using computational singular perturbation analysis. J. Geophys. Res. Atmos. 111(013302) (2006)... 

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