Asymptotics and Perturbations

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I - Asymptotic Methods and Perturbation Theory, Springer, Berlin, 1999. 

Because of the complex nature of the Painleve transcendents and of the resulting difficulties in satisfying the boundary conditions we shall not proceed with the exact analytical solution of b.v.p. (5.3.6) (5.3.8) any further, but rather we turn to an asymptotic and numerical study of this singular perturbation problem. 

With simplified chemical kinetics, perturbation methods are attractive for improving understanding and also for seeking quantitative comparisons with experimental results. Two types of perturbation approaches have been developed, Damkohler-number asymptotics and activation-energy asymptotics. In the former the ratio of a diffusion time to a reaction time, one of the similarity groups introduced by Damkohler [174], is treated as a large parameter, and in the latter the ratio of the energy of activation to the thermal... 

This section presents [after references (j) and ( )] perturbation expressions for several definitions of reactivity, as well as the equivalent expressions required for the experimental determination of reactivities of exactly the same definition. The reactivities considered are the static reactivity (the reactivity most commonly calculated) and three special cases of Henry s time-dependent reactivity (5) the asymptotic-period reactivity, the promptmode reactivity, and the source-multiplication reactivity. 

QCD has the important property of asymptotic freedom - that at very high energies (and, hence, short distances) the interactions between quarks tend to zero as the distance between them tends to zero. Because of asymptotic freedom, perturbation theory maybe used to calculate the high energy aspects of strong interactions, such as those described by the parton modeL... 

Using scaling analysis and perturbation methods, we have been able to derive approximate expressions for the momentum and energy flux in dilute gases and liquids. These methods physically involve formal expansions about local equilibrium states, and the particular asymptotic restrictions have been formally obtained. The flux expressions now involve the dependent transport variables of mass or number density, velocity, and temperature, and they can be utilized to obtain a closed set of transport equations, which can be solved simultaneously for any particular physical system. The problem at this point becomes a purely mathematical problem of solving a set of coupled nonlinear partial differential equations subject to the particular boundary and initial conditions of the problem at hand. (Still not a simple matter see interlude 6.2.)... 

Lam, S.H., Goussis, D.A. Conventional asymptotics and computational singular perturbation for simplified kinetics modeling. In Smooke, M.O. (ed.) Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames. Springer Lecture Notes, vol. 384, pp. 227-242. Springer, Berlin (1991)... 

Maz ya V.G., Nazarov S.A. (1987) Asymptotics of energy integrals for small perturbations of the boundary near corner and conical points. Trudy Moscow Math. Soc. 50, 79-129 (in Russian). 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180. 

Since xi0 is the known solution and xt is the perturbed solution, an important case arises when <( ) - 0 for t - oo. In such a case the stability is asymptotic. 

Inequality (1.88) defines the domain where rotational relaxation is quasi-exponential either due to the impact nature of the perturbation or because of its weakness. Beyond the limits of this domain, relaxation is quasi-periodic, and t loses its meaning as the parameter for exponential asymptotic behaviour. The point is that, for k > 1/4, Eq. (1.78) and Eq. (1.80) reduce to the following ... 

Fig. 5.15. Theoretical dependences of HWHM on the rate of rotational energy relaxation perturbation theory asymptotics (1), classical weak-collision. /-diffusion model (2), quantum theory without (3) and with (4) adiabatic correction.

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981) and Lager-strom, P. A., Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag, New York (1988). 

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