Asymptotic expansions

The divergence factor (DF) introduced by the asymptotic expansion, accounts for the deformation of the refracted wavefront (initially spherical in the coupling medium). It ensures, under the GO approximation, the energy conservation of a ray-pencil propagating... 

This statement is not exactly true - the slightly different system of ODEs is defined by an asymptotic expansion in powers of At which is generally divergent. 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

Lagerstrom, P. A. Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag (1988). 

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. 

FIG. 2 Phase diagram in the M-z plane for a square lattice (MC) and for a Bethe lattice q = A). Dashed lines Exact results for the Bethe lattice for the transition lines from the gas phase to the crystal phase, from the gas to the demixed phase and from the crystal to the demixed phase full lines asymptotic expansions. Symbols for MC transition points from the gas phase to the crystal phase (circles), from the gas to the demixed phase (triangles) and from the crystal to the demixed phase (squares). (Reprinted with permission from Ref. 190, Fig. 7. 1995, American Physical Society.)... 

For different regions in the flow field in front of an expanding piston, separate solutions in the form of asymptotic expansions may be developed. An overall solution can be constructed by matching these separate solutions. This mathematical technique was employed by several authors including Guirao et al. (1976), Gorev and Bystrov (1985), Deshaies and Clavin (1979), Cambray and Deshaies (1978), and Cambray et al. (1979). 

In addition to a near-shock and an acoustic region, Deshaies and Clavin (1979) distinguished a third—a near-piston region—where nonlinear effects play a role as well. As already pointed out by Taylor (1946), the near-piston flow regime may be well approximated by the assumption of incompressibility. For each of these regions, Deshaies and Clavin (1979) developed solutions in the form of asymptotic expansions in powers of small piston Mach number. These solutions are supposed to hold for piston Mach numbers lower than 0.35. 

The approxmations reviewed so far were all developed for the low-piston Mach number regime. Cambray and Deshaies (1978), on the other hand, developed a solution of the similarity equations by asymptotic expansions in powers of high-piston Mach numbers. These solutions are supposed to hold for piston Mach numbers higher than 0.7. Finally, Cambray et al. (1979) suggested an interpolation formula to cover the intermediate-piston Mach number range. 

Figure 4.14. Flow-field parameter distributions in front of an expanding piston. Soiution by matched asymptotic expansions by Guirao et al. compared to exact similarity solutions for various piston Mach numbers.

However, serious difficulties appeared later when efforts were made to attack more general problems not necessarily of the nearly-linear character. In terms of the van der Pol equation this occurs when the parameter is not small. Here the progress was far more difficult and the results less definite moreover there appeared two distinct theories, one of which was formulated by physicists along the lines of the theory of shocks in mechanics, and the other which was analytical and involved the use of the asymptotic expansions (Part IV of this chapter). The latter, however, turned out to be too complicated for practical purposes, and has not been extended sufficiently to be of general usefulness. 

Since, for obvious reasons, the series of Poincar6 cannot be applied here, attempts were made to use the so-called asymptotic expansions (this approach is to a great extent also due to Poincar6) which are not analytic and, for that reason, seem to be adequate when a trajectory has to turn around a sharp comer, as was explained in Section 6.28. 

Having included line shifts in the component frequencies, it can be regarded that Im f(0) = 0 and the asymptotic expansion of the denominator... 

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

With this relation in view, it is not difficult to derive the asymptotic expansion... 

When providing current manipulations, the solution u = u(x,t) and the available data of the original problem are preassumed to be smooth enough and sufficient for the existence of the asymptotic expansion... 

The error function erfx has a power series expansion for small x and an asymptotic expansion for large x... 

Estimates for the discretization error are derived in the appendix. Unlike the estimates (2.6) these are not obtained as strict inequalities, but rather as leading terms of asymptotic expansions. For the integral (2.12a) with the integration limits —oo to oo the discretization error is (for large n and sufficiently small h, see appendix... 

From this asymptotic expansion in powers of n no conclusions on the radius of convergence of ed h) are possible, but there are some hints that the radius of convergence is that ofcosech anh ), i.e.the series (A.4) probably converges for... 

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). 

This term can only control regioselectivity if the transition state occurs relatively early along the reaction path (so that the asymptotic expansion about the separated reagent limit is still relevant) and if the extent of electron-transfer is large compared to the electrostatic interactions between the reagents. The importance of Equation 18.27 for explaining the utility and scope of the Fukui function was first noted by Berkowitz in 1987 [59]. 

Equations (257) through (266) provide us with a closed set of equations which allow us, in principle, to calculate yaB and w . However, an exact solution of these equations is very difficult to obtain and is moreover not very useful. Indeed, we expect our macroscopic description to be valid only at very small ionic concentrations and it is thus not necessary to derive an exact result only the leading term in an asymptotic expansion at small C will be relevant. The following approximations will thus be used ... 

Asymptotic Expansions of Bessel Functions. In certain physical problems it is desirable to know the value of a It ess el function for large values of its argument. In this section we shall derive the asymptotic expansion of the Bessel function of the first kind Jn %) and merely indicate the results for the other Bessel occurring in mathematical physics. 

As only one of us has but a single joint artiele [1] with Jens, not surprisingly beeause we are not quantum chemists, we are honoured to have been invited to contribute to this special issue in his honour. In all the 25 or so years we have known him, being professors of applied mathematics, we have had to listen to cheap jibes about special functions, asymptotic expansions and other things of beauty, so this invitation gives us the opportunity for some revenge, because, knowing how conscientious Jens is, we are sure that he will read the article. 

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