Asymptotic approximation expansion

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of the form... 

In the previous section we demonstrated the application of asymptotic expansion techniques to obtain the high- and low-frequency limits of the velocity field for flow in a circular tube driven by an oscillatory pressure gradient. In the process, we introduced such fundamental notions as the difference between a regular and a singular asymptotic expansion and, in the latter case, the concept of matching of the asymptotic approximations that are valid in different parts of the domain. However, all of the presentation was ad hoc, without the benefit of any formal introduction to the properties of asymptotic expansions. The present section is intended to provide at least a partial remedy for that shortcoming. We note, however,... 

As a starting point, we recall that the limit a/R = 0 corresponds to a straight circular tube, with the flow described by the Poiseuille flow solution w = (1 — r2), u = v = 0. In the present context, we consider small, but nonzero, values of a/R, and recognize the Poiseuille flow solution as a first approximation in an asymptotic approximation scheme. In particular, if we assume that a solution exists for u in the form of a regular asymptotic expansion,... 

To obtain a valid approximate solution for heat transfer from a sphere in a uniform streaming flow at small, but nonzero, Peclet numbers, we must resort to the method of matched (or singular) asymptotic expansions.4 In this method, as we have already seen in Chap. 4, two (or more) asymptotic approximations are proposed for the temperature field at Pe 1, each valid in different portions of the domain but linked in a so-called overlap or matching region where it is required that the two approximations reduce to the same functional form. The approximate forms of (9-1), from which these matched expansions are derived, can be obtained by nondimensionalization by use of characteristic length scales that are appropriate to each subdomain. 

Higher-order terms to 0(Re3) were obtained by Chester and Breach.11 It is disappointing, but fairly typical of asymptotic approximations, that the calculation of many terms achieves a relatively small increase in the range of Reynolds number in which the drag can be evaluated accurately compared with (9 122). We may recall that asymptotic convergence is achieved by taking the limit Re 0 for a fixed number of terms in the expansion, rather than an increasing number of terms for some fixed value of Re. 

W. Dahmen and C.A. Micchelli, Using the Refinement Equation for Evaluating Integrals of Wavelets, SIAM Journal of Numerical Analyst, 30 (1993). 507-537. W. Sweldens and R. Piessens, Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions. SIAM Journal of Numerical Analyses, 31 (1994). 1240-1264. 

Another technique, that is more flexible than the rescaled expansion and appears to converge slightly better, is weighted truncation [11]. This method is based on the idea of an optimal asymptotic approximation [36]. A typical characteristic of divergent asymptotic expansions is that their partial sums at first steadily approach the correct value but then, after a certain point, become steadily worse. An estimate of the error in the nth partial sum is given by the term in the expansion of order n - -1, that is The partial sum for... 

We call this technique weighted truncation, since each of these approximants is the optimal asymptotic approximation of a series that has been weighted by subtracting out the expansion of a pole with the residue chosen so as to minimize the asymptotic error of the truncation. 

The presence of a singularity in E(S) at = 0 implies that any of the summation methods described in Section A that employ partial sums will eventually diverge. The optimal asymptotic approximations are reached quite early, and these methods cannot utilize the expansion coefficients beyond that point. A major advantage of dimensional perturbation theory over other types of perturbation theory is the relative ease with which the expansion coefficients can be calculated [6-10]. For this reason, the development of summation methods that continue to work at large order is a central goal. 

In the mean-field approximation, when the effects of the critical fluctuations are neglected, eq 10.1 reduces to an asymptotic Landau expansion ... 

From eq 10.20, we note that for the lattice gas is to be identified with the critical part of the Helmholtz-energy density. From eq 10.11, we see that in the classical mean-field approximation ci has an asymptotic Landau expansion of the form ... 

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... 

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. 

Using the impact approximation presented in Chapter 6, they may easily be found for any rotational band even if rotational-vibrational interaction is nonlinear in J. In 1954 R W. Anderson proved as a theorem [104] that expansion of the spectral wings in inverse powers of frequency is controlled by successive odd derivatives of the correlation function at the origin. In impact approximation the lowest non-zero derivative of this type is the third and therefore asymptotics G/(co) is described by the power expansion [20]... 

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 ... 

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