Approximations asymptotic

The cumulative uptake can easily be calculated by numerical integration of Eq. (29). Comparison of the numerical and analytical solutions for a range of nutrient parameters showed very close agreement, indicating that the asymptotic approximations were valid. 

Smooke, M. D., Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames Lecture Notes in Physics 384 . Springer-Verlag, New York, 1991. 

Smooke, M. D. 1991. Reduced kinetic mechanisms and asymptotic approximations for methane-air flames. New York Springer-Verlag. 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

Figure 2. Equation 15 (circles) (a) asymptotic approximation for small values of a (eq 16), continuous line (b) asymptotic approximation for large values of a (eq 20), continuous line.

The error introduced by use of the Wien equation is less than 1 percent when XT < 3000 pm K. The Wien equation has significant practical value in optical pyrometry for T < 4600 K when a red filter (X = 0.65 pm) is employed. The long-wavelength asymptotic approximation for Eq. (5-102) is known as the Rayleigh-Jeans formula, which is accurate to within 1 percent for XT > 778,000 pm-K. The Raleigh-Jeans formula is of limited engineering utility since a blackbody emits over 99.9 percent of its total energy below the value of XT = 53,000 pm-K. 

Calculations with these formulas proved tedious, so asymptotic approximations were sought. DeMoivre (1733) obtained the following analytic asymptotes [see Feller (1968), Chapter 7] ... 

F.A. Williams, Influences of Detailed Chemistry on Asymptotic Approximations for Flame Structure, in Mathematical Modeling in Combustion and Related Topics (Nijhoff, Dordrecht, 1988) pp. 315-341. 

It turns out, very fortunately, that this asymptotic approximation is also an exact solution of the Schrddinger equation Eq (7.29) with = 0, Just what happened for the harmonic-oscillator problem in Chapter 5. The solutions are designated Rnlit), where the label n is known as the principal quantum number, as well as by the angular momentum i, which is a parameter in the radial equation. The solution (7.32) corresponds to Rioir)- This should be normalized according to the condition... 

In the procedure of asymptotic approximation no arbitrary approximation functions are used, instead a series of functions... 

Also, we have been very loose about the sense in which our formulas approximate the true solutions. The relevant notion is that of asymptotic approximation. For introductions to asymptotics, see Lin and Segel (1988) or Bender and Orszag (1978). 

One important class of problems for which we can obtain significant results at the first level of approximation is the motion of fluids in thin films. In this and the subsequent chapter, we consider how to analyze such problems by using the ideas of scaling and asymptotic approximation. In this chapter, we consider thin films between two solid surfaces, in which the primary physics is the large pressures that are set up by relative motions of the boundaries, and the resulting ideas about lubrication in a general sense. 

Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

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,... 

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