Singular asymptotic expansion

The proper singular perturbation treatment has thus to take care of this initial stage. Probably the simplest way to do this is via a matched asymptotic expansion procedure, with the outer solution of the type (5.2.13), (5.2.14), valid for t = 0(1), matched with an initial layer solution that has an internal layer at x = 0. 

A.B. Vasilieva and V.F. Butuzov, Asymptotic Expansion for Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian). 

The theory of singular perturbations leads us to seek asymptotic expansions of x(t) and y (f) of the form... 

ASYMPTOTIC EXPANSIONS FOR ORDINARY DIFFERENTIAL EQUATIONS, Wolfgang Wasow. Outstanding text covers asymptotic power series, Jordan s canonical form, turning point problems, singular perturbations, much more. Problems. 3B4pp. 5K x 8X. 65456-7 Pa. 8.95... 

Prom Eqn. (2.6.65), it is apparent that this is a singular perturbation problem (as the highest derivative term is multiplied by the small parameter) and then one can use matched asymptotic expansion to obtain (f> by describing the solution in terms of outer and inner solutions. 

The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 

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,... 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

The difference in scaling between the central core of the thin cavity (6-122) and the vicinity of the end walls (6-123) means that the asymptotic solution for s <dimensionless equations and a different form for the asymptotic expansion for e <[Pg.387]

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. 

Unlike the regular perturbation expansion discussed earlier, the method of matched asymptotic expansions often leads to a sequence of gauge functions that contain terms like Pe2 In Pe or Pe3 In Pe that are intermediate to simple powers of Pe. Thus, unlike the regular perturbation case, for which the form of the sequence of gauge functions can be anticipated in advance, this is not generally possible when the asymptotic limit is singular In the latter case, the sequence of gauge functions must be determined as a part of the matched asymptotic-solution procedure. 

S. Kaplan, Low Reynolds number flow past a circular cylinder, J. Math. Meek 6, 595-603 (1957) S. Kaplan and R A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers, J. Math. Mech. 6, 585-93 (1957). These and other published and previously unpublished works of Kaplan are reproduced in the following book S. Kaplan, Fluid Mechanics and Singular Perturbations P. A. Lagerstrom, L. N. Howard, and C. S. Lin (eds.). (Academic, New York, 1957). 

We can rule out a simple square-root branch point for the singularity at the origin since that would lead to half-integer powers of 8 in the asymptotic expansion, Eq. (9) the 6 expansions of the wave-function and of the Hamiltonian do have half-integer powers, but they cancel exactly leaving Eq. (9) as the correct form for the energy ex-... 

As long as 8 is positive, the scaled coupling constant g is also positive and is a stable well, in the shape of the solid curve in Fig. 8. Since the behavior of the asymptotic expansion depends on the singularity structure within an infinitely small circle in the 8 plane centered at the origin, we must also consider what happens as 8 approaches zero from below. If 8 is negative, then g is also negative and Vejj looks like the dashed curve in Fig. 8. The dashed curve is not stable at the bottom of the well since the particle can tunnel out to oo. Hikami and Brezin... 

This asymptotic expansion works for all eigenvalues (mr), except the first one, that is, = 0- For this eigenvalue, the second and subsequent terms are more singular than the first one. This problem of growing in singularity is a common problem in perturbation methods and will be dealt with in the next homework problem. 

Now we can proceed with the delineation of the formal structure for a singular perturbation approach. For given x and t, we write y and y as asymptotic expansions, and the first few terms are... 

---