Asymptotic solutions regular

We saw in the previous example lor R, <approximate solution, (4-4), which we obtained by taking the limit R, —> 0 in the exact equation, (4-1), was just the first term in an asymptotic solution for Rr approximate equation and solution (4-17) that we obtain by taking the limit Rm -> oo in (4-16) will also be the first term in a formal asymptotic expansion of H for Rm y>> 1. Assuming this expansion is regular, it will take the form... 

To pursue this solution, we follow the prescription of the preceding section and seek a regular asymptotic solution. Thus, we seek a solution in the form... 

In the analysis of stability that follows, we seek an approximate asymptotic solution of this equation in which the small parameter is the dimensionless magnitude of the initial departure from the equilibrium radius RE. Hence we seek a solution in the form of a regular perturbation expansion ... 

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

Let us suppose, then, that an asymptotic solution exists for Pe <regular perturbation expansion ... 

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. 

We now seek a solution of (9 7) and (9-8) for small values of the Peclet number, Pe , by using the matched asymptotic expansion procedure that was detailed for uniform flow past a sphere in Section C. Although the reader may not immediately see that the derivation of an asymptotic solution for this new problem necessitates use of the matched asymptotic expansion technique, an attempt to develop a regular expansion for 9 for Pe 1 leads to a Whitehead-type paradox similar to that encountered for the uniform-flow problem. 

The asymptotic solution (6.15) differs only slightly from the base solution, which is the solution when = 0. This is the regular perturbation solution. The asymptotic solution (6.20) deviates substantially from the base solution it is the singular perturbation solution. 

Applying the regular perturbation to this new equation to show that the asymptotic solution will have the form... 

Here u = E o e iiiix, y) is the regular part of the asymptotic solution that can be determined in exactly the same manner as in Section VI. The notation II is used for boundary layer functions playing an important role near the sides of the rectangle, and the notation P is used for corner boundary functions important in the vicinities of the vertices of the rectangle. Corresponding to the sides of the rectangle, the II part of the asymptotics can be represented as a sum of four terms ... 

In a similar way an asymptotic solution can be constructed for the case of system (7.12) containing both fast variables (function u) and slow variables (function u). However, the addition of the slow variables leads to a situation where the asymptotics can be constructed only to the zeroth- and first-order approximations. This is related to the fact that the initial and boundary conditions for the function V2(x, t) [the coefficient of in the regular series for v(x, t, e)] are not matched at the corner points (0,0) and (1,0). As a result, the function V2(x, t) is not smooth in Cl. The derivatives and d V2/dx are unbounded in the vicinities of the... 

This stage of the process refers to the regular behavior. When the solution of a diiference scheme for problem (1) also possesses the properties similar to (2) and (3), the scheme is said to be asymptotically stable. We now deal with the scheme with weights... 

Asymptotic stability. Let us describe rather mild conditions under which the behavior of the solution yJ of the difference problem (4) becomes regular if one makes tj — jr large enough ... 

Such a full investigation is virtually impossible except in some very simple cases, and is even then usually very difficult. In particular for quadratically convergent methods, the convergence region is usually bounded by a fractal instead of a regular curve. Out of necessity, the convergence properties studied are usually some necessary criterion on f and c near the desired solution, and the influence of that criterion on the asymptotic error. 

In Eq. (77), zR and Zj are the logarithmic derivatives (matrices) of the regular solution and the asymptotic (Jost) solution, respectively. 

We fit to a vanishing boundary condition at R (it is Ri R) = 0) and must fit also at the atomic sphere. Only the ), is regular at the origin, so if the potential had the same constant value within the sphere, the eorrcct solution would be simply /, that is, the phase shift <5, would be zero. The effect of a scattering potential is simply to introduce a nonzero phase shift, which indeed simply shifts the phase of the sinusoidal oscillations in the asymptotic form given in Eq. (20-21). 

Examining these solutions, we see that the temperature becomes nonuniform and the velocity profile nonlinear. These results are to be expected from a qualitative point of view. In a sense, the most important conclusion is that the regular asymptotic expansion in terms of the small parameter Br provides a method to obtain an approximate solution of the highly nonlinear boundary-value problem to evaluate the influence of weak dissipation, which can clearly be applied to other problems. 

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,... 

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e << 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

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