Chapter 1

Asymptotic Approaches

Asymptotic analysis is a constantly growing branch of mathematics which influences the development of various pure and applied sciences. The famous mathematicians Friedrichs [109] and Segel [217] said that an asymptotic description is not only a suitable instrument for the mathematical analysis of nature but that it also has an additional deeper intrinsic meaning, and that the asymptotic approach is more than just a mathematical technique; it plays a rather fundamental role in science. And here it appears that the many existing asymptotic methods comprise a set of approaches that in some way belong rather to art than to science. Kruskal [151] even introduced the special term “asymptotology” and defined it as the art of handling problems of mathematics in extreme or limiting cases. Here it should be noted that he called for a formalization of the accumulated experience to convert the art of asymptotology into a science of asymptotology.

Asymptotic methods for solving mechanical and physical problems have been developed by many authors. We can mentioned excellent monographs by Eckhaus [96], [97], Hinch [133], Holms [134], Kevorkian and Cole [147], Lin and Segel [162], Miller [188], Nayfeh [62], [63], Olver [197], O'Malley [198], Van Dyke [244], [246], Verhulst [248], Wasov [90] and many others [15], [20], [34], [71], [72], [110], [119], [161], [169], [173]-[175], [216], [222], [223], [250], 251]. The main feature of the present book can be formulated as follows: it deals with new trends and applications of asymptotic approaches in the fields of Nonlinear Mechanics and Mechanics of Solids. It illuminates developments in the field of asymptotic mathematics from different viewpoints, reflecting the field's multidisciplinaiy nature. The choice of topics reflects the authors' own research experience and participation in applications. The authors have paid special attention to examples and discussions of results, and have tried to avoid burying the central ideas in formalism, notations, and technical details.

1.1 Asymptotic Series and Approximations

1.1.1 Asymptotic Series

As has been mentioned by Dingle [92], theory of asymptotic series has just recently made remarkable progress. It was achieved through the seminal observation that application of asymptotic series is tightly linked with the choice of a summation procedure. A second natural question regarding the method of series summation emerges. It is widely known that only in rare cases does a simple summation of the series terms lead to satisfactory and reliable results. Even in the case of convergent series, many problems occur, which increase essentially in the case of a study of divergent series [64]. In order to clarify the problems mentioned so far, let us consider the general form of an asymptotic series widely used in physics and mechanics [65]:

1.1

where denotes an integer, and is a Gamma function (see [2], Chapter 6).

The quantity is often referred to as a singulant, and denotes a modifying factor. The sequence tends to a constant for and yields information on the slowly changed series part, whereas the constant is associated with the first singular point of the initially studied either integral or differential equation linked to the series (1.1).

In what follows we recall the classical definition: a power type series is the asymptotic series regarding the function , if for a fixed and essentially small , the following relation holds

where the symbol denotes the accuracy order of (see Section 1.2).

In other words we study the interval for , .

Although series (1.1) is divergent for , its first terms vanish exponentially fast for . This underscores an important property of asymptotic series, related to a gamebetween decaying terms and factorial increase of coefficients. An optimal accuracy is achieved if one takes a smallest term of the series, and then the corresponding error achieves , where is the constant, and is the small/perturbation parameter. Therefore, a truncation of the series up to its smallest term yields the exponentially small error with respect to the initial value problem. On the other hand, sometimes it is important to include the above-mentioned exponentially small terms from a computational point of view, since it leads to improvement of the real accuracy of an asymptotic solution [52], [53], [64], [65], [226], 230].

Let us consider the following Stieltjes function (see [65]):

1.2

Postulating the approximation

1.3

and putting series (1.3) into integral (1.2) we get

1.4

where

1.5

Computation of integrals in Equation (1.4) using integration by parts yields

If tends to infinity, then we get a divergent series. It is clear, since the under integral functions have a simple pole in the point , therefore series (1.3) is valid only for . The obtained results cannot be applied in the whole interval .

Let us estimate an order of divergence by splitting the function into two parts, i.e.

Since for , the following estimation is obtained: .

Therefore, the exponential decay of the error is observed for decreasing , which is a typical property of an asymptotic series.

Let us now estimate an optimal number of series terms. This corresponds to the situation in which the term in Equation (1.4) is a minimal one, which holds for . For we observe the divergence, and this yields the following estimation: , where […] denotes an integer part of the number. The optimally truncated series is called the super-asymptotic one [65], whereas the hyperasymptotic series [52], 53] refers to the series with the accuracy barrier overcome. It means that after the truncation procedure one needs novel ideas to increase accuracy of the obtained results. Problems regarding a summation of divergent series are discussed in Chapters 1.3–1.5.

One may, for instance, transform the series part

1.6

into the PA, i.e. into a rational function of the form

1.7

where constants , are chosen in a such a way that first terms of the MacLaurin series (1.7) coincide with the coefficients of series (1.6). It has been proved that a sequence of PA (1.7) is convergent into a Stieltjes integral, and the error related to estimation of decreases proportionally to .

The definition of an asymptotic series indicates a way of numerical validation of an asymptotic series [62]. Let us for instance assume that the solution is the asymptotic of the exact solution , i.e.

One may take as a numerical solution. In order to define , usually graphs of the dependence versus for different values of are constructed. The associated relations should be closed to linear ones, whereas the constant can be defined using the method of least squares. However, for large the asymptotic property of the solution is not clearly exhibited, whereas for small values it is difficult to get a reliable numerical solution. Let us study an example of the following integral

for large values of . Although the infinite series

is divergent for all values of , series parts

1.8

are asymptotically equivalent up to the order of with the error of for . In Figure 1.1 the dependence vs. , where , is reported (curves going down correspond to decreasing values of ).

Figure 1.1 Asymptotic properties of partial sums of (1.8)

It is clear that curve slopes are different. However, results reported in Table 1.1 of the least square method fully prove the high accuracy of the method applied.

Table 1.1 Slope coefficient as the function of defined via the least square method

Let us briefly recall the method devoted to finding asymptotic series, where the function values are known in a few points. Let a numerical solution be known for some values of the parameter : , , . If we know a priori that the solution is of an asymptotic-type, and its general properties are known (for instance it is known that the series corresponds only to integer values of ), then the following approximation holds

and the coefficients can be easily identified. The latter approach can be applied in the following briefly addressed case. In many cases it is difficult to obtain a solution regarding small values of , whereas it is easy to find it for of order 1. Furthermore, assume that we know a priori the solution asymptotic for , but it is difficult or unnecessary to define it analytically. In this case the earlier presented method can be applied directly.

1.1.2 Asymptotic Symbols and Nomenclatures

In this section we introduce basic symbols and a nomenclature of the asymptotic analysis considering the function for . In the asymptotic approach we focus on monitoring the function behavior for . Namely, we are interested in finding another arbitrary function being simpler than the original (exact) one, which follows for with increasing accuracy. In order to compare both functions, a notion of the order of a variable quantity is introduced accompanied by the corresponding relations and symbols.

We say that the function is of order for , or equivalently

if there is a number , such that in a certain neighborhood of the point we have .

Besides, we say that is the quantity of an order less than for , or equivalently

if for an arbitrary we find a certain neighborhood of the point , where .

In the first case the ratio is bounded...