Asymptotic Analysis

Exact analytical solutions cannot be found for most differential and integral equations which arise in practical situations. By an exact solution we mean one that is given in terms of functions whose properties are known or tabulated: Bessel functions, trigonometric functions, Legendre functions, exponentials, and so on are typical examples. Such a solution may not be particularly useful, however, from either a computational or analytical point of view. For example, a solution which involves a slowly convergent infinite series of Bessel functions is of little use computationally, or even analytically, if we are interested in the dependence of the solution on some parameter of the problem, as is frequently the case. Even within the class of linear equations with linear boundary conditions, to which transform techniques are applicable, the integral representing the inverse of the transform solution may not be integrable in terms of suitable, in the sense of useful, functions. In general asymptotic analysis is that branch of analysis which is concerned with both developing techniques and obtaining approximate analytical solutions to such problems when a parameter or some variable in the equation or integral becomes either large or small or is in the vicinity of a parameter value or point where the solution is not analytic. The ideas developed below are equally applicable under appropriate conditions to differential and integral equations, difference equations, integral evaluation in general, and the evaluation of functions which are represented by series which may be, strictly speaking, divergent, but nevertheless can be used for calculating the function to a high degree of accuracy.

We have already seen in §1.1 that integration by parts is one way of finding asymptotic approximations to integrals. It is one of the simplest procedures but it is rather limited in its applicability. The procedure is essentially to integrate by parts and then show that the resulting series is asymptotic by estimating the remainder which is in the form of an integral: this is exactly what was done in §1.1 to obtain (1.11) for Ei(x) as x → ∞. We include in this procedure the technique where the integrand is expanded as a series and the asymptotic series is obtained by integrating term by term: the asymptotic nature of the resulting series again depends on the estimation of an integral remainder (see, for example, §1.1 exercise 9). A survey of these methods, presented by way of specific examples, is given in the book by Copson (1965).

The method of steepest descents or saddle-point method is essentially a generalization of Laplace’s method to integrals in the complex plane. The method originated with Riemann†. It was developed in its present form independently by Debye ‡. Various extensions and rigorous proofs of some of the procedures have been given since their work and some of these are in the books cited. The basic idea and method which still has the widest applicability, are discussed in this section, with several illustrative worked examples being given in §3.2.

The integrals for which this method was originally developed by Stokes† and Kelvin‡ are of the general form where a, b, g(t), h(t), t, and λ are all real. Asymptotic expansions are sought as λ → ∞. It should be said here that if g(t) and h(t) can be suitably analytically continued off the real axis then the class of integrals (4.1) can be treated, as discussed briefly below, by the method of steepest descents in §3.1. However, the original method of stationary phase antecedes the steepest descents one. There are three main reasons for considering it separately. First, the physical idea and motivation behind the original exposition of the method are interesting and instructive. Secondly, we would like to be able to deal with such integrals by considering the integration along the real axis only. Thirdly, the fact that integrals like (4.1) play such a fundamental role in the study of general wave motion, as will be seen in §4.2 below, is in itself a valid reason for obtaining the asymptotic expansion. In §4.2 a brief introduction is given to dispersive wave motion which is of current interest and practical importance: the most exciting developments in the subject have appeared since about 1960.

Linear boundary value problems are often solved by integral transform methods†. The final step in such a procedure is the transform inversion which involves, in general, the evaluation of integrals of the form where t is real, K(z, t) is some given kernel, a function of two variables z and t, C is a given finite or infinite contour in the complex z-plane, and F(z) is known, or at least its singularity properties are given. Here the function F(z) is the integral transform of f(t) and (5.1) defines its inverse.

The methods in the previous sections provide asymptotic approximations for functions defined by integrals. Many of these functions are solutions of specific differential equations: the Bessel function is an example. In the case of functions defined as the solutions of differential equations which cannot be solved explicitly, these integral methods described above naturally cannot be used. (We group in the class of explicit solutions those given in the form of an integral.) In these circumstances we must resort to differential equation methods to find asymptotic approximations for the functions. The subject of asymptotic methods for solving differential equations is large. In this and the following sections we give a brief introduction to the subject. In the case of ordinary differential equations the book by Wasow (1965) discusses several aspects with detail and rigour. A large part of each of the books by Erdelyi (1956) and Jeffreys (1966) is devoted to ordinary differential equations. In the partial differential equation area the books by Van Dyke (1964) and Cole (1968)† are of importance in the particular area of asymptotic analysis called singular perturbation theory.