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A First Look at Perturbation Theory
By James G. Simmonds, James E. Mann Jr.
Dover Publications, Inc.Copyright © 1998 James G. Simmonds and James E. Mann, Jr.
All rights reserved.
Introduction and Overview
Perturbation theory is the study of the effects of small disturbances. If the effects are small, the disturbances or perturbations are said to be regular; otherwise, they are said to be singular. The basic idea in perturbation theory is to obtain an approximate solution of a mathematical problem by exploiting the presence of a small dimensionless parameter—the smaller the parameter, the more accurate the approximate solution.
Regular perturbations are assumed nearly every time we construct a mathematical model of a real world phenomenon. Our choice of language reflects this: the flow is almost steady, the density varies essentially with altitude only, the conductivity is virtually independent of temperature, the spring is nearly linear, the friction is practically negligible.
Singular perturbations are probably less familiar. Fig. 1.1 illustrates two examples. The top in Fig. 1.1a is set spinning rapidly about a vertical axis. During one revolution, the effects of aerodynamic drag and the friction at the tip are small (regular perturbations). Eventually, though, the top falls and comes to rest in a position far from its initial one. Thus, over a long period of time, the perturbations are singular. Models such as this are characterized by what may be called a singularity in the domain. For the top, this means that we are interested in what happens for all time after its release, that is, for all times t on the semi-infinite (and therefore singular) domain t > 0.
Fig. 1.1b shows a hemispherical elastic shell under an internal pressure p. The shell has been clamped at its edge, which prevents displacement or rotation there. Since the shell is symmetric, the stresses depend only on the polar angle φ. The maximum stress occurs in the outer fibers of the shell and can be found by solving nonhomogeneous ordinary differential equations. The solutions contain constants determined by regularity conditions at the pole (φ = 0) and edge conditions at the equator (φ = π/2). The differential equations contain the small parameter h/R, where h is the shell thickness and R is the radius of the shell midsurface. If h/R is set to zero in these equations, then we obtain the equation for a membrane, i.e., a shell with no bending stiffness. The solutions of these simplified equations predict a maximum stress of pR/2h everywhere. The defect of these simplified equations is that their solutions cannot meet the edge conditions. This fact signals a singularity in the model.
In Fig. 1.1b we have plotted, as a function of φ, the maximum dimensionless stress (2hs/pR) as predicted by shell and membrane theory for a typical value of h/R. Except near the equator, the results are virtually identical. However, in a narrow zone near the equator—the boundary layer—the dimensionless maximum stress predicted by shell theory dips below 1 and then rises to a value of 2. The key feature of this graph is that no matter how small the parameter h/R, the rise of the stress by a factor of 2 at the edge never diminishes. This is a singular perturbation phenomenon. The "shell versus membrane" solutions reflect what may be called a singularity in the model. Setting h/R = 0 leads to an over-simplified model that fails to predict the nonnegligible stress rise at the boundary. This failure of the membrane theory occurs in a narrow region near the boundary; the width of the failure region depends on the size of h/R.
Perturbation problems arising from a singularity in the domain were first studied systematically by Poincaré, who encountered them in orbital mechanics. The first extensive analysis of problems involving a singularity in the model (boundary layer problems) was done by Prandtl in his study of low viscosity fluids flowing over solid objects. Although the problems attacked by Poincaré and Prandtl are too elaborate for this book, we can explore many aspects of perturbation theory by working with simple equations, many of which can be applied to common phenomena.
The First Quantitative Step. We begin with the following problem from the theory of quadratic equations:
Determine how the roots of z2 - 2z + [member of] change as [member of] is perturbed slightly away from zero.
If [member of] = 0, the roots are, by inspection, z1 = 0 and z2 = 2. For other values of [member of] we have, as a result of the quadratic formula,
z1([member of]) = 1 - [square root of (1 - [member of])] (1.1)
z2([member of]) = 1 - [square root of (1 - [member of])]. (1.2)
The Numerics of a Regular Problem . Using a hand calculator, we readily construct from (1.1) and (1.2) the following table. Our numerical calculations suggest that a perturbation about [member of] = 0 is regular but teach us little else. Moreover, we took no advantage of the fact that the roots for [member of] = 0 came with little effort. We shall remedy this lack of analysis presently.
The Numerics of a Singular Problem. If we switch [member of] with the coefficient of z2, we have the polynomial [member of] z2 - 2z + 1. If [member of] = 0, z1 = 1/2 is its only root. If [member of] ≠ 0, there are two:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)
Again, using a hand calculator, we construct the following table. What is notable about Table 1.2? First, while one of the roots, z1([member of]), approaches that of 2z - 1 as [member of] [right arrow] 0, the other, z2([member of]) goes to infinity. This is a manifestation of singular behavior. On a more fundamental level, changing [member of] from 0 to an arbitrarily small number has changed the number of solutions for z (from one to two). Notice that the equation changes from linear ([member of] = 0) to quadratic ([member of] [≠ 0). Such a change in the order of an equation characterizes many singular perturbation problems.
A second feature of Table 1.2 is that as [member of] [right arrow] 0, z1 ([member of]) at first approaches 0.5 steadily but then, for very small values of [member of] , begins to exhibit small fluctuations. These are caused by round-off errors produced by computingdifferences of nearly equal terms in (1.3). While there is a simple way to remedy this in the present problem-multiply numerator and denominator in (1.3) by 1 + [square root of (1 - [member of])]—the diagnosis of round-off error in more complicated problems, much less the cure, is not so simple. In cases such as these in which numerics falter, perturbation theory can sometimes save the day.
Analysis of the Regular Problem . To find approximate solutions that are accurate and easy to use, we study the effect of a small parameter. First, let us find approximate formulas for the roots of z2 - 2z + [member of] when [member of] is small. Unlike most perturbation problems, this one can be analyzed completely. We analyze precisely the simplest problem of a class with the hope of inferring patterns or principles which can aid in the attack on more complicated problems.
Infinite Series . The formulas (1.1) and (1.2) for the roots of z2 - 2z + [member of] are exact. However, the smallness of [member of] simplifies the problem. We can certainly assume that |[member of]| < 1. This not only rules out complex roots, but, more importantly, it allows us to expand [square root of (1 - [member of])] in a power series in [member of]. Recall the binomial expansion:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
Setting a = 1, b = -[member of], and m = 1/2, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.6)
This is a formal series, so-called because we as yet have made no attempt to ask what it means to try to add together an infinite number of terms.
The way to study an infinite series is to study its sequence of partial sums. If the sequence converges, then we say that the series converges, and if we are able to compute the limit, S, of this sequence, we say that the series sums to S. One of the simplest tests for convergence is the ratio test, which states that [summation] uk converges if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
From (1.5) and (1.6) we find that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)
Thus the right side of (1.6) converges if |[member of]| < 1, as do the two series
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
that we obtain upon substituting (1.6) into (1.1) and (1.2).
The first few terms of these series must, to be useful, yield close approximations to the roots. Taking only those terms displayed explicitly in (1.8) and [member of] = 0.1, we find z1(.1) ≈ 0.5(.1) + 0.125(.1)2 = 0.05125 and z2(.1) ≈ 2 - 0.5(.l) = 1.95. The exact values are z1(.l) = 0.05131 ... and z2(.1) = 1.94.... Are these roots accurate enough? This decision is made from external information in the context of your own problem.
In a more elaborate problem, there may not be a formula for the exact solution(s) with which one can check the accuracy of an approximation, or, if there is an exact formula, it may be difficult to apply. As an example of the latter, consider finding the roots of
z3 - 5z2 + 4z + [member of] = 0 (1.9)
when [member of] is small. Though there is an exact formula for the roots of this (and any other) cubic, it is difficult to use. However, if [member of] = 0, we have
z3 - 5z2 + 4z = 0 (1.10)
which has the roots
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)
Presently, we shall develop a technique which exploits the smallness of [member of] to produce an acceptable approximation of the roots of any polynomial P]member of](z) when the roots of the "nearby" or reduced polynomial P0(z) are known. But first we need a way of making error estimates that depends on the approximation process itself.
Taylor's expansion with a remainder uses the value of a function and its first few derivatives at a point at which they are easily found to estimate the value of the function at a nearby point where the function is difficult to calculate. More precisely, if a function f ([member of]) and its first n derivatives are continuous on a closed interval |[member of]| ≤ a, and if the (n+l)st derivative of f ([member of]) exists on the open interval |[member of]| < a, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)
Here f(n)(0) denotes the nth derivative of f ([member of]) evaluated at [member of] = 0 and the remainder after n+1 terms, Rn+1([member of]), has the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.13)
The number x depends on [member of] but is otherwise unknown.
The important facts about the expansion (1.12) are:
1. it allows us to approximate f ([member of]) by a (Taylor) polynomial in [member of], namely the right side of (1.12) without Rn+1([member of]).
2. if we can find an upper bound on Rn+1([member of]) then we have an upper bound on the error of our polynomial approximation.
3. it is almost incidental in applications whether the associated infinite series f (0) + f' (0)[member of] + l/2f" (0)[member of]2 + ... converges. The size of Rn+1([member of]) is the salient fact.
With f ([member of]) = (1 - [member of])1/2, we have, if n = 1,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)
and, if n = 2,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.15)
Substituting (1.15) into (1.1) and (1.14) into (1.2), we find that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.16)
These expressions offer a way of estimating the errors we make when truncating the infinite series expansions for z1 ([member of]) and z2([member of]) after two terms.
If p > 0, and |x| < 1, then (1 - x)-p is largest when x is as close to 1 as possible. Thus, if |x| < 1 - d where d is any fixed number such that 0 < d < 1,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.17)
Exercise 1.1 . Earlier, we used the terms displayed explicitly in (1.8) to approximate z1(.1) and z2(.1). Use (1.16) and (1.17) to obtain an upper bound on the errors we made.
The Order Symbols . Using (1.17), we may rewrite (1.16) in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.19)
The symbols 0([member of]3) and 0([member of]2), to be read "big 'O' of [member of] cubed" and "big 'O' of [member of] squared" are used to sweep all irrelevant algebraic details under the rug.
In general, g ([member of]) = O([member of]p) means that, for [member of] sufficiently small, there exists a positive constant K independent of [member of], such that |g([member of])| < K|[member of]|p. In (1.18) and (1.19), "sufficiently small" means |[member of]| < 1-δ, and the K's from (1.17) are (1/16)δ-5/2 and (1/8)δ-3/2 respectively. In more complicated problems, however, we can rarely pin down the words "sufficiently small" and "there exists." Thus, in practice, we may have to view a statement such as f ([member of]) = O([member of]2) as simply implying that f grows no faster than the square of [member of] when [member of] is small.
Analysis of the Singular Problem. With what we have learned we can now quickly analyze the singular problem of finding simple, approximate formulas for the roots of [member of]z2 -2z + 1 when [member of] is small. Substituting the Taylor formula for [square root of (1 - [member of])] into (1.3) and (1.4), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.21)
The terms displayed give an approximation to z1(10-6) of 0.5 + .125(10-6) = 0.5000000125, and an approximation to z2(10-6) of 2(106) - 0.5 = 1999999.5. The approximation to z1(10-6) has been improved dramatically from the value found earlier with a calculator, but z2([member of]) still approaches infinity as [member of] approaches zero. This behavior is inherent in the problem and is not a numerical artifact. Our analysis of z2 ([member of]) has, nevertheless, provided useful information: we now know that z2([member of]) behaves like 2/[member of] as [member of] [right arrow] 0. For larger values of [member of] we might need more terms in the Taylor polynomial for z1([member of]) and z2([member of]) to obtain sufficiently accurate approximations.
Exercise 1.2 . Make upper bound estimates of the remainders and determine the smallest Taylor polynomials that can produce approximations to the roots of [member of]z2 - 2z + 1 with an absolute error < 10-3 for all |[member of]| < 0.2.
Note that when we finally obtained useful numerical formulas for the roots of z2 - 2z + [member of]—(1.18) and (1.19)—each was of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.22)
The right side of — is called an asymptotic or regular perturbation expansion. It is ideal for assessing, numerically or theoretically, the effect of a small perturbation in [member of] about zero. Though an asymptotic expansion need not converge as N [right arrow] ∞, we do require that [member of]-NRN [right arrow] 0 as [member of] [right arrow] 0 for fixed N. Any function with a representation of the form (1.22) is called regular because z ([member of]) approaches a finite value as [member of] approaches 0.
Excerpted from A First Look at Perturbation Theory by James G. Simmonds, James E. Mann Jr.. Copyright © 1998 James G. Simmonds and James E. Mann, Jr.. Excerpted by permission of Dover Publications, Inc..
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