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CHAPTER 10

Introduction to Perturbation Methods

Perturbation theory is an approximate method of solving equations which contain a parameter
that is small in some sense. The method should result in an approximate solution that may
be termed “precise” in the sense that the error (the difference between the approximate and
exact solutions) is understood and controllable and can be made smaller by some rational
technique. Perturbation methods are particularly useful in obtaining solutions to equations that
are nonlinear or have variable coefficients. In addition, it is important to note that if the method
yields a simple, accurate approximate solution of any problem it may be more useful than an
exact solution that is more complicated.

10.1 EXAMPLES FROM ALGEBRA
We begin with examples from algebra in order to introduce the ideas of regular perturbations
and singular perturbations. We start with a problem of extracting the roots of a quadratic
equation that contains a small parameter ε
1.

10.1.1 Regular Perturbation
Consider, for example, the equation

2
x + εx − 1 = 0 (10.1)
The exact solution for the roots is, of course, simply obtained from the quadratic formula:

ε ε 2
x =− ± 1 + (10.2)
2 4
which yields exact solutions

x = 0.962422837

and

x =−1.062422837