2.1 Introduction

Most systems in the natural and engineering sciences, if their motions are bounded, oscillate. Often the oscillations are periodic or eventually become periodic. Thus there is a need to understand how oscillatory functions can be represented. The general area of mathematics concerned with this and related issues is Fourier analysis. The main purpose of this chapter is to introduce the elements of Fourier series and also include some interesting results on the various functional relations from trigonometry, and the Fourier and Laplace transforms. We generally only state the relevant theorems and provide no proofs of them. Such details are provided by the books listed in the bibliography given at the end of the chapter.

An illustration as to how the techniques of this chapter can be applied is given by the example of the following differential equation which models a very nonlinear oscillator,

Defining, y = dx/ dt, and using

we obtain

Integrating once gives,

In the ( x, y) phase-plane, y( x) is a closed curve and, as will be seen in later chapters of this book, such a closed curve corresponds to x( t) and y( t) being periodic functions of t. Several questions and issues naturally arise:

1. Can an explicit solution to Eq. (2.1.1) be found?

2. If no simple explicit solution exists, can a suitable analytical approximation be constructed?

3. Related to (ii) is the question: How, in fact, can such an analytical approximation...