Introduction

We examined in Chapter 1 Fourier's work on heat conduction. In addition to developing a general theory for heat flow, Fourier discovered a method for solving the initial boundary value problem he derived. His solution led him to propose the bold idea that any real valued function defined on a closed interval can be represented as a series of trigonometric functions. This is known today as the Fourier expansion. D'Alembert and the Swiss mathematician Daniel Bernoulli (1700–1782) had actually proposed a similar idea before Fourier. They claimed that the vibrations of a finite string can be formally represented as an infinite series involving sinusoidal functions. They failed, however, to see the generality of their observation.

Fourier's method for solving the heat equation provides a convenient method that can be applied to many other important linear problems. The method also enables us to deduce several properties of the solutions, such as asymptotic behavior, smoothness, and well-posedness. Historically, Fourier's idea was a breakthrough which paved the way for new developments in science and technology. For example, Fourier analysis found many applications in pure mathematics (number theory, approximation theory, etc.). Several fundamental theories in physics (quantum mechanics in particular) are heavily based on Fourier's idea, and the entire theory of signal processing is based on Fourier's method and its generalizations.

Nevertheless, Fourier's method cannot always be applied for solving linear differential problems. The method is applicable only for problems with an appropriate symmetry.