ABSTRACT
This chapter introduces the theory of Fourier transforms and utilizes it to find solutions of partial differential equation on infinite domains. It proves some properties of Fourier transforms under various assumptions, so that the reader will have some basis for believing that the formal manipulations are likely to lead to a correct solution. The chapter suggests that the reader or instructor skip the more difficult proofs during the first reading, and concentrate on the examples. It applies Fourier transform methods to the heat problem on the infinite rod, thereby formally obtaining the formula, which is then rigorously shown to solve. The chapter shows how Fourier transform methods can be applied to problems for the wave equation and Laplace's equation. The choice of which transform to use depends on the nature of the problem at hand. Laplace transforms are ideally suited for initial-value problems for linear systems of ordinary differential equation.
