Introduction

In real-world applications if is often the case that a purely analytical approach to transform calculus is insufficient. This chapter explores two techniques that extend the utility of transform methods by allowing a purely numerical treatment. We can explore numerical methods for both Fourier and Laplace transforms – in principle any method could be applied to either type of transform, since one is a rotation of the other. However, in practice, two types of problem appear to be of most importance:

(1) An essentially numerical approach to forward and backward Fourier transforms;

(2) The inversion of a complicated Laplace transform given in analytical form, for which there is no known analytical inverse.

An entire book could be written about the first topic, which is at the heart of many problems in applied mathematics, physics and engineering. It is of particular importance for signal and image processing. We shall briefly explore Mathematica's Fourier and InverseFourier functions. It should also be clear than any Fourier transform could be worked out numerically by the use of NIntegrate. In this chapter this approach will be explored in detail only for Laplace transforms.

The second topic is particularly important for generating solutions to partial differential equations. As we have seen in Chapters 18 and 19, certain partial differential equations may be simplified by transforming with respect to one or more of the independent variables, leading to a solvable algebraic or ordinary differential equation.