Spectral Methods

Abstract

In this chapter we introduce spectral methods. The principle is familiar to anyone who has used Fourier¹ series to solve a differential equation: we represent the unknown but presumed-to-exist solution as the summation of a (possibly infinite) number of constants times base functions, and convert the problem of solving a differential equation into solving a set of equations for the coefficients.

Jean Baptiste Joseph, Baron de Fourier, 1768 – 1830, developed his series in the attempt to explain heat as a fluid permeating space. He narrowly escaped execution for political offenses during the post-Revolutionary Reign of Terror, and joined Napoleon Bonapartes army in Egypt in 1798. There he administrated the political and scientific organisations Napoleon set up while trapped in Egypt by the British Navy. [288]