PART C
Fourier Analysis. Partial Differential Equations (PDEs)
Chapter 11 and Chapter 12 are directly related to each other in that Fourier analysis has its most important applications in modeling and solving partial differential equations (PDEs) related to boundary and initial value problems of mechanics, heat flow, electrostatics, and other fields. However, the study of PDEs is a study in its own right. Indeed, PDEs are the subject of much ongoing research.
Fourier analysis allows us to model periodic phenomena which appear frequently in engineering and elsewhere—think of rotating parts of machines, alternating electric currents or the motion of planets. Related period functions may be complicated. Now, the ingeneous idea of Fourier analysis is to represent complicated functions in terms of simple periodic functions, namely cosines and sines. The representations will be infinite series called Fourier series.1 This idea can be generalized to more general series (see Sec. 11.5) and to integral representations (see Sec. 11.7).
The discovery of Fourier series had a huge impetus on applied mathematics as well as on mathematics as a whole. Indeed, its influence on the concept of a function, on integration theory, on convergence theory, and other theories of mathematics has been substantial (see [GenRef7] in App. 1).
Chapter 12 deals with the most important partial differential equations (PDEs) of physics and engineering, such as the wave equation, the heat equation, and the Laplace equation. These equations can model a vibrating string/membrane, temperatures on a bar, and electrostatic potentials, respectively. PDEs are very important in many areas of physics and engineering and have many more applications than ODEs.
1JEAN-BAPTISTE JOSEPH FOURIER (1768–1830), French physicist and mathematician, lived and taught in Paris, accompanied Napoléon in the Egyptian War, and was later made prefect of Grenoble. The beginnings on Fourier series can be found in works by Euler and by Daniel Bernoulli, but it was Fourier who employed them in a systematic and general manner in his main work, Théorie analytique de la chaleur (Analytic Theory of Heat, Paris, 1822), in which he developed the theory of heat conduction (heat equation; see Sec. 12.5), making these series a most important tool in applied mathematics.