The Fourier Transform, Heat Conduction, and the Wave Equation

Abstract

The numerous examples of Chapters 2 and 3 demonstrate how the classical method of separation of variables is used to generate solutions of Laplace’s and the heat equation over specified finite domains. Utilizing the modern methods inherent in Green’s functions and Green’s Theorem, solutions of Poisson’s equation over an arbitrary finite domain are established. Can such modern methods be applied to the heat equation? Unlike Laplace’s or Poisson’s equations which are static in time, the heat equation evolves in time. Indeed, the heat equation is the prototype for what are commonly called evolution equations. Nonlinear evolution equations are examined in Chapter 6 and a special nonlinear system is detailed in Chapter 7. For now, the focus will be on the linear heat and wave equations. Instead of Green’s Theorem, one of the most powerful ideas in modern mathematics is applied: The Fourier transform. The program of study for this chapter then is to define the Fourier transform, develop its properties, apply it to the heat and wave equations, and derive analytic solutions.