Chapter 4: The Heat Equation

In this chapter solutions to one of the basic equations of mathematical physics will be presented. The previous material on separation of variables (Sec. 1.6) and Fourier series (Chap. 3) will be used extensively. The heat equation also arises in situations where a quantity (whether it is heat energy, a chemical, or something more abstract such as “information”) diffuses through a continuous medium, thus an understanding of its properties and solution is important to any applied mathematician or engineer. The sections of this chapter are organized according to the spatial domains and the boundary conditions imposed on the heat equation. The first section will treat the heat equation on a closed, bounded (i.e., compact) interval. After three different types of homogeneous boundary conditions on the closed, bounded interval have been studied, nonhomogeneous boundary conditions, such as boundary conditions which depend on time will be studied. One of the most important theoretical results related to the heat equation is that under relatively mild assumptions the maximum and minimum properties of the solution can be described and the uniqueness of the solution to the initial boundary value problem can be proved. Section 4.3 covering the Maximum Principle and the uniqueness of solutions can be skipped on a first reading. In Sec. 4.4 the boundedness of the spatial domain will be relaxed and solutions on infinite and semi-infinite intervals will be explored. The chapter will close with an extension of the one-dimensional results to solutions of the heat equation on a two-dimensional rectangular spatial domain and suggestions for further explorations of the heat equation.