The Heat Equation

In this chapter, we consider formulating the physical phenomena of heat conduction. We start with the derivation of the governing equation, the heat equation. This is followed by the introduction of the method of integral transform, a method typically used to solve PDE problems defined in infinitely large domains. We will also revisit the method of separation of variables, motivated by the need to find solutions for the initial–boundary value problems for the heat equation. Along this direction, the concept of the Sturm–Liouville differential equation system is then presented. In the end of this chapter, we discuss the maximum principle for the heat equation which underlies the diffusive feature of heat conduction. From a mathematical viewpoint, the maximum principle is presented to investigate the properties of solutions for problems governed by the heat equation.