4.1 Maximum-minimum Principle for the Diffusion Equation

In this section we consider the homogeneous one-dimensional diffusion (heat) equation

(4.1)

which appears in the study of heat conduction and other diffusion processes. As a model for equation (4.1), we consider a thin metal bar of length l whose sides are insulated. Denote by u ( x, t) the temperature of the bar at the point x at the time t. The constant k = ? ² is known as the themnal conductivity. The parameter k depends only on the material from which the bar is made. The units of k are (length) ²/time. Some values of k are as follows: Silver 1.71, Copper 1.14, Aluminium 0.86, Water 0.0014. In order to determine the temperature in the bar at any time t we need to know:

(1) initial temperature distribution

where ? ( x) is a given function.

(2) boundary conditions at the ends of the bar.

For instance, we assume that the temperatures at the ends are fixed

However it turns out that it suffices to consider the case T ₁ = T ₂ = 0 only. We can also assume that the ends of the bar are insulated, so that no heat can pass through them, which implies

A well posed problem for a diffusion process is

(4.2)

where u ( x, t) satisfies the initial condition