The heat equation.

  In this section we will consider unsteady heat conduction in a uniform bar, as shown in figure 1. We will assume that the temperature T depends only on the position x along the bar and the time t. In other words, we assume that the temperature variations over the cross sections are negligible. We also assume that the heat conduction satisfies Fourier's law, so that the heat flux in the x-direction is given by the  

$$q = k \d Tx$$ (1)

where we assume that k is a constant.

  

Figure 1: Heat conduction in a bar.