The boundary conditions.

  More information is needed to solve the problem. For clearly, the evolution of the temperature will depend on what we do at the ends of the bar. If we heat or cool an end, it will affect the temperature inside the bar. The additional information about what happens at the ends of the bar is referred to as the boundary conditions.

If we assume that the heat flows much more easily through the bar than through the surrounding medium, than at the right hand end in figure 1, the heat flux must be zero. For, heat cannot flow out of the bar into the nonconducting medium, or out of the medium into the bar. According to Fourier's law (1), this implies that  

$$\d Tx (\ell,t) = 0$$ (3)

A boundary condition on the derivative in the direction normal to a boundary such as this is called a Neumann condition.

At the other end, the bar of figure 1 is attached to a wall. If the wall is made of a well conducting material, it may be assumed that the temperature variations throughout the wall are small, especially since the cross section through which heat can flow in the bar is much less than in the wall. And if the wall is sufficiently voluminous, the heat flowing out of the bar should not raise the wall temperature significantly. So the wall will stay pretty much at its initial temperature, and with it, the left hand end of the bar. In other words, the left hand temperature is a given constant;  

$$T(0,t)= T_{\mbox{wall}}.$$ (4)

A condition such as this on the dependent variable T itself is called a Dirichlet boundary condition.

A more mathematical way of understanding why there should be two boundary conditions is from looking at the steady case. Without the time derivative, the heat equation (2) becomes a second order ordinary differential equation for any arbitrary given time. Note that a fixed time corresponds to a horizontal line in the x,t-plane figure 2. When solving the ordinary differential equation for such a fixed time, we need two additional conditions to find the integration constants. Those two additional conditions correspond to the two boundary conditions at the given time. (If you do not like the assumption of steadiness, instead use a Laplace transform in time to get rid of the time derivative.)