1.

Determine the units of the thermal conductivity k from (2.6).

2.

In the CRC Handbook of Chemistry and Physics [35], there is a table labeled "Heat Capacity of Selected Solids," which "gives the molar heat capacity at constant pressure of representative metals as a function of temperature in the range 200 to 600 K" ([35], page 12 190). For example, the entry for iron is as follows:

As this table indicates, the specific heat of a material depends on its temperature. How would the heat equation change if we did not ignore the dependence of the specific heat on temperature?

3.

Verify that the integral in (2.1) has units of energy.

4.

Suppose u represents the temperature distribution in a homogeneous bar, as discussed in this section, and assume that both ends of the bar are perfectly insulated.

1. What is the IBVP modeling this situation?

2. Show (mathematically) that the total heat energy in the bar is constant with respect to time. (Of course, this is obvious from a physical point of view. The fact that the mathematical model implies that the total heat energy is constant is one confirmation that the model is not completely divorced from reality.)

5.

Suppose we have a means of "pumping" heat energy into a bar through one of the ends. If we add r Joules per second...