We have a horizontal rod of length L and examined the left side at a higher temperature than the right side.

Assume the situation to be steady state and that the only mechanism of heat transfer is conduction. Set up a balance over an arbitrary, small strip and compute what happens to the flow of heat.

And, we have the same rod, but the two temperatures on left and right are equal (T0), but heat production inside the rod: Q‴ (W/m^3).

What is the temperature distribution in steady state of this rod if we again assume heat transfer by conduction and the outside of the rod – all around the rod – to be perfectly isolated?