Solution to the heat equation from a source

[Adel Makram]

What is the best solution of the heat equation that described a transmission of heat from a source kept at certain temperature to a reservoir with an initial constant temperature (lower than the source) where its ends are not insulated from the surroundings and the surrounding is kept at a fixed temperature. For example, heating one end of a metallic rod while the other end is not insulated but exposed to the room of a constant temperature. What if the room temperature is allowed to rise like in case of a heater?

 

[mfb]

Write down a differential equation for the heat flow in your setup, solve it.
Heat transfer to air can be complicated due to convection.