Heat always conducts from warmer objects to cooler objects. The composition of a material effects its conduction rate. If a copper rod and an iron rod are joined together end to end, and the ends placed in heat sources, the heat will conduct through the copper end more quickly than the iron end because copper has a K value of 92, whereas, iron has a K value of 11.
Fourier (1768-1830) considered the above properties and summarized them in Fourier's law of heat conduction.
Which states that the heat flux (
) is proportional to the
temperature differences per unit length.
The proportionality constant is the Thermal conductivity K.
Substituting Fourier's law into the conservation of heat law, the following is obtained by
Which is the integral form of the heat equation. This can be written as by
In order to complete the model some boundary conditions must be specified
at each end of the rod, and an initial state of the temperature
distribution is also required.
The side conditions can take several forms.
The value of the temperature or the value of the flux can be specified.
Consider the case in which one side of the rod is embedded in ice while
the other is at an open flame at 
.
The rod is initially at ambient temperature 
.
These are expressed as:
