The Heat Equation

Our aim is to construct a mathematical model that describes temperature distribution in a body via heat conduction. There are other forms of heat transfer such as convection and radiation that will not be considered here. Basically, heat conduction in a body is the exchange of heat from regions of higher temperatures into regions with lower temperatures. This exchange is done by a transfer of kinetic energy through molecular or atomic vibrations. The transfer does not occur at the same rate for all materials. The rate of transfer is high for some materials and low for others. This thermal diffusivity depends mainly on the atomic structure of the material. To interpret this mathematically, we first need to recall the notion of flux (that you might have seen in multivariable calculus). Suppose that a certain physical quantity Q flows in a certain region of the 3-dimensional space R. For example Q could represent a mass (think of flowing water), or could represent energy (think of heat), or an electric charge. The flux corresponding to Q is a vector-valued function q⃗ whose direction indicates the direction of flow of Q and whose magnitude |q⃗| the rate of change of Q per unit of area per unit of time. If for example Q measures gallons of water, then the units for the flux could be gallons per meter per minute. One way to understand the relationship between Q and q⃗ is a as follows. Let m0 = (x0, y0, z0) be a point in R with the standard canonical basis of orthonormal vectors i⃗, j⃗, k⃗. Consider a small rectangular surface S1 centered at m0 and parallel to the yz-plane. So the unit vector i⃗ is normal to S1. Assume that S1 has side lengths ∆z and ∆y (see Figure 1.)