ABSTRACT
Heat may be transferred by conduction, convection, and radiation. In conduction, the heat is transferred locally by impacts of molecules with adjacent molecules. The function can also have the interpretation of being the concentration of a chemical or dye in a liquid without currents, and hence the heat equation is often called the diffusion equation. This chapter discusses variations of solutions with respect to variations in initial and boundary data. It motivates and utilizes Duhamel's method to handle the inhomogeneous heat equation and time-dependent boundary conditions. The law that heat flows from hot to cold regions is statistical in the sense that it can be violated, but only improbably; in the mathematically ideal limit, the probability of violation approaches zero. The concept of "temperature at a point" is a mathematical idealization that might be achieved by taking a limit as the regions become smaller and as the size of the molecules decreases, while the number of molecules increases.
