# 3Diffusion and Heat Conduction

## 3.1 Derivation of Diffusion Equation

The diffusion equation is one of the most important PDE applications, so let's see how it is derived. We let be the density (mass per unit volume) of a diffusing substance X, and let be any small subregion of the region where diffusion is occurring. Then the total mass of X within the subregion is , and

This equation says that the rate of change of the total mass of X in is equal to the net rate at which X is entering from outside plus the net rate at which X is being created internally due to sources and sinks. The flux vector represents the net flow of X, in mass per unit area per unit time due to diffusion or convection and can be thought of as the density times the average velocity of the particles. Thus the dot ...

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