Brian Blais ’ Homemade Guide to the Diffusion Equation

What we are looking for are functions ρ(x, t) such that, when they are placed in Equation 1.1 the equation holds true. There may be many such functions (such as ρ(x, t) = 0), but we are only interested in the physically meaningful or interesting (non trivial) solutions. A rule of thumb for solving physics problems is to know the answer before you start, otherwise you don’t know if you are doing something incorrectly. Well, we won’t be able to know the complete solution without doing the math, but we can try to figure out what kinds of solutions we expect because we happen to know the physics behind it. The diffusion equation is used to describe the spreading out of some quantity of stuff. That stuff could be temperature spreading through a medium or a drop of food coloring spreading in a glass of milk, for example. The ρ in Equation 1.1 is the density of the quantity that we are talking about, so we would expect that the curve of ρ(x, t) would flatten out as time goes on, representing the thinning out of the substance over space. We expect, then, that any solution must decay in time. Notice also that the ∇2 is like the curvature of the function ρ(x, t) (in one dimension it would be ∂2/∂x2) which suggests that the diffusion equation is flattening out the density function by making functions of high curvature change faster than ones of lower curvature. Now that we understand the physics behind the equation, and have a feel for what it is trying to tell us, we are ready to find the solutions of it. We will focus primarily on the one dimensional case, for clarity, and then move onto higher dimensional forms later.