The previous chapter introduced the technique of separation of variables for finding some special solutions to the diffusion equation. This technique works for other equations as well, and the discussion here is meant to provide another, slightly different example. In this example, we are going to model the distribution of a chemical (say a protein) in a developing embryo. (It is known that the concentration of some proteins at a given cell in an embryo determines whether or not a particular gene is expressed in that cell.)

This fairly schematic model is set up as follows: The embryo is modeled as a square of side length L (measured in appropriate units). Thus, the embryo encompasses those points in the (x, y) plane with 0 ≤ x ≤ L and 0 ≤ y ≤ L. In this model, the thickness of the embryo is assumed to be unimportant. We let u(t, x, y) denote the density of the protein in question at time t and position (x, y) in the embryo. Thus, we will restrict our attention to where both 0 ≤ x ≤ L and 0 ≤ y ≤ L. I will suppose that the protein molecules move randomly through the embryo (sometimes this not a good assumption as proteins commonly bind to other molecules; indeed, the various molecules to which a given protein binds determines its effect in a cell or embryo) and that the protein is broken down in the cells of the embryo at some rate r.