In this chapter, we begin to consider equations that can be used to predict the behavior of quantities that depend on both time and space. That is, we wish to predict the values of some function u(t, x) that is a function of t time and x space. Here is a simple example for the sort of analysis that lies ahead: Suppose that the wind is blowing from west to east at a constant velocity of 3 meters per second. Suppose that an explosion in a chemical warehouse has pumped particulate pollution into the air. Suppose that the particles fall out of the air at a constant percentage rate r. What do we need to know to predict the particle concentration east and west of the explosion as a function of time and distance from the explosion?

What Comes In Must Go Out

Let u(t, x) be the function that gives the density (number of particles per meter) of particulate matter in the air at time t and point x along the west/east line through the warehouse. (Make the origin at the warehouse.) The following considerations lead to an equation for u: First, fix a point x and a small distance Δx.