In Chapter 1, we saw that there are many first-order PDEs which we can already solve. In particular, the linear equation

a(x, y)ux + b(x, y)u = f(x, y) (5.1)

can be treated as an ODE. What about more general linear, first-order PDEs? And why would we be interested in these equations? The PDE aux + buy = 0 often is called the convection equation (as

in “convey”) or advection equation (where advection is a synonym of the noun transport). Imagine a very narrow stream flowing at constant velocity v. Suppose there is a chemical that has polluted the stream and that this chemical is carried downstream without diffusing at all. Let

u(x, t) = concentration of chemical per unit length, at point x

along the stream, at time t

(we are assuming that the stream is narrow enough so that the concentration is the same along any line across its width, so we may treat the stream as one-dimensional). See Figure 5.1. What can we say about the function u? Consider a short length of the stream, from x to x + Δx, and calculate the change in the amount of chemical present from time t to time t+ Δt. First, the amount present at time t is approximately

u(x, t)Δx,

so the change is approximately

[u(x, t+Δt)− u(x, t)]Δx.