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Chapter 4

Flows, Vibrations and Diffusions

The subject of PDEs was practically a branch of physics until the twentieth century. In
this chapter we present a series of examples of PDEs as they occur in physics. They
provide the basic motivation for all the PDE problems we study in the rest of the book.
We shall see that most often in physical problems the independent variables are those of
space x, y, z and time t.

4.1 Simple Transport

Consider a fluid, water, say, flowing at a constant rate c along a horizontal pipe of fixed
cross section in the positive x direction. A substance, say a pollutant, is suspended in the
water. Let u(x, t) be its concentration in grams/centimeter at time t. Then

u t + cu x = 0. (4.1)

(That is, the rate of change u t of concentration is proportional to the gradient u x . Dif-
fusion is assumed to be negligible.) Solving this equation as above, we find that the
concentration is a function of (x−ct) only. This means that the substance is transported
to the right at a fixed speed c. Each individual particle moves to the right at speed c;
that is, in the xt-plane, it moves precisely along a characteristic line (see Figure 4.1).

y
t = 1 t = 2 t = 3

x

Figure 4.1: Motion of particle

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