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1. Setting the Scene
12

t
x = x(t)

x
x 0

FIGURE 1.5. The characteristic starting at x = x 0.

or

u x(t),t = φ(x 0 ). (1.23)
This means that if, for a given a = a(x, t), we are able to solve the ODE
given by (1.22), we can compute the solution of the Cauchy problem (1.20)–
(1.21). Let us consider two simple examples illustrating the strength of this
technique.
Example 1.1 Consider the Cauchy problem

u t + au x =0, x ∈ R,t > 0,
(1.24)
u(x, 0) = φ(x),x ∈ R,
where a is a constant. For this problem, the ODE (1.22) takes the form

x (t)= a, x(0) = x 0 ,
and thus
x = x(t)= x 0 + at. (1.25)

Since, by (1.23), we have

u(x, t)= u x(t),t = φ(x 0 ),
and by (1.25) we have

x 0 = x − at,