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1. Setting the Scene
14
Here a, b, and φ are given smooth functions. Again we define the charac-
teristic by

x (t)= a x(t),t ,

x(0) = x 0 ,
and study the evolution of u along x = x(t),
dx
d

u x(t),t = u t + u x
dt
dt
= u t + a(x, t)u x (1.30)

= b x(t),t .
Hence, the solution is given by
t

u x(t),t = φ(x 0 )+ b x(τ),τ dτ (1.31)
0
along the characteristic given by x = x(t). So the procedure for solving
(1.29) by the method of characteristics is to first find the characteristics
defined by (1.30) and then use (1.31) to compute the solutions along the
characteristics.
Example 1.3 Consider the following nonhomogeneous Cauchy problem:
u t + u x = x, x ∈ R,t > 0
(1.32)
u(x, 0) = φ(x),x ∈ R.
Here, the characteristics defined by (1.30) are given by
x(t)= x 0 + t,
and along a characteristic we have
t

u x(t),t = φ(x 0 )+ x(τ)dτ
0
1
= φ(x 0 )+ x 0 t + t ;
2
2
cf. (1.31). Since x 0 = x − t,weget

t
u(x, t)= φ(x − t)+ x − t.
2