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By comparing Equations 2.20 and 2.21 we obtain ordinary differential equations for the
characteristics x(t) and the solution along the characteristics u(x(t), t).

dx du
= a(x, t, u), = 0
dt dt
Suppose an initial condition is specified, u(x, 0) = f(x). Then we have ordinary differen-
tial equation, initial value problems.

dx
= a(x, t, u), x(0) = x 0
dt
du
= 0, u(0) = f(x 0 )
dt
We see that the solution is constant along the characteristics. The solution of Equa-
tion 2.20 is a wave moving with velocity a(x, t, u).

Example 2.3 Consider the inviscid Burger equation,

u t + uu x = 0, u(x, 0) = f(x).

We write down the differential equations for the solution along a characteristic.

dx
= u, x(0) = x 0
dt
du
= 0, u(0) = f(x 0 )
dt

First we solve the equation for u. u = f(x 0 ). Then we solve for x. x = x 0 + f(x 0 )t.
This gives us an implicit solution of the Burger equation.

u(x 0 + f(x 0 )t, t) = f(x 0 )

The method of characteristics is used to solve first order quasi-linear partial differential
equations. Consider the partial differential equation

au x + bu y = u, u(0, y) = f(y).

The derivative of u with respect to some variable, s, is

du ∂x ∂y
= u x + u y .
ds ∂s ∂s
Comparing the above two equations we see that the partial differential equation is equiv-
alent to the system of ODE’s

∂x ∂y du
= a = b = u
∂s ∂s ds

The initial condition corresponds to

x(s = 0) = 0 y(s = 0) = α u(s = 0) = f(α)

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