### Example 1.9.1.

Consider the equation

\begin{equation*}
u_t + \alpha u_x = 0, \qquad u(x,0) = f(x) .
\end{equation*}

This particular equation, \(u_t + \alpha u_x = 0\text{,}\) is called the

*transport equation*.The data will propagate along curves called characteristics. The idea is to change to the so-called

*characteristic coordinates*. If we change to these coordinates, the equation simplifies. The change of variables for this equation is
\begin{equation*}
\xi = x - \alpha t , \qquad s = t .
\end{equation*}

Let’s see what the equation becomes. Remember the chain rule in several variables.

\begin{equation*}
\begin{aligned}
& u_t = u_\xi \xi_t + u_s s_t = - \alpha u_\xi + u_s , \\
& u_x = u_\xi \xi_x + u_s s_x = u_\xi .
\end{aligned}
\end{equation*}

The equation in the coordinates \(\xi\) and \(s\) becomes

\begin{equation*}
\underbrace{(- \alpha u_\xi + u_s)}_{u_t} + \alpha
\underbrace{(u_\xi)}_{u_x} = 0 ,
\end{equation*}

or in other words

\begin{equation*}
u_s = 0 .
\end{equation*}

That is trivial to solve. Treating \(\xi\) as simply a parameter, we have obtained the ODE \(\frac{d u}{d s} = 0\text{.}\)

The solution is a function that does not depend on \(s\) (but it does depend on \(\xi\)). That is, there is some function \(A\) such that

\begin{equation*}
u = A(\xi) = A(x - \alpha t) .
\end{equation*}

The initial condition says that:

\begin{equation*}
f(x) = u(x,0) = A(x - \alpha 0) = A(x) ,
\end{equation*}

so \(A=f\text{.}\) In other words,

\begin{equation*}
u(x,t) = f(x-\alpha t) .
\end{equation*}

Everything is simply moving right at speed \(\alpha\) as \(t\) increases. The curve given by the equation

\begin{equation*}
\xi = \text{constant}
\end{equation*}

is called the characteristic. See Figure 1.20. In this case, the solution does not change along the characteristic.

In the \((x,t)\) coordinates, the characteristic curves satisfy \(t = \frac{1}{\alpha} ( x- \xi)\text{,}\) and are in fact lines. The slope of characteristic lines is \(\frac{1}{\alpha}\text{,}\) and for each different \(\xi\) we get a different characteristic line.

We see why \(u_t + \alpha u_x = 0\) is called the transport equation: everything travels at some constant speed. Sometimes this is called

*convection*. An example application is material being moved by a river where the material does not diffuse and is simply carried along. In this setup, \(x\) is the position along the river, \(t\) is the time, and \(u(x,t)\) the concentration the material at position \(x\) and time \(t\text{.}\) See Figure 1.21 for an example.