Then the solution of the advection equation (1.2.1) with the initial condition (1.2.2) and the boundary condition (1.2.3) is given by the relations

(1.2.4) $\qquad \displaystyle 
u(x, \,t) \,\,\, = \,\, u_0(x - at) \,\,, \qquad \, x-at \,> \, 0 \,$

(1.2.5) $\qquad \displaystyle 
u(x, \,t) \,\,\, = \,\, v_0 \Bigl( t \,- {{x}\over{a}}\Bigr) \,\,, \qquad x-at \, < \,
0 \,. \,$

The initial condition $\,\, u_0({\scriptstyle \bullet}) \,\,$ is advected towards space-time point $\, (x,\,t) \,$ when $\, x-at > 0 \,$ and the boundary condition $\,\, v_0({\scriptstyle \bullet}) \,\,$ is activated for $\, x-at < 0 \,. \,$

$\bullet \qquad \,\,\, $ [**Proof of Proposition 1.2.**]{}

In order to solve the problem (1.2.1)-(1.2.3), we use the method of characteristics. We fix a point $\,\, (x,\,t) \,$ of space-time domain that satisfies $\,\, x>0 ,\,
t>0 \,\,$ and we go upstream in time with the help of the characteristic line that goes through this point (see Figure 1.2) :

(1.2.6) $\qquad \displaystyle 
 x(\lambda) \,= \, x - a \lambda \,,\,\qquad  t(\lambda) \,=\, t - \lambda \,.\,$

$\bullet \qquad \,\,\, $ First case : $\,\, x-at \,> \,0 \,. \, $ When we take the particular value $\, 