of the solution of the advection equation

at time $\, t=T .\,$

[**Figure 1.4.**]{} Evolution of the solution at the particular point $\, x = X .\,$

[1.3]{} $ \,\,\,\,  $ [ **Inflow and outflow for the advection equation.** ]{}

$\bullet \qquad \,\,\, $ We still suppose that celerity $\,a \,$ is positive and we consider the resolution of the advection (1.2.1) in the space-time domain

(1.3.1) $\qquad \displaystyle 
0 \,< \, x \, < \, L \,, \qquad t \, > \, 0 \,. \,$

The relations (1.2.4) and (1.2.5) can still be applied because the proof of Proposition 1.2 remains unchanged in this particular case. As a consequence of the previous property, we remark that [**no boundary condition**]{} is necessary at the particular position $ \,\, x=L \,\,$ for solving the advection problem in the space-time domain defined in relations (1.3.1). The initial condition (1.2.2) has simply to be restricted in domain $\,\, ]0,\, L [ \,\,$ :

(1.3.2) $\qquad \displaystyle 
u(x, \,0) \,\,= \,\, u_0(x) \,,\qquad \quad 0 < x < L  \,,\,$

and the boundary condition (1.2.3) at $\,\, x=0 \,\, $ remains unchanged :

(1.3.3) $\qquad \displaystyle 
u(0, \,t) \,\,\, = \,\, v_0(t) \,,\qquad \quad 