using Laplace transform for advection equation I use Laplace transform to solve an advection-diffusion equation with given boundary and initial conditions. I am stuck on the special case that only advection is considered. The advection equation is,
$$ 
\frac {\partial{T}}{\partial{t}}+u\frac{\partial{T}}{\partial{x}}=0
$$
with initial condition $T(x,t=0)$ and boundary condition $T=T_0$ at $x=0$.
Using Laplace transform,
$$
s\overline{T}-T(x,t=0)+u\frac{\partial{\overline{T}}}{\partial{x}}=0
$$
The analytical solution in Laplace domain is,
$$
\overline{T}=T_0(e^{-sx/u}+T(x,t=0)/s)
$$
And the solution in real domain is thus,
$$
T=T_0(\delta(t-x/u)+T(x,t=0))
$$ 
However, the solution seems incorrect. I think the correct solution should be a stepwise function. For $x<ut$,$T=T_0$, and for $x>ut$,$T=T(x,t=0)$.Any comments are appreciated.

$$
s\overline{T}-T(x,t=0)+u\frac{\partial{\overline{T}}}{\partial{x}}=0 \qquad \text{is OK.}
$$
with condition $\overline{T(0,t)}=\frac{T_0}{s}$
The mistake is in the solving of this equation. 
HINT : To make it more clear, let 
$\begin{cases}
f(x)=T(x,t=0) \quad\text{a given function,}\\
\overline{T}=y(x) \quad\text{for each } s \\
\text{Condition : } y(0)=\frac{T_0}{s} 
\end{cases}
\quad\to\quad sy+u\frac{dy}{dx}=f(x) $
$$y=\frac{1}{u}e^{-\frac{s}{u}x} \left( \int_0^x f(\xi)e^{\frac{s}{u}\xi}d\xi +\frac{u}{s}T_0 \right)$$
Back to the original symbols :
$$\overline{T}=\frac{1}{u}e^{-\frac{s}{u}x}\left(\int_0^x T(\xi,t=0)e^{\frac{s}{u}\xi}d\xi +\frac{u}{s}T_0\right)$$ 
