Yeming is right. I will post the result here. Correct me if I'm wrong.
Since the question asks a Dirichlet condition for the convection, in problem 3(c) we have shown the transformation $u(x,t) = U(x-ct,t)$ is not appropriate since $u(0,t) =U(-ct,t) =0$ is not a boundary condition for $U$.
Thus we use transformation $u(x,t) = v(x,t)e^{\alpha x + \beta t}$. Note in this case we have
\begin{equation}
u(0,t) = v(0,t)e^{\beta t} = 0 \rightarrow v(0,t) = 0
\end{equation}
This is a valid boundary condition for v.
Now we figure out how $g(x)$ changed here.
\begin{equation}
u(x,0) = v(x,0)e^{\alpha x} = g(x) = e^{-\epsilon |x|} \rightarrow v(x,0) = e^{-\epsilon |x| - \alpha x}
\end{equation}
Where $\alpha = \frac{c}{2k}$
Then general solution of Dirichlet problem gives
\begin{equation}
u(x,t) = \int_0^\infty (G(x,y,t) - G(x,-y,t))e^{-(\epsilon + \frac{c}{2k}) y}dy
\end{equation}
We can take $y$ out from $|y|$ since its all positive.
Do the same thing Catch has been doing. The final answer is
\begin{equation}
u(x,t) = \frac{e^{c^2-2cx}}{2}(1+erf(\frac{x-ct}{\sqrt{4kt}})) - \frac{e^{c^2+2cx}}{2}(1+erf(\frac{x+ct}{\sqrt{4kt}}))
\end{equation}
