1. ## Initial value problem

Hello! Can somebody help me find a solution of:
$u_{tt}-u_{xx}-a(u_t-u_x)=0$ on $\mathbb{R} \times (0, \infty)$
$u=g, u_t=h$ on $\mathbb{R}\times \{t=0\}$ with $a \in \mathbb{R}$

I do not really have an idea how to solve this.

2. Looks like separation of variables would work. Let $u(x,y)=X(x)T(t)$. When I plug that in I get:

$\frac{T''}{T}-a\frac{T'}{T}=\frac{X''}{X}+a\frac{X'}{X}$

You can go from there right?

3. Hi! Thanks for this answer. This works out, but i think it should be $\frac{T''}{T}+a\frac{T'}{T}=\frac{X''}{X}+a\frac{X '}{X}$, so one only has to solve the ODE $\frac{T''}{T}+a\frac{T'}{T}=const$, right?

Regards

4. Ok, thanks for that correction. However maybe this is not the way to go since once you solve the ODEs I'm not sure how you would then go on to form some combination of them to satisfy the initial and boundary values like is done with the heat or wave equation not containing the single partials. Maybe Fourier Transforms are the way to go since one of the bounds is infinite. Not sure. Sorry. I'm rusty with this.

5. Oh that's bad. One can just solve the pde but the initial values get into one's way. And i do not see how that can be repaired.
Has anyone another idea how to approach this problem?