# Initial value problem

• Dec 8th 2008, 02:50 AM
gammafunction
Initial value problem
Hello! Can somebody help me find a solution of:
$\displaystyle u_{tt}-u_{xx}-a(u_t-u_x)=0$ on $\displaystyle \mathbb{R} \times (0, \infty)$
$\displaystyle u=g, u_t=h$ on $\displaystyle \mathbb{R}\times \{t=0\}$ with $\displaystyle a \in \mathbb{R}$

I do not really have an idea how to solve this.
• Dec 8th 2008, 05:50 AM
shawsend
Looks like separation of variables would work. Let $\displaystyle u(x,y)=X(x)T(t)$. When I plug that in I get:

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

You can go from there right?
• Dec 8th 2008, 09:36 AM
gammafunction
Hi! Thanks for this answer. This works out, but i think it should be $\displaystyle \frac{T''}{T}+a\frac{T'}{T}=\frac{X''}{X}+a\frac{X '}{X}$, so one only has to solve the ODE $\displaystyle \frac{T''}{T}+a\frac{T'}{T}=const$, right?

Regards
• Dec 8th 2008, 11:27 AM
shawsend
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.
• Dec 9th 2008, 01:00 PM
gammafunction
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?