Q3-T0101

[Victor Ivrii]

• Find solution
  \begin{equation*}
  \left\{ \begin{aligned}
  & u_{tt}-c^2u_{xx}=0, &&&t > 0, x > 0,  \\
  &u|_{t=0}= \phi (x),   &&u_t|_{t=0}= c\phi'(x) &x > 0, \\
  &(u_x+\alpha u)|_{x=0}=0,  &&&t > 0
  \end{aligned}
  \right.
  \end{equation*}
  (separately in $x>ct$, and $0<x<ct$).
• In particular, consider $\phi(x)=e^{ikx}$.
  

[Jingxuan Zhang]

Write $u=\varphi(x+ct)+\psi(x-ct)$ then Instantly by D'Alembert' s on $x>ct$
$$\varphi=\phi,\psi=0$$
\begin{equation}u=\phi(x+ct),x>ct\end{equation}
On $0<x<ct$ solution $\varphi=\phi$ still works, whereas from boundary condition
$$\phi(-s)'+\psi'(s)+\alpha\phi(-s)+\alpha\psi(s)=0 \implies (e^{\alpha s}\psi(s))'=(e^{\alpha s}\phi(-s))'-2\alpha\phi(-s)
\implies \psi(s)=\phi(-s)+2\alpha e^{-\alpha s}\int^s e^{-\alpha s'}\phi(s)\, ds'.$$
So that
\begin{equation}u=\phi(ct+x)+\phi(ct-x)+2\alpha e^{-\alpha (ct-x)}\int^{ct-x} e^{-\alpha s}\phi(s)\, ds,0<x<ct.\end{equation}
In particular, if $\phi(x)=e^{ikx}$, then
\begin{equation}u=\left\{\begin{aligned}
&e^{ik(x+ct)}&x>ct\\
&e^{ik(ct+x)}+e^{ik(ct-x)}-2\frac{ik\alpha+\alpha^2}{k^2+\alpha^2}e^{(ik-2\alpha) (ct-x)}&0<x<ct
\end{aligned}\right.
\end{equation}
Provided, indeed, $k^2+\alpha^2\neq 0$.

[George Lu]

I took a crack at the first part, but I'm not entirely confident in my steps and I was in the other section.

As always in the wave eqn, we have $u=\varphi(x+ct)+\psi$

I first solved in the region $x>ct$, where we simply have $\varphi(x)+\psi(x)=\phi(x)$ and $\varphi'(x)-\psi'(x)=\phi'(x)$, giving us $\varphi(x)=\phi(x)$ and $\psi(x)=0$.

For the region $0<x<ct$:

$\varphi'(ct)+\psi'(-ct)+\alpha\varphi(ct)+\alpha\psi(-ct)=0$

Let $t=-\frac{x}{c}$, so $\varphi'(-x)+\psi'(x)+\alpha\varphi(-x)+\alpha\psi(x)=0$

Then $\psi'(x)+ \alpha\psi(x) =-(\varphi'(-x)+\alpha\varphi(-x))$

We can see then, that $(e^{\alpha x}\psi(x))'=(e^{\alpha x}\varphi(x))'$

Integrating tells us that $\psi(x)=e^{-\alpha x}\int e^{\alpha x}\varphi(x)dx$

So $u(x,t)=\varphi(x+ct)$ for $x>ct$

And $u(x,t)=\varphi(x+ct)+e^{-\alpha x}\int_0^{ct-x} e^{\alpha x'}\varphi(x')dx'$ for $0<x<ct$

In particular, I'm not sure where the $-2\alpha\phi(-s)$ outside of the e^ derivatives in Jingxuan's solutions come from.

[Tristan Fraser]

I got something similar, but quite the same as Jingxan:

Noteably, I still had

$$u = \phi (x+ct) , x> ct $$

On the region $ 0<x<ct $ , we get from the boundary condition:

$$ 0 = \Theta '(ct) + \Psi' (-ct) + \alpha (\Theta(ct) + \Psi(-ct)) $$

Apply that $ x = -ct $, and this prompts for me to rearrange and treat each side as an ODE:

$$ \Psi'(x) + \alpha \Psi(x) =  -(\Theta'(-x) + \alpha \Theta(-x)) $$
$$ (e^{\alpha x} \Psi(x) ) ' =  -(\Theta'(-x) + \alpha \Theta(-x)) $$
And, critically:
$$ e^{\alpha x} \Psi(x)) = \Theta(-x) + \alpha  \int_{0}^{-x} \Theta(x')dx' + c $$

$$\Psi(x) =  e^{-\alpha x}(\Theta(-x) +  \alpha  \int_{0}^{-x} \Theta(x')dx' + c $$

Leading to: (note $\Theta = \phi$ in this problem)

$$ u (x,t) = \phi(x+ct) + e^{\alpha (ct-x)}(\phi(ct-x) +  \alpha  \int_{0}^{ct-x} \Theta(x')dx' + k )$$

Examining part b: $ \phi(x) = e^{ikx} $ :

$$ u(x,t) =  e^{ik(x+ct)} + e^{\alpha( ct - x)}( e^{ik(ct-x)} + \alpha  \int_{0}^{ct-x} e^{ikx'}dx' + c) $$

$$ u(x,t) =  e^{ik(x+ct)} + e^{\alpha( ct - x)}( e^{ik(ct-x)} + \alpha  (\frac{e^{ik(ct-x)} - 1}{ik} + c) $$

Which can be expanded into sines and cosines:

$$ u(x,t) = cosk(x+ct) + i sink(x+ct) + e^{\alpha} cos(k(ct-x)) + \alpha e^{\alpha}(\frac{cosk(ct-x) + isink(ct-x) - c}{ik}) $$

A few notes:

I share the same concerns as George about the $-2\alpha\phi(-s) $, and I'm not quite sure if my above solution is correct. What I do know about part b is that we are probably expected to take the mess of sines and cosines we get, and rearrange them into some form of $ sin( a \pm b) $ or $cos(a \pm b) $ or we could maybe break it into $Re$ and $Im$ components?

[Victor Ivrii]

You do not need to go to $\sin$ and $\cos$

All "mathoperators" (LaTeX terminology) like $\sin$, $\tan$, $\exp$, $\max$, $\sup$,... should be upright, so they carry escape character \: \sin

[Victor Ivrii]

It looks like all of you got wrong (but in different ways) from ($s<0$)
\begin{align*}
&\psi'(s)+\alpha \psi(s)+ \phi'(-s)+ \alpha \phi(-s)=0\implies\\
&\psi'(s)+\alpha \psi(s)= (\phi (-s))'- \alpha \phi(-s)\implies\\
&(\psi (s)e^{\alpha s})'=(\phi (-s)e^{\alpha s})-2\alpha \phi(-s)e^{\alpha s}\implies
&\psi (s)= \phi (-s) -2\alpha e^{\alpha s}\int_0 ^s e^{\alpha s'}\phi(-s')\,ds' +C =
\phi (-s) + 2\alpha e^{\alpha s}\int_0 ^{-s} e^{-\alpha s'}\phi(s')\,ds'+C
\end{align*}
where we do change of variable $s'\mapsto -s'$. We also need a constant here so that $\psi(0^+)=\psi(0^-)$, but the former is $0$, so $C=-\phi(0)$.