Toronto Math Forum

In C part, shouldn't it be sinh(ηx) instead of sin(ηx)?

I guess yous should apply the method of continuation here.

Just have edited the original post

So, after some more time spent on this problem as a result of integration I get
$$
 \Re(\beta(|u_t|^2))(x=L) -  \Re(\alpha(|u_t|^2))(x=0).
$$
Does this imply that the answers should be

a) $\Re(\alpha)=0, \Re(\beta)=0$
b) $\Re(\alpha)>0, \Re(\beta)<0$

P.S. Zarak, Thanks for editing!

Question: is in (c) $u(x,t)=c$ is a solution for any constant $c$?

No. From Robin conditions we can see that  $u(x,t)=c$ is a solution only when $c=0$.

Maximum principle states that U(x,t) takes maximum values only when at least one of the following holds: t=0 or x=0 or x=L. So your aim is to find where U(x,t) is at maximum. After you find the maximum point you will see that smth is wrong

P.S. When the author asks "where precisely the proof of maximum principle breaks down", he means that there is a standard way to prove the maximum principle for heat equation. And the question is "At which step exactly, having this equation, we cannot continue moving, while we could have continued moving if we had heat equation.

I will expand this question a little bit:
So, can we use the formula of general solution for wave equation or we should prove/derive it?

P.S. This is my first post here, so I am not sure if I am asking an appropriate question