Problem5

[Murat Gokmen]

I thought for a heat equation there is max principle in general. Is this happening because the coefficient is x but not another random variable? Thanks

[Victor Ivrii]

I thought for a heat equation there is max principle in general. Is this happening because the coefficient is x but not another random variable? Thanks

The truth is that it is not a heat equation as coefficient at $u_{xx}$ is not everywhere positive

[Shu Wang]

This may be a stupid question, but could you clarify how the "proof of maximum" breaks down while we're asked to find the maximum? Suppose if we can prove a way to find some maxima.

[Levon Avanesyan]

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.

[Victor Ivrii]

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

You are formulating a more demanding problem. The problem as stated does not require to find a maximum but to prove that it is not at one of the segments comprising the boundary (without upper  lid) $\Gamma$. It is sufficient to find maximum on $\Gamma$ or even to prove that it does not exceed $M$ and then to show a point inside or on the upper lid where value is larger (and there is no need to prove that it is a maximum).

Exactly like: to disprove a statement that Mr. Johns is the tallest person in the world you don't need to bring the tallest person in the world and compare with Mr. Johns, someone who is taller than Mr. Johns would be sufficient.

[Shu Wang]

Haha, I see what you mean now, and the the maxima is definitely messed up due to the du/dx.
By the way, when is the problem set due? since we didn't have the lecture on Monday.

[Victor Ivrii]

By the way, when is the problem set due? since we didn't have the lecture on Monday.

ROTFL: it is on the top of the HA3!!!

[Levon Avanesyan]

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

You are formulating a more demanding problem.

Sorry, I was sleepy while writing that
EDIT
After rereading the problem 5, I noticed that part b straightly asks us to find the maximum point, so I guess I am was not stating a more demanding problem...

[Jinlong Fu]

q5

[Victor Ivrii]

PLease use b/w scan settings

[Bowei Xiao]

Should it be Ut- xUxx = -2*x*epsilon instead of -2xk? At the last three line of the picture you attached?