APM346-2012 > Home Assignment 3

Problem5

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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:

--- Quote from: murat_gokmen on October 01, 2012, 08:16:53 PM ---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

--- End quote ---

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:

--- Quote from: Levon Avanesyan on October 09, 2012, 01:49:02 AM ---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 ;)


--- End quote ---


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. :D

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