# Toronto Math Forum

## APM346-2012 => APM346 Math => Home Assignment 3 => Topic started by: Murat Gokmen on October 01, 2012, 08:16:53 PM

Title: Problem5
Post by: 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
Title: Re: Problem5
Post by: Victor Ivrii on October 01, 2012, 10:42:19 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

The truth is that it is not a heat equation as coefficient at $u_{xx}$ is not everywhere positive
Title: Re: Problem5
Post by: Shu Wang on October 08, 2012, 10:56:01 PM
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.
Title: Re: Problem5
Post by: 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 ;)

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.
Title: Re: Problem5
Post by: Victor Ivrii on October 09, 2012, 04:33:05 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 ;)

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
Title: Re: Problem5
Post by: Shu Wang on October 09, 2012, 05:09:42 AM
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.
Title: Re: Problem5
Post by: Victor Ivrii on October 09, 2012, 05:12:49 AM
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!!!
Title: Re: Problem5
Post by: Levon Avanesyan on October 09, 2012, 01:02:49 PM
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...
Title: Re: Problem5
Post by: Jinlong Fu on October 10, 2012, 09:40:57 PM
q5
Title: Re: Problem5
Post by: Victor Ivrii on October 11, 2012, 04:46:37 AM
PLease use b/w scan settings
Title: Re: Problem5
Post by: Bowei Xiao on October 16, 2012, 03:16:13 PM
Should it be Ut- xUxx = -2*x*epsilon instead of -2xk? At the last three line of the picture you attached?