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**Home Assignment 5 / Problem 1**

« **on:**October 29, 2012, 12:14:01 PM »

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

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

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So, first start with whole interval and then as before examples, make it in the 0<x<+âˆž; Right?I think so

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I guess yous should apply the method of continuation here.

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Thanks for your help Prof. Ivrii

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Just have edited the original post

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by sqr I mean "to the power of 2".

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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!

$$

\Re(\beta(|u_t|^2))(x=L) - \Re(\alpha(|u_t|^2))(x=0).

$$

Does this imply that the answers should be

P.S. Zarak, Thanks for editing!

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Here are the answers I got.

a) $alpha$ = $beta$

b) $beta$ < $alpha$

Are they correct Prof. Ivrii?

P.S. Dont post the solution because a) dont know how to type math, b) dont have a scanner c) it is written in a very messy style

a) $alpha$ = $beta$

b) $beta$ < $alpha$

Are they correct Prof. Ivrii?

P.S. Dont post the solution because a) dont know how to type math, b) dont have a scanner c) it is written in a very messy style

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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 $c=0$.

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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...

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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.

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

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In this problem it is stated that we should use formulas (1)-(2). But shouldn't we have an initial condition in order to apply these formulas.

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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

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

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