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**Home Assignment 3 / Re: problem 3**

« **on:**December 18, 2012, 12:45:51 PM »

Solution to question3 part b is attached!

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Solution to question3 part b is attached!

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http://weyl.math.toronto.edu:8888/APM346-2011F-forum/index.php?topic=208.0

In last year final exam, problem 1, when you solved it by method of continuation, I was wondering if the first term will be:

u(x,t)=1/2**(-Ï•(xâˆ’ct)+Ï•(x+ct))**+ 1/2câˆ«ct+x0Ïˆ(s)ds +1/2câˆ«ctâˆ’x0Ïˆ(s)ds

instead of what you wrote:

"In our case correct formula is different

u(x,t)=1/2**(Ï•(xâˆ’ct)âˆ’Ï•(x+ct))**+ 1/2câˆ«ct+x0Ïˆ(s)ds +1/2câˆ«ctâˆ’x0Ïˆ(s)ds"

In last year final exam, problem 1, when you solved it by method of continuation, I was wondering if the first term will be:

u(x,t)=1/2

instead of what you wrote:

"In our case correct formula is different

u(x,t)=1/2

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I was wondering if Cn has to be sin(nÎ¸) instead of cos(nÎ¸)?

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So, An is not zero when n=4k.

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Please check the attachment!

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Solution to (d) and (e) is attached!

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In heat equation: We know IFT of e^ (-(Î¾^2)/2) is (âˆš2pi)e (-(x^2)/2). Then when we try to find IFT of e^ (-(ktÎ¾^2)), we scale Î¾ to âˆš(2kt)Î¾ and therefore x scale to âˆš(2kt)x. But in the lecture note, it is mentioned that x scale to (2kt)^(**-**1/2)x. I believe it should be just square root of 2kt and not (-square root) of 2kt.

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Aida, please next time provide a link, like this one

http://weyl.math.toronto.edu:8888/APM346-2011F-wiki/index.php/Home_Assignment_8#5.4.12.I am just wondering in this question when P.T. wrote Parseval's equality, it shouldn't contain "integral |sin(nÏ€x/l)|^2"; Right?

Yes, except $\int_0^l |sin(nÏ€x/l)|^2\,dx =l/2$, so you don't need this integral. Ditto for 5.4.13, 5.4.15.

And who is P.T.?

"Paul Tan" Last year student.

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I am just wondering in this question when P.T. wrote Parseval's equality, it shouldn't contain "integral |sin(nÏ€x/l)|^2"; Right?

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Aida: How did you do the integral?

He didn't. He used a fourier transform pair from part 1(a).

Yes, That's right!

By the way Zarak, I guess it is clear from my name that I am female!