Author Topic: TT2-P3  (Read 1631 times)

Victor Ivrii

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TT2-P3
« on: November 18, 2015, 08:39:01 PM »
Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a, <y<b\}$ with Dirichlet boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u_{x=0}=u_{x=a}=u_{y=0}=u_{y=b}=0.\label{3-2}
\end{align}

Catch Cheng

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Re: TT2-P3
« Reply #1 on: November 19, 2015, 12:43:05 AM »
Please correct if something is wrong, thank you.

Rong Wei

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Re: TT2-P3
« Reply #2 on: November 19, 2015, 01:05:57 AM »
furthermore, we will have λn for U = (pi*n/a)^2 + (pi*n/b)^2
     

Emily Deibert

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Re: TT2-P3
« Reply #3 on: November 19, 2015, 01:52:18 AM »
Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!

Bruce Wu

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Re: TT2-P3
« Reply #4 on: November 19, 2015, 01:58:00 AM »
I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Rong Wei

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Re: TT2-P3
« Reply #5 on: November 19, 2015, 01:58:33 AM »
Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!
I guess she means y, but clerical error

Emily Deibert

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Re: TT2-P3
« Reply #6 on: November 19, 2015, 01:59:03 AM »
I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

Emily Deibert

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Re: TT2-P3
« Reply #7 on: November 19, 2015, 01:59:30 AM »
Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!
I guess she means y, but clerical error

Yes, you must be right! For a second I was worried that I had done everything wrong!

Bruce Wu

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Re: TT2-P3
« Reply #8 on: November 19, 2015, 02:02:19 AM »
I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

You're right, but then in the end shouldn't the final eigenvalues be $\lambda_n=(\frac{n\pi}{a})^{2}+(\frac{n\pi}{b})^{2}$?
« Last Edit: November 19, 2015, 02:04:11 AM by Fei Fan Wu »

Emily Deibert

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Re: TT2-P3
« Reply #9 on: November 19, 2015, 02:03:25 AM »
I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

You're right, but then in the end shouldn't the final eigenvalues be $\lambda_n=\left(\frac{n\pi}{a}\right)^2+\left(\frac{n\pi}{b}\right)^2$?

Indeed, good point.

Rong Wei

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Re: TT2-P3
« Reply #10 on: November 19, 2015, 02:22:49 AM »
I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

You're right, but then in the end shouldn't the final eigenvalues be $\lambda_n=(\frac{n\pi}{a})^{2}+(\frac{n\pi}{b})^{2}$?
I think so, maybe she lost that step