Toronto Math Forum
APM346-2015F => APM346--Tests => Test 2 => Topic started by: Victor Ivrii on November 18, 2015, 08:39:01 PM
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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}
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Please correct if something is wrong, thank you.
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furthermore， we will have λn for U = (pi*n/a)^2 + (pi*n/b)^2
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Please correct if something is wrong, thank you.
Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!
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I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?
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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
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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.
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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!
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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 $\lambda_n=(\frac{n\pi}{a})^{2}+(\frac{n\pi}{b})^{2}$?
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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.
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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