Toronto Math Forum

APM346-2015F => APM346--Home Assignments => HA6 => Topic started by: Yumeng Wang on October 26, 2015, 02:55:53 PM

Title: hm6 Q1
Post by: Yumeng Wang on October 26, 2015, 02:55:53 PM

http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter4/S4.2.P.html

Title: Re: hm6 Q1
Post by: Yumeng Wang on October 26, 2015, 02:58:45 PM

This is my answer for question1(a), the prove part.

Title: Re: hm6 Q1
Post by: Yumeng Wang on October 26, 2015, 03:24:56 PM

This is my answer for prove part of (b).

Title: Re: hm6 Q1
Post by: Yumeng Wang on October 26, 2015, 03:55:27 PM

(d)

Title: Re: hm6 Q1
Post by: Bruce Wu on October 28, 2015, 04:06:50 PM

I don't see why $A_{n}=\omega_{n}$

Title: Re: hm6 Q1
Post by: Emily Deibert on October 28, 2015, 06:07:27 PM

After speaking to Professor Ivrii about it we found out that you simplify choose the value of $A_n$ for convenience. You do not need to plug in any tangent term.

Title: Re: hm6 Q1
Post by: Rong Wei on October 28, 2015, 07:32:13 PM

Quote from: Yumeng Wang on October 26, 2015, 03:55:27 PM

(d)

YU MENG,
I think in the step By Foundamental Theorem of Calculus, it should be d(-Xn'Xm + XnXm')/dx
the signs of your solution is wrong, but anyway, it doesn't influence the right answer.

Title: Re: hm6 Q1
Post by: Emily Deibert on October 29, 2015, 11:27:54 AM

Could someone clarify what we are supposed to do in part c? It seems like there is no question being asked.

Title: Re: hm6 Q1
Post by: Yeming Wen on October 29, 2015, 12:00:25 PM

I think the question is asking the case where $\lambda =0$.

Title: Re: hm6 Q1
Post by: Victor Ivrii on October 29, 2015, 12:35:02 PM

Quote from: Yeming Wen on October 29, 2015, 12:00:25 PM

I think the question is asking the case where $\lambda =0$.

It is only part of the story. Consider plane $(\alpha,\beta)$. After you found when $\lambda=0$ is an e.v. you got an equation to $(\alpha,\beta)$ and it describes a hyperbola, which breaks the plane into 3 zones. Since $\lambda_n=\lambda_n(\alpha,\beta)$ depend on $(\alpha,\beta)$ continuously in each of those zones the number of negative eigenvalues is the same. This number changes when you go from one zone to another and some eigenvalue crosses $0$.

One can use hint: $\lambda_n$ monotone with respect to each of arguments. If $alpha>0,\beta>0$ then there are no negative e.v.