Toronto Math Forum
APM346-2012 => APM346 Math => Home Assignment 4 => Topic started by: Calvin Arnott on October 18, 2012, 04:39:28 PM
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It seems to me that question 1 part c) doesn't ask any question and instead makes a statement. Is there anything I'm missing there?
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It seems to me that question 1 part c) doesn't ask any question and instead makes a statement. Is there anything I'm missing there?
Justify statements.
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for part a, should we add a condition that alpha and beta are real? Or they have to be real since we are assuming all eigenvalues are real?
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for part a, should we add a condition that alpha and beta are real? Or they have to be real since we are assuming all eigenvalues are real?
Right, I put it explicitly. Thanks for checking and asking questions (on behalf of everyone).
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What does OX stand for?
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What does OX stand for?
$x$ axis
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For X_n expression in a), if why don't we have w_n in front of the cosine, instead we have w?
Also for e) can we assume there are no degeneracy in the eigenfunctions/states? Otherwise they would be orthogonal with equal eigenvalue.
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For X_n expression in a), if why don't we have w_n in front of the cosine, instead we have w?
Corrected (btw, it is $\omega$, not $w$
Also for e) can we assume there are no degeneracy in the eigenfunctions/states? Otherwise they would be orthogonal with equal eigenvalue.
Not sure what you mean.
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Hi, I have a question. To solve this question, should we substituting the Xn given into the initial conditions to justify whether it is right ? Or we just use the initial conditions to get the 2 equation given in the problem?
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Hi, I have a question. To solve this question, should we substituting the Xn given into the initial conditions to justify whether it is right ? Or we just use the initial conditions to get the 2 equation given in the problem?
Since we have two conditions (one at each end) we call them boundary conditions.
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Solution to Problem 1(d)
To show that eigenfunctions corresponding to different eigenvalues are orthogonal, we evaluate the following:
$$(\lambda_{n}-\lambda_{m})\intop_{0}^{l}X_{n}(x)X_{m}(x)dx$$
Notice that we can make a simple substitution, apply the Fundamental Theorem of Calculus using the boundary conditions. Then,
$$(\lambda_{n}-\lambda_{m})(X_{n}(x)X_{m}(x))=X_{n}"(x)X_{m}-X_{n}(x)X"_{m}(x)=(X_{n}'(x)X_{m}(x)-X_{n}(x)X'_{m}(x))'$$
Plugging into the original integral, we obtain:
$$\intop_{0}^{l}(X_{n}'(x)X_{m}(x)-X_{n}(x)X'_{m}(x))'dx=X_{n}'(l)X_{m}(l)-X_{n}(l)X'_{m}(l)-X_{n}'(0)X_{m}(0)+X_{n}(0)X'_{m}(0)=0$$
Therefore, the eigenfunctions corresponding to different eigenvalues are orthogonal.
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You may assume that all eigenvalues are real (which is the case).
Justify examples 6--7 of
Lecture 13 (http://www.math.toronto.edu/courses/apm346h1/20129/L13.html)
Consider eignevalue problem with Robin boundary conditions
\begin{align*}
& X'' +\lambda X=0 && 0<x<l,\\[3pt]
& X'(0)=\alpha X(0), \quad X'(l)=-\beta X(l)
\end{align*}
$\alpha, \beta \in \mathbb{R}$.
- (a) Prove that positive eigenvalues are $\lambda_n=\omega_n^2 $ and the corresponding eigenfunctions are $X_n$ where $\omega_n>0$ are roots of
\begin{align*}
& \tan (\omega l)= \frac{(\alpha+\beta)\omega}{\omega^2-\alpha\beta};\\
& X_n= \omega_n \cos (\omega_n x) +\alpha \sin (\omega_n x);
\end{align*}
($n=1,2,\ldots$).
Solve this equation graphically. - (b) Prove that negative eigenvalues if there are any are $\lambda_n=-\gamma_n^2$ and the corresponding eigenfunctions are $Y_n$ where $\gamma_n>0$ are roots of
\begin{align*}
& \tanh (\gamma l )= {-\frac{(\alpha + \beta)\gamma }{\gamma ^2 + \alpha\beta}},\\
& Y_n(x) = \gamma_n \cosh (\gamma_n x) + \alpha \sinh (\gamma_n x).
\end{align*}
Solve this equation graphically. - (c) To investigate how many negative eigenvalues are, consider the threshold case of eigenvalue $\lambda=0$: then $X=cx+d$ and plugging into b.c. we have $c=\alpha d$ and $c=-\beta (d+lc)$; this system has non-trivial solution $(c,d)\ne 0$ iff $\alpha+\beta+\alpha \beta l =0$. This hyperbola divides $(\alpha,\beta)$-plane into three zones:
(http://www.math.toronto.edu/courses/apm346h1/20129/L13-1.png)
Check above arguments and justify that in the described zones there are really no, one, two negative eigenvalues respectively. - (d) Prove</strong> that eigenfunctions corresponding to different eigenvalues are orthogonal:
\begin{equation}
\int_0^l X_n(x)X_m (x)\,dx =0\qquad\text{as } \lambda_n\ne \lambda_m
\label{eq-ort}
\end{equation}
where we consider now all eigenfunctions (no matter corresponding to positive or negative eigenvalues). - (e) Bonus Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.
We have proof of (d) but I definitely want (a)--(c) and (e) (I have seen that they were solved in what was submitted). In (c) I allow to use the following fact (which is due to simple variational arguments which unfortunately we have no time to study): Since the problem is symmetric (which implies that e.v. are real) and quadratic form in the right-hand expression of (13.29) is monotone increasing with respect to $\alpha,\beta$, so do eigenvalues $\lambda_n =\lambda_n(\alpha,\beta)$.[/list]
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Part (a) proof:
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Part (b) proof:
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Thanks for parts a b d above. As no one has posted part c yet, I just post solution attached.
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In addition to Part c solution I posted above, here is Problem 1 Part e Bonus solution to prove eigenvalues are simple.
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In each of three zones number of negative eigenvalues stays the same as they can cross to positives only on the borders. So in fact one can go along line $\alpha=\beta$ which intersects all of them. Then things are slightly simpler to analyze.