Toronto Math Forum
APM3462012 => APM346 Math => Home Assignment 4 => Topic started by: Calvin Arnott on October 18, 2012, 04:39:28 PM

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?

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.

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?

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).

What does OX stand for?

What does OX stand for?
$x$ axis

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.

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.

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?

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.

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.

You may assume that all eigenvalues are real (which is the case).
Justify examples 67 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 nontrivial 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/L131.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{eqort}
\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 righthand expression of (13.29) is monotone increasing with respect to $\alpha,\beta$, so do eigenvalues $\lambda_n =\lambda_n(\alpha,\beta)$.[/list]

Part (a) proof:

Part (b) proof:

Thanks for parts a b d above. As no one has posted part c yet, I just post solution attached.

In addition to Part c solution I posted above, here is Problem 1 Part e Bonus solution to prove eigenvalues are simple.

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.