
### APM 346 (2012) Home Assignment 4

Deadline Wednesday, October 24; solution posting is allowed after 21:30 of this day

"Solve equation graphically" means that you plot a corresponding function and points $(z_n,0)$ where it intersects with $OX$ will give us all the frequencies $\omega_n=\omega (z_n)$.

"Simple solution" $u(x,t)=X(x)T(t)$.

You may assume that all eigenvalues are real (which is the case).

#### Problem 1

Justify examples 6--7 of Lecture 13: 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*} $\alpha, \beta \in \mathbb{R}$.

1. 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.

2. 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.

3. 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:

4. Check above arguments and justify that in the described zones there are really no, one, two negative eigenvalues respectively.
5. Prove that eigenfunctions corresponding to different eigenvalues are orthogonal: $$\int_0^l X_n(x)X_m (x)\,dx =0\qquad\text{as } \lambda_n\ne \lambda_m \label{eq-ort}$$ where we consider now all eigenfunctions (no matter corresponding to positive or negative eigenvalues).

6. Bonus Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

#### Problem 2

Oscillations of the beam which with both its ends having fixed positions and fixed directions (bricked into the walls: it is called ''clamped'') are described by an equation
\begin{equation*} u_{tt} + K u_{xxxx}=0, \qquad 0<x< l, \end{equation*} with $K>0$ and the boundary conditions \begin{equation*} u(0,t)=u_{x}(0,t)=u(l,t)=u_{x}(l,t)=0. \end{equation*}

1. Find equation describing frequencies and corresponding eigenfunctions (You may assume that all eigenvalues are real and positive).

2. Solve this equation graphically.

3. Prove that eigenfunctions corresponding to different eigenvalues are orthogonal (see (\ref{eq-ort})).

4. Bonus Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

#### Problem 3

Consider wave equation with the Dirichlet boundary condition on the left and "weird" b.c. on the right: \begin{align*} & u_{tt}-c^2u_{xx}=0 && 0<x<l,\\ & u(0,t)=0,\\ & (u_x + i \alpha u_t) (l,t)=0 \end{align*}

with $\alpha \in \mathbb{R}$.
1. Separate variables;
2. Find "weird" eigenvalue problem for ODE;
3. Solve this problem;
4. Find simple solution $u(x,t)$.