$\renewcommand{\Re}{\operatorname{Re}}$
$\renewcommand{\Im}{\operatorname{Im}}$
$\newcommand{\erf}{\operatorname{erf}}$
$\newcommand{\dag}{\dagger}$
$\newcommand{\const}{\mathrm{const}}$
$\newcommand{\arcsinh}{\operatorname{arcsinh}}$
###[Problems to Sections 4.1, 4.2](id:sect-4.2.P)
> 1. [Problem 1](#problem-4.2.P.1)
> 2. [Problem 2](#problem-4.2.P.2)
> 3. [Problem 3](#problem-4.2.P.3)
> 4. [Problem 4](#problem-4.2.P.4)
> 5. [Problem 5](#problem-4.2.P.5)
> 6. [Problem 6](#problem-4.2.P.6)
> 7. [Problem 7](#problem-4.2.P.7)
"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.](id:problem-4.2.P.1)**
Justify [Example 4.2.6](./S4.2.html#example-4.2.6) and [Example 4.2.7](./S4.2.html#example-4.2.7): Consider eignevalue
problem with Robin boundary conditions
\begin{align}
& X'' +\lambda X=0 && 0\< x\< l,\label{p}\\\\[3pt]
& X'(0)=\alpha X(0),\label{q}\\\\
& X'(l)=-\beta X(l),\label{r}
\end{align}
with $\alpha, \beta \in \mathbb{R}$.
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}
with $n=1,2,\ldots$. **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.
d. **Prove** 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
\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.
**[Problem 2.](id:problem-4.2.P.2)**
Analyse the same problem albeit with Dirichlet condition on the left end, :
$X(0)=0$.
**[Problem 3.](id:problem-4.2.P.3)**
Oscillations of the beam are described by equation
\begin{equation}
u\_{tt} + K u\_{xxxx}=0, \qquad 0\< x\< l.
\label{a}
\end{equation}
with $K\>0$.
If both ends clamped (that means having the fixed positions and directions) then the boundary conditions are
\begin{align}
&u(0,t)=u\_{x}(0,t)=0,\label{b}\\\\
&u(l,t)=u\_{x}(l,t)=0.\label{c}
\end{align}
a. **Find** equation describing frequencies and corresponding
eigenfunctions (You may assume that all eigenvalues are real and
positive).
b. **Solve** this equation graphically.
c. **Prove** that eigenfunctions corresponding to different eigenvalues
are orthogonal.
d. **Bonus** Prove that eigenvalues are simple, i.e. all eigenfunctions
corresponding to the same eigenvalue are proportional.
*Hint.* Change coordinate system so that interval becomes $[-L,L]$ with $L=l/2$; consider separately even and odd eigenfunctions.
**[Problem 4.](id:problem-4.2.P.4)**
Consider oscillations of the beam with both ends free:
\begin{align}
&u\_{xx}(0,t)=u\_{xxx}(0,t)=0,\label{d}\\\\
&u\_{xx}(l,t)=u\_{xxx}(l,t)=0.\label{e}
\end{align}
Follow previous problem but also consider eigenvalue $0$.
**[Problem 5.](id:problem-4.2.P.5)**
Consider oscillations of the beam with the clamped left end and the free right end. Then boundary conditions are (\ref{a}) and (\ref{e}).
*Note.* In this case due to the lack of symmetry you cannot consider separately even and odd eigenfunctions.
**[Problem 6.](id:problem-4.2.P.6)**
Consider wave equation with the Neumann boundary condition on the left
and "weird" b.c. on the right:
\begin{align}
& u\_{tt}-c^2u\_{xx}=0 && 0\< x\< l, \\\\
& u\_x (0,t)=0, \\\\
& (u\_x + i \alpha u\_t) (l,t)=0
\end{align}
with $\alpha \in \mathbb{R}$.
a. **Separate** variables;
b. **Find** "weird" eigenvalue problem for ODE;
c. **Solve** this problem;
d. **Find** simple solution $u(x,t)=X(x)T(t)$.
*Hint.* You may assume that all eigenvalues are real (which is the case).
**[Problem 7.](id:problem-4.2.P.7)**
Consider energy levels of the particle in the "rectangular well"
\begin{equation}
-u\_{xx}+V u =\lambda u
\end{equation}
with $V(x)=\left\\{\begin{aligned} -&H && |x|\le L,\\\\
&0 &&|x|>0\end{aligned}\right.$
*Hint.* Solve equation for $|x|\< L$ and for $|x| \> L$ and solution must be continous (with its first derivative) as $|x|=L$: $u(L-0)=u(L+0)$,
$u\_x (L-0)=u\_x (L+0)$ and the same at $-L$.
*Hint.* All eigenvalues belong to interval $(-H,0)$.
*Hint.* Consider separately even and odd eigenfunctions.
______________
[$\Uparrow$](../contents.html) [$\uparrow$](./S4.2.html) [$\Rightarrow$](./S4.3.html)