$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###Problems to Sections 4.1, 4.2 > 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.** 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 $\alpha, \beta \in \mathbb{R}$. 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} with $n=1,2,\ldots$. **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. **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). 5. **Bonus** Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional. **Problem 2.** Analyse the same problem albeit with Dirichlet condition on the left end, : $X(0)=0$. **Problem 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} 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. 4. **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.** 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.** 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.** 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 $\alpha \in \mathbb{R}$. 1. **Separate** variables; 2. **Find** "weird" eigenvalue problem for ODE; 3. **Solve** this problem; 4. **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.** 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)