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Problems to Sections 4.1, 4.2

"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 and 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.