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\centerline{{\sc Problem set 5}} \centerline{Math 207B, Winter 2016} \centerline{Due: Fri, Feb. 19}


\bigskip\noindent {\bf 1.} Suppose that $p :[a,b] \to \Rl$ is a continuously differentiable function such that $p > 0$, and $q, r :[a,b] \to \Rl$ are continuous functions such that $r > 0$, $q \ge 0$. Define a weighted inner product on $L^2(a,b)$ by \[ \langle u,v\rangle_r = \int_a^b r(x) \overline{u(x)} v(x)\, dx. \] Let $A : D(A) \subset L^2(a,b)\to L^2(a,b)$ be the operator \[ A = \frac{1}{r(x)}\left[-\frac{d}{dx} p(x) \frac{d}{dx} + q(x)\right] \] with Dirichlet boundary conditions and domain \[ D(A) = \left\{u\in H^2(a,b) : u(a)=0, u(b)=0\right\}. \] (a) Show that \[ \langle u, Av\rangle_r = \langle Au,v\rangle_r\qquad \text{for all $u,v\in D(A)$}, \] meaning that $A$ is self-adjoint with respect to $\langle \cdot,\cdot\rangle_r$.

\smallskip\noindent(b) Show that the eigenvalues $\lambda$ of the weighted Sturm-Liouville eigenvalue problem \[ -(pu')' + q u = \lambda r u,\qquad u(a)=0,\quad u(b)=0 \] are real and positive and eigenfunctions associated with different eigenvalues are orthogonal with respect to $\langle \cdot,\cdot\rangle_r$.

\bigskip\noindent {\bf 2.} A nonuniform string of length one with wave speed $c_0(x) = \sqrt{T/\rho_0(x)} >0$ is fixed at each end, with zero initial displacement and nonzero initial velocity. The transverse displacement $y=u(x,t)$ of the string satisfies the IBVP \begin{align*} &u_{tt} = c_0^2(x)u_{xx}\qquad 0<x<1, \quad t>0, \\ &u(0,t) = 0,\qquad u(1,t) = 0\qquad t>0, \\ &u(x,0) = 0\qquad 0<x<1, \\ &u_t(x,0) = g(x)\qquad 0<x<1, \end{align*} Find the solution in terms of the eigenvalues $\lambda_n$ and eigenfunctions $\phi_n(x)$ of the weighted Sturm-Liouville problem \[ -c_0^2\phi_n'' = \lambda_n\phi_n,\qquad \phi_n(0) = 0,\quad \phi_n(1)=0,\qquad n=1,2,3,\dots \]


\bigskip\noindent {\bf 3.} The Fourier solution of the initial value problem \begin{align*} &u_{tt} = u_{xx}\qquad 0<x<1, \quad t>0, \\ &u(0,t) = 0,\qquad u(1,t) = 0\qquad t>0, \\ &u(x,0) = \begin{cases} 2x &\text{if $0\le x \le1/2$} \\ 2(1-x) & \text{if $1/2 < x < 1$}, \end{cases} \\ &u_t(x,0) = 0\qquad 0\le x \le1, \end{align*} is given by \[ u(x,t) = \frac{8}{\pi^2} \sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1)^2}\sin \left[(2n-1)\pi x\right]\cos \left[(2n-1)\pi t\right] \] (a) Show that the Fourier series converges to a continuous function. What order of spatial (weak) $L^2$-derivatives does $u(x,t)$ have?

\smallskip\noindent (b) Verify from the Fourier solution that \[ \int_0^1 \left[u_t^2 (x,t)+ u_x^2(x,t)\right]\, dx = \text{constant}\qquad \text{for $-\infty< t < \infty$.} \]

\smallskip\noindent (c) Use \textsc{matlab} (or another program) to compute the partial sum \[ u_N(x,t) = \frac{8}{\pi^2} \sum_{n=1}^N \frac{(-1)^{n+1}}{(2n-1)^2}\sin \left[(2n-1)\pi x\right]\cos \left[(2n-1)\pi t\right] \] at $t=0.25$ for $N=5$ and $N=50$.

\smallskip\noindent (d) Use the addition formula for sines to shows that the Fourier solution can be written in the form of the d'Alembert solution as \[ u(x,t) = F(x-t) + F(x + t) \] for a suitable function $F:\Rl \to\Rl$. What is $F$?



\bigskip\noindent {\bf 4.} Suppose that $u(x,t)$ is a smooth solution of the wave equation \begin{align*} &u_{tt} = c_0^2 \Delta u, \end{align*} where $x\in \Rl^n$, the wave speed $c_0>0$ is a constant.

\smallskip\noindent (a) Show that $u$ satisfies the energy equation \[ \frac{1}{2}\left(u_t^2 + c_0^2 |\nabla u|^2\right)_t - \nabla\cdot\left(c_0^2u_t\nabla u\right) = 0. \]
(b) For $T > 0$, let $\Omega_T\subset \Rl^{n+1}$ be the space-time cone \[ \Omega_T  = \left\{(x,t)\in \Rl^{n+1} : \text{$|x| < c_0 (T-t)$, $0<t < T$}\right\},\]
and for $0\le t \le T$, let $B(T-t)$ be the spatial cross-section of $\Omega_T$ at time $t$
\begin{align*}
B(T-t) &= \left\{ x\in \Rl^n : |x| < c_0(T-t)\right\}.
\end{align*}
Define
\begin{align*}
e_T(t) &= \frac{1}{2}\int_{B(T-t)}\left(u_t^2 + c_0^2 |\nabla u|^2\right)\, dx,
\end{align*}
and show that $e_T(t) \le e_T(0)$.

\smallskip\noindent
(c) Suppose that $u_1$, $u_2$ are smooth solution of the wave equation such that \[ u_i(x,0) =f_i(x), \quad u_{it}(x,0) = g_i(x)\qquad \text{$i=1,2$} \] where $f_1=f_2$, $g_1=g_2$ in $|x| \le c_0T$, show that $u_1=u_2$ in $\Omega_T$.

\smallskip\noindent
\textsc{hint}. For (b), apply the divergence theorem in space-time to the equation in (a) over the truncated cone $\{(x,t') \in \Omega_T : 0<t'<t\}$, and note that the space-time normal to the side of the cone $\Omega_T$ is $N=(\hat{x},c_0)/\sqrt{1+c_0^2}$ where $\hat{x} = x/|x|$. For (c), consider $u=u_1-u_2$.
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