Toronto Math Forum

Let $u$ solve the initial value problem for the wave equation in one dimension
\begin{equation*}
\left\{\begin{aligned}
& u_{tt}-  u_{xx}= 0 ,\qquad&& ~{\mbox{in}} ~\mathbb{R} \times (0,\infty),\\[3pt]
&u (0,x) = f(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} ,\\[3pt]
&u_t(0,x)= g(x),  \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} .
\end{aligned}\right.
\end{equation*}
Suppose $f(x)=g(x)=0$ for all $|x|>1000.$ The  kinetic energy is
$$
k(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_t^2 (t,x) dx
$$
and the potential energy is
$$
p(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_x^2 (t,x) dx.
$$
 Prove

• $k(t)+ p(t)$ is constant with $t$ (so does not change as $t$ changes),
• $k(t)=p(t)$ for all large enough times $t$.

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problem 4 part (a)

Consider a $2\pi$-periodic function $f$ with full Fourier series
$$
\sum_{n \in \mathbb{Z}} c_n e^{i n x}.
$$
 Suppose that the Fourier coefficients decay fast enough to satisfy
$$
\sum_{n \in \mathbb{Z}} |n| \cdot |c_n| < 17.
$$
Prove that $f'$ is bounded.

includes solution to part (d) and what I think is the graph to part (e).

Solutions to problem 5.

Do you want the solution in the complex form or real form of the Fourier series?

I think Qitan Cui is correct. In my haste I must have messed up the integration of the bonus part.

My solution. Please check there might be mistakes. Sorry for quality of scan and handwriting.

I solved the question in the same way as Aida Razi did. Of course for part b, C(x) > 0  for all x.