$\newcommand{\const}{\mathrm{const}}$

Deadline Monday, October 1, 9 pm

Consider equation with the initial conditions \begin{align} & u_{tt}-4u_{xx}=0,\qquad &&t>0, x>vt, \label{eq-1}\\ &u|_{t=0}= e ^{-x}, \qquad &&x>0, \label{eq-2}\\ &u_t|_{t=0}= e^{-x}, \qquad &&x>0, \label{eq-3} \end{align}

Let $v=3$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

- None,
- $u|_{x=vt}=0$ ($t>0$),
- $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

Let $v=1$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

- None
- $u|_{x=vt}=0$ ($t>0$),
- $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

Let $v=-3$. Find which of these conditions (a)-(c) at $x=vt$, $t>0$ could be added to (\ref{eq-1})-(\ref{eq-3}) so that the resulting problem would have a unique solution:

- None
- $u|_{x=vt}=0$ ($t>0$),
- $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).

Solve the problem you deemed as a good one.

A spherical wave is a solution of the three-dimensional wave equation of the form $u(r, t)$, where r is the distance to the origin (the spherical coordinate). The wave equation takes the form \begin{equation} u_{tt} = c^2 \bigl(u_{rr}+\frac{2}{r}u_r\bigr) \qquad\text{(“spherical wave equation”).} \label{eq-4} \end{equation}

Change variables $v = ru$ to get the equation for $v$: $v_{tt} = c^2 v_{rr}$.

Solve for $v$ using \begin{equation} v = f(x+ct)+g(x-ct) \label{eq-5} \end{equation} and thereby solve the spherical wave equation.

Use \begin{equation} v(r,t)=\frac{1}{2}\bigl[ \phi (r+ct)+\phi (r-ct)\bigr]+\frac{1}{2c}\int_{r-ct}^{r+ct}\psi (s)\,ds \label{eq-6} \end{equation} with $\phi(r)=v(r,0)$, $\psi(r)=v_t(r,0)$ to solve it with initial conditions $u(r, 0) = \Phi (r)$, $u_t(r, 0) = \Psi(r)$.

Find the general form of solution continuous as $r=0$.

By method of continuation combined with D'Alembert formula solve each of the following four problems (a)--(d).

- \begin{equation} \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=1, \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \end{equation}
\begin{equation} \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=1, \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \end{equation}

\begin{equation} \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=x, \qquad &&x>0,\\ &u|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \end{equation}

\begin{equation} \left\{\begin{aligned} &u_{tt}-4u_{xx}=0, \qquad &&x>0,\\ &u|_{t=0}=0, \qquad &&x>0,\\ &u_t|_{t=0}=x, \qquad &&x>0,\\ &u_x|_{x=0}=0, \qquad &&t>0. \end{aligned}\right. \end{equation}

- Show that \begin{equation} \frac{\partial e}{\partial t} = \frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = \frac{\partial e}{\partial x} . \label{eq-11} \end{equation}
- Show that both $e(x, t)$ and $p(x,t)$ also satisfy the wave equation.