Problems to Section 2.8

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

### Problems

Problem 1. Find the characteristics and write down the general solutions to the systems $U_u+AU_x=0$ with \begin{align} A&=\begin{pmatrix} 2 &3\\3 &2 \end{pmatrix},\\ A&=\begin{pmatrix} 2 &-3\\1 &-2 \end{pmatrix},\\ A&=\begin{pmatrix} 1 &-1\\2 &4 \end{pmatrix},\\ A&=\begin{pmatrix} -1 &-1\\2 &-4 \end{pmatrix},\\ A&=\begin{pmatrix} 3 &2\\0 & -1 \end{pmatrix},\\ A&=\begin{pmatrix} 3 &0\\ 2 & -1 \end{pmatrix} \end{align} Problem 2. For each system from Problem 1 in $\{ x>0, t>0\}$ determine which of the following IVBPs is well-posed and find solution ($U= \begin{pmatrix}u\\ v\end{pmatrix}$): \begin{align} &u|_{t=0}=f(x), v|_{t=0}=g(x); \\ &u|_{t=0}=f(x), v|_{t=0}=g(x); &&u|_{x=o}=\phi(t),\\ &u|_{t=0}=f(x), v|_{t=0}=g(x); &&u|_{x=o}=\phi(t), v|_{x=0}=\psi(t) \end{align} Problem 3. Find the characteristics and write down the general solutions to the systems $U_u+AU_x=0$ with \begin{align} A&=\begin{pmatrix} 3 &2 &1\\0 &2 &1\\ 0 &0&1 \end{pmatrix},\\ A&=\begin{pmatrix} 3 &2 &1\\0 &2 &1\\ 0 &0&-1 \end{pmatrix},\\ A&=\begin{pmatrix} 3 &2 &1\\0 &-2 &1\\ 0 &0&-1 \end{pmatrix},\\ A&=\begin{pmatrix} -3 &2 &1\\0 &-2 &1\\ 0 &0&-1 \end{pmatrix},\\ A&=\begin{pmatrix} 1 &2 &3\\2 &0 &3\\ 2 &3&0 \end{pmatrix}. \end{align} Problem 4. For each system from Problem 3 in $\{ x>0, t>0\}$ determine which of the following IVBPs is well-posed and find solution ($U= \begin{pmatrix}u\\ v\\w\end{pmatrix}$): \begin{align} &u|_{t=0}=f(x), v|_{t=0}=g(x), w|_{t=0}=h(x); \\ &u|_{t=0}=f(x), v|_{t=0}=g(x), w|_{t=0}=h(x),\notag\\ &u|_{x=0}=\phi(t);\\ &u|_{t=0}=f(x), v|_{t=0}=g(x), w|_{t=0}=h(x),\notag\\ &u|_{x=0}=\phi(t), v|_{x=0}=\psi(t);\\ &u|_{t=0}=f(x), v|_{t=0}=g(x), w|_{t=0}=h(x), \notag\\ &u|_{x=0}=\phi(t), v|_{x=0}=\psi(t), w|_{x=0}=\chi(t). \end{align}