Problems to Section 2.1

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

### Problems

Problem 1.

1. Draw characteristics and find the general solution to each of the following equations \begin{gather} 2 u_t +3u_x=0; \\ u_t+ t u_x=0; \\ u_t +xu_x=0; \\ u_t+x^2u_x=0 \\ u_x+x^3u_x=0. \end{gather}

2. Consider IVP problem $u|_{t=0}=f (x)$ as $-\infty< x < \infty$ ; does solution always exists? If not, what conditions should satisfy $f(x)$?

3. Where this solution is uniquely determined?

4. Consider this equation in $\{t>0, x>0\}$ with the initial condition $u|_{t=0}=f(x)$ as $x>0$; where this solution defined? Is it defined everywhere in $\{t>0, x>0\}$ or do we need to impose condition at $x=0$? In the latter case impose condition $u|_{x=0}=g(t)$ ($t>0$) and solve this IVBP;

5. Consider this equation in $\{t> 0, x<0\}$ with the initial condition $u|_{t=0}=f(x)$ as $x<0$; where this solution defined? Is it defined everywhere in $\{t>0, x< 0\}$ or do we need to impose condition at $x=0$? In the latter case impose condition $u|_{x=0}=g(t)$ ($t>0$) and solve this IVBP;

6. Consider problems (d) as $t<0$;

7. Consider problems (e) as $t<0$;

Problem 2.

1. Find the general solution to each of the following equations \begin{gather} xu_x+ yu_y=0,\\ xu_x-yu_y=0 \end{gather} in $\{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$? Explain the difference between these two cases;

2. Find the general solution to each of the following equations \begin{equation} yu_x+ xu_y=0,\\ yu_x-xu_y=0 \end{equation} in $\{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$? Explain the difference between these two cases;

Problem 3. In the same way consider equations \begin{gather} (x^2+1)yu_x+(y^2+1)xu_y=0; \\ (x^2+1)yu_x-(y^2+1)xu_y=0. \end{gather}

Problem 4. Find the solution of \begin{equation} \left\{\begin{aligned} &u_x+3u_y=xy,\\ &u|_{x=0}=0. \end{aligned} \right. \end{equation}

Problem 5. Find the general solutions to each of \begin{gather} yu_x-xu_y=x; \\ yu_x-xu_y=x^2; \\ xu_x+xu_y=x; \\ yu_x+xu_y=x^2; \end{gather} In one instance solution does not exist. Explain why.

Problem 6. Solve IVP \begin{align} &u_t+uu_x=0,\qquad t> 0; \\ &u|_{t=0}=f(x) \end{align} and describe domain in $(x,t)$ where this solution is properly defined with one of the following initial data \begin{align} f(x)&=\hphantom{-}\tanh (x); \\[5pt] f(x)&=-\tanh (x);\\ f(x)&=\left\{\begin{aligned} -1& && x<-a,\\ x/a& && -a\le x \le a,\\ 1& && x>a; \end{aligned}\right. \\ f(x)&=\left\{\begin{aligned} 1& && x<-a,\\ -x/a& && -a\le x \le a,\\ -1& && x>a; \end{aligned}\right. \\ f(x)&=\left\{\begin{aligned} -1& && x<0,\\ 1& && x> 0;\\ \end{aligned}\right. \\ f(x)&=\hphantom{-}\sin(x). \\ f(x)&=\left\{\begin{aligned} &\sin (x) && |x|<\pi,\\ &0 && |x|>\pi, \end{aligned}\right.\\ f(x)&=\left\{\begin{aligned} &-\sin (x) && |x|<\pi,\\ &0 && |x|>\pi, \end{aligned}\right. \end{align} Here $a>0$ is a parameter.