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

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4
  5. Problem 5
  6. Problem 6

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 - t u_x=0;\\ u_t+ t^2 u_x=0;\\ u_t +x u_x=0;\\ u_t +t x u_x=0;\\ u_t+x^2u_x=0;\\ u_t+(x^2+1)u_x=0;\\ u_t+(t^2+1)u_x=0;\\ (x+1)u_t +u_x=0;\\ (x+1)^2u_t+u_x=0;\\ (x^2+1)u_t+u_x=0;\\ (x^2-1)u_t+u_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 $f(x)$? Consider separately $t > 0$ and $t < 0$.
  3. Where this solution is uniquely determined? Consider separately $t > 0$ and $t < 0$.
  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{gather} yu_x+ xu_y=0,\\ yu_x-xu_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;

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 solutions of \begin{align} & \left\{\begin{aligned} & u_x+3u_y=xy,\\ & u|_{x=0}=0; \end{aligned} \right.\\[3pt] & \left\{\begin{aligned} & u_x+3u_y=u,\\ & u|_{x=0}=y. \end{aligned} \right. \end{align}

Problem 5. Find the general solutions to each of \begin{gather} yu_x-xu_y=x;\\[3pt] yu_x-xu_y=x^2;\\[3pt] yu_x+xu_y=x;\\[3pt] yu_x+xu_y=x^2. \end{gather} In one instance solution does not exist. Explain why.

Problem 6. Solve IVP \begin{equation} \left\{\begin{aligned} & u_t+uu_x=0,\qquad t > 0; \\ & u|_{t=0}=f(x) \end{aligned}\right. \end{equation} 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.


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