Problems

• Problem 1
• Problem 4
• Problem 7
• Problem 10

Problem 1.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c^2u_{xx}=0, &&t > 0, x > 0, \\\ &u|_{t=0}= \phi (x), &&x > 0, \\ &u_t|_{t=0}= c\phi'(x), &&x > 0, \\ &u|_{x=0}=\chi(t), &&t > 0. \end{aligned} \right. \end{equation} (separately in $x > ct$ and $0 < x < ct$).
2. \begin{equation} \left\{ \begin{aligned} & u_{tt}-c^2u_{xx}=0, &&t > 0, x > 0, \\\ &u|_{t=0}= \phi (x), &&x > 0, \\ &u_t|_{t=0}= c\phi'(x), &&x > 0, \\ &u_x|_{x=0}=\chi(t), &&t > 0. \end{aligned} \right. \end{equation} separately in $x > ct$ and $0 < x < ct$.

Problem 2.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c_1^2u_{xx}=0, &&&t > 0, x > 0, \\ & u_{tt}-c_2^2u_{xx}=0, &&&t > 0, x < 0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c_1\phi'(x) &x > 0, \\ &u|_{t=0}=0, &&u_t|_{t=0}=0, &x < 0,\\ &u|_{x=+0}=\alpha u|_{x=-0}, &&u_x|_{x=+0}=\beta u_x|_{x=-0} &t > 0 \end{aligned} \right. \end{equation} (separately in $x > c_1 t$, $0 < x < c_1 t$, $-c_2t < x < 0$ and $x < -c_2t$).
2. Discuss reflected wave and refracted wave. In particular, consider $\phi(x)=e^{ik x}$.

Problem 3.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c_1^2u_{xx}=0, &&&t > 0, x > 0, \\ & v_{tt}-c_2^2v_{xx}=0, &&&t > 0, x > 0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c_1\phi'(x) &x > 0, \\ &v|_{t=0}=0, &&v_t|_{t=0}=0, &x > 0,\\ &(u+\alpha v)|_{x=0}=0, &&(u_x+\beta v_x)|_{x=0}=0 &t > 0 \end{aligned} \right. \end{equation} (for $u$ separately in $x > c_1t$, and for $v$ separately in $0 < x < c_2t$).
2. Discuss two reflected waves. In particular, consider $\phi(x)=e^{ik x}$.
3. Discuss connection to Problem 2.

Problem 4.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c^2u_{xx}=0, &&&t > 0, x > 0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c\phi'(x) &x > 0, \\ &(u_x+\alpha u)|_{x=0}=0, &&&t > 0 \end{aligned} \right. \end{equation} (separately in $x > ct$, and $0 < x < c t$).
2. Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.

Problem 5.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c^2u_{xx}=0, &&&t > 0, x > 0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c\phi'(x) &x > 0, \\ &(u_x+\alpha u_{t})|_{x=0}=0, &&&t > 0 \end{aligned} \right. \end{equation} (separately in $x > c t$, and $0 < x < c t$).

2. Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.

Problem 6.

1. Find solution \begin{equation} \left\{ \begin{aligned} & u_{tt}-c ^2u_{xx}=0, &&&t > 0, x > 0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c \phi'(x) &x > 0, \\ &(u_x-\alpha u_{tt})|_{x=0}=0, &&&t > 0 \end{aligned} \right. \end{equation} (separately in $x > c t$, and $0 < x < c t$).
2. Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.

Problem 7. Consider equation with the initial conditions \begin{equation} \left\{ \begin{aligned} & u_{tt}-c^2 u_{xx}=0,\qquad &&t > 0, x > vt,\\ &u|_{t=0}= f(x), \qquad &&x > 0, \\ &u_t|_{t=0}= g(x), \qquad &&x>0. \end{aligned} \right. \end{equation}

1. Find which of these conditions (a)-(d) at $x=vt$, $t > 0$ could be added to (8) so that the resulting problem would have a unique solution and solve the problem you deemed as a good one:
   1. None,
   2. $u|_{x=vt}=0$ ($t > 0$),
   3. $(\alpha u_x +\beta u_t)|_{x=vt}=0$ ($t > 0$),
   4. $u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).
   Consider cases $v > c$, $-c < v < c$ and $v < -c$. In the case condition (C) find necessary restrictions to $\alpha,\beta$.
2. Find solution in the cases when it exists and is uniquely determined; consider separately zones $x > ct$, $-ct < x < ct $ and $x > ct$ (intersected with $x > vt$).

Problem 8.

By the method of continuation combined with D'Alembert formula solve each of the following twelve problems (9)--(20). \begin{align} &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=0, \qquad &&x > 0,\\ &u_t|_{t=0}=\cos (x), \qquad &&x > 0,\\ &u|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=0, \qquad &&x > 0,\\ &u_t|_{t=0}=\cos (x), \qquad &&x > 0,\\ &u_x|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=0, \qquad &&x > 0,\\ &u_t|_{t=0}=\sin(x), \qquad &&x > 0,\\ &u|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=0, \qquad &&x > 0,\\ &u_t|_{t=0}=\sin(x), \qquad &&x > 0,\\ &u_x|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=\cos (x), \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=\cos(x), \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u_x|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=\sin(x), \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=\sin(x), \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u_x|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=1, \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u_x|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0, \qquad &&x > 0,\\ &u|_{t=0}=1, \qquad &&x > 0,\\ &u_t|_{t=0}=0, \qquad &&x > 0,\\ &u|_{x=0}=0, \qquad &&t > 0. \end{aligned}\right.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{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.\\[4pt] &\left\{\begin{aligned} &u_{tt}-c^2u_{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{align}

Problem 9.

Solve \begin{equation} \left\{\begin{aligned} &(t^2+1)u_{tt}+tu_t-u_{xx}=0,\\[3pt] &u|_{t=0}=0, \qquad u_t|_{t=0}=1. \end{aligned}\right. \end{equation} Hint: Make a change of variables $x=\frac{1}{2}(\xi+\eta)$, $t=\sinh (\frac{1}{2}(\xi-\eta))$ and calculate $u_\xi$, $u_\eta$, $u_{\xi\eta}$.

Problem 10. Consider problem: \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2u_{xx}=f(x,t),&&x>0, t>0,\\ &u|_{t=0}=g(x),\\ &u_t|_{t=0}=h(x),\\ &u|_{x=0}=p(t). \end{aligned}\right. \label{Q} \end{equation} and another problem \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2u_{xx}=f(x,t),&&x>0, t>0,\\ &u|_{t=0}=g(x),\\ &u_t|_{t=0}=h(x),\\ &u_x|_{x=0}=q(t). \end{aligned}\right. \label{R} \end{equation}

Assuming that all functions $g,h$ and either $p$ or $q$ are sufficiently smooth, find conditions necessary for solution $u$ be

1. $C$ in ${x>0,t>0}$ (was on the lecture);
2. $C^1$ in ${x>0,t>0}$;
3. $C^2$ in ${x>0,t>0}$;
4. $C^3$ in ${x>0,t>0}$; where $C^n$ is the class on $n$-times continuously differentiable functions.

Hint: These compatibility conditions are on $f,g,h$ and either $p$ or $q$ and may be their derivatives as $x=t=0$. Increasing smoothness by $1$ ads one condition. You do not need to solve problem, just plug and may be differentiate.

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