Problems to Section 2.7

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Problems

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

Problem 1. For equation \begin{equation} u_{tt}-c^2 u_{xx}+ f(u)=0, \qquad x> 0 \end{equation} prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int_0^\infty \bigl(u_t^2+ c^2 u_{x}^2+F(u)\bigr)\,dx \end{equation} under Dirichlet ($u|_{x=0}=0$) or Neumann ($u_x|_{x=0}=0$) boundary condition; here $F$ is a primitive of $f$.

Problem 2. For beam equation \begin{equation} u_{tt}+K u_{xxxx}=0, \qquad x> 0,\\ \end{equation} prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ K u_{xx}^2)\,dx \end{equation} under each of the pair of the boundary conditions: \begin{align} &u|_{x=0}=u_x |_{x=0}=0;\\ &u|_{x=0}=u_{xxx} |_{x=0}=0;\\ &u_{x}|_{x=0}=u_{xx} |_{x=0}=0. \end{align}

Problem 3. a. For problem \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2 u_{xx}=0, &&x >0,\\ &(\alpha_0 u_x+\alpha_1 u)|_{x=0}=0 \end{aligned}\right. \end{equation} find energy conservation law \begin{equation} E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ c^2 u_{x}^2)\,dx + \frac{1}{2}a u(0)^2 \end{equation} (you need to calculate $a$).

b. For problem \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2 u_{xx}=0, &&0< x < l,\\ &(\alpha_0 u_x+\alpha_1 u)|_{x=0}=0,\\ &(\beta_0 u_x-\beta_1 u)|_{x=l}=0 \end{aligned}\right. \end{equation} find energy conservation law \begin{equation} E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ c^2 u_{x}^2)\,dx + \frac{1}{2}a u(0)^2+ \frac{1}{2}b u(l)^2 \end{equation} (you need to calculate $a$, $b$).

Problem 4. For problem \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2 u_{xx}=0, &&x >0,\\ &(u_x-i\alpha u_t)|_{x=0}=0 \end{aligned}\right. \end{equation} with real $\alpha$ prove energy conservation law \begin{equation} E(t)=\frac{1}{2}\int_0^\infty (|u_t|^2+ c^2 |u_{x}|^2)\,dx. \end{equation}

Problem 5. Consider Problem 2.6.2. \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} Let \begin{align} &E_1(t)=\frac{m_1}{2}\int_0^\infty (u_t^2 + c_1^2 u_x^2)\,dx,\\ &E_2(t)=\frac{m_2}{2}\int_{-\infty}^0 (u_t^2 + c_2^2 u_x^2)\,dx. \end{align}

  1. Find ratio $m_1:m_2$ such that $E(t)=E_1(t)+E_2(t)$ is conserved.
  2. In this case prove that if $\phi (x)=0$ for $x > L$ then for $t > L/c_1$: $E_1(t)= k_1 E(0)$ and $E_2(t)= k_2 E(0)$.
    Calculate $k_1,k_2$ and prove that $k_1+k_2=1$.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$