Web bonus problem : Week 4 (#4)

[Victor Ivrii]

Problem 3 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.7.P.html#problem-2.7.P.3

[Zaihao Zhou]

a)
\begin{equation}     \frac{\partial E(t)}{\partial t} = \int_0^\infty (u_tu_{tt} +c^2u_xu_{xt}) dx + au(0)u_t(0)   \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = \int_0^\infty (c^2u_tu_{xx} +c^2u_xu_{xt}) dx + au(0)u_t(0)   \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = c^2\int_0^\infty (u_tu_x)_x dx + au(0)u_t(0)   \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = u_t(au - c^2u_x)|_{x=0}    \end{equation}
We need
\begin{equation}     \frac{\alpha_1}{\alpha_0} = \frac{a}{-c^2} \ \ \   \rightarrow \ \ \  a = - \frac{\alpha_1}{\alpha_0}c^2  \end{equation}

b)
\begin{equation}     \frac{\partial E(t)}{\partial t} = \int_0^l (u_tu_{tt} +c^2u_xu_{xt}) dx + auu_t|_{x=0} + buu_t|_{x=l}   \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} =u_t( c^2u_x +bu)|_{x=l} + u_t(au - c^2u_x)|_{x=0}   \end{equation}
Thus we need
\begin{equation}     -\frac{\beta_1}{\beta_0} = \frac{b}{c^2} \ \ \   \rightarrow \ \ \  b = - \frac{\beta_1}{\beta_0}c^2  \end{equation}
\begin{equation}     -\frac{\beta_1}{\beta_0} = -\frac{a}{c^2} \ \ \  \rightarrow \ \ \  a =  \frac{\beta_1}{\beta_0}c^2  \end{equation}