Math 6410 • ODE

A problem in the dynamics of the overhead power wire for an electric locomotive leads
to the model
\[
  \frac{\partial^2u}{\partial t^2}
  =\frac{\partial^2u}{\partial x^2}
  \qquad\text{for } x\ne X(t),\ t>0,
\]
where \(X\) is a prescribed smooth function with \(0<X'(t)<1\). Here \(X(t)\) is the
locomotive position and \(u\) is the displacement of the wire. Across \(x=X\) there
are prescribed discontinuities
\[
  \left[\frac{\partial u}{\partial x}\right]_{X^-}^{X^+}
  =-V(X(t),t),\qquad [u]_{X^-}^{X^+}=0.
\]
Suppose that \(u=u_t=0\) at \(t=0\).
\begin{enumerate}
\item[(a)] By differentiating along the curve \(x=X(t)\), show that
\[
  \left[\frac{\partial u_\pm}{\partial t}\right]_{X^-}^{X^+}
  =-X'(t)\left[\frac{\partial u_\pm}{\partial x}\right]_{X^-}^{X^+}.
\]
\item[(b)] For fixed \((\xi,s)\), consider the domain of dependence
\(M=\{(x,t)\mid 0<t<s,\ \xi-s<x<\xi+s\}\). Partition the domain into the subdomains
\(M_1\) and \(M_2\), \(M_1\cup M_2\), as shown in the figure below. By constructing a weak
solution that takes into account the discontinuities across \(X\), show that
\[
  u(x,t)=u(P)=\int_0^{\tau_0}V(X(\tau),\tau)(1-X'(\tau)^2)\dd\tau,
\]
where the range of \(\tau\) is taken so that \(y=X(\tau)\) lies within the range
\[
  \tau-t<y-x<t-\tau,\qquad \tau>0.
\]
[Hint: Within each subdomain the wave equation is satisfied, so apply integration by
parts to the subdomains \(M_1\) and \(M_2\) separately. Then combine your results and
use part (a).]
\begin{center}
\begin{tikzpicture}[scale=0.88,>=Latex,line cap=round,line join=round]
  \draw[->] (-0.2,0) -- (6.3,0) node[right] {\(x\)};
  \draw[->] (0,0) -- (0,4.5) node[above] {\(t\)};
  \coordinate (B) at (0.7,0);
  \coordinate (C) at (2.1,0);
  \coordinate (A) at (5.35,0);
  \coordinate (Cp) at (2.32,1.75);
  \coordinate (P) at (3.95,2.85);
  \draw (B) -- (P) -- (A);
  \draw[very thick] (C) -- (2.6,4.2) node[above left] {\(X(t)\)};
  \fill (B) circle (2pt) node[above left] {\(B\)};
  \fill (C) circle (2pt) node[below] {\(C\)};
  \fill (A) circle (2pt) node[above right] {\(A\)};
  \fill (Cp) circle (2pt) node[right] {\(C'\)};
  \fill (P) circle (2pt) node[above] {\((\xi,s)\)};
  \node[below] at (P) {\(P\)};
  \node at (1.8,0.95) {\(M_1\)};
  \node at (4.1,1.2) {\(M_2\)};
  \node[below] at (0.7,-0.05) {\(x(0)=\xi-cs\)};
  \node[below] at (5.35,-0.05) {\(x(0)=\xi+cs\)};
\end{tikzpicture}
\end{center}
\end{enumerate}