Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 2014 • Question 9

Question

Math 6410 • ODE Fall 2014 • Question 9

A problem in the dynamics of the overhead power wire for an electric locomotive leads to the model \[ \frac {\partial ^2u}{\partial t^2} =\frac {\partial ^2u}{\partial x^2} \quad \text {for }x\ne X(t),\ t>0, \] 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 \(t=0\). By constructing a weak solution that takes into account the discontinuities across \(X\), show that \[ u(x,t)=\int _0^{\tau _0}V(X(\tau ),\tau )\left (1-X'(\tau )^2\right )d\tau , \] 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: you will need to derive a relationship between discontinuities in \(\partial _tu\) and in \(\partial _xu\) along \(X(t)\). You will also need to consider the domain of dependence in the \((x,t)\)-plane].

A problem in the dynamics of the overhead power wire for an electric locomotive leads
to the model
\[
  \frac{\partial^2u}{\partial t^2}
  =\frac{\partial^2u}{\partial x^2}
  \quad\text{for }x\ne X(t),\ t>0,
\]
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 \(t=0\). By constructing a weak solution that takes into
account the discontinuities across \(X\), show that
\[
  u(x,t)=\int_0^{\tau_0}V(X(\tau),\tau)\left(1-X'(\tau)^2\right)\dd\tau,
\]
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: you will need to derive a relationship between discontinuities in \(\partial_tu\)
and in \(\partial_xu\) along \(X(t)\). You will also need to consider the domain of
dependence in the \((x,t)\)-plane].

Topic Tags

Distributions and Weak Solutions Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions Sobolev Spaces and Energy Estimates