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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2020 • Question 6

Question

Math 6410 • ODE Spring 2020 • Question 6

  1. Show that the wave equation \(u_{tt}=c^2u_{xx}\) \((-\infty <x<\infty ,\ t>0)\) conserves energy \[ E=\int _{-\infty }^{\infty }\left [(u_t)^2+c^2(u_x)^2\right ]d x. \]
  2. Show the uniqueness: If two solutions of the wave equation satisfy the same initial conditions, then the solutions are identical.

Raw LaTeX

\begin{enumerate}
\item[(a)] Show that the wave equation \(u_{tt}=c^2u_{xx}\)
\((-\infty<x<\infty,\ t>0)\) conserves energy
\[
  E=\int_{-\infty}^{\infty}\left[(u_t)^2+c^2(u_x)^2\right]\dd x.
\]
\item[(b)] Show the uniqueness: If two solutions of the wave equation satisfy the
same initial conditions, then the solutions are identical.
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Poincare Maps and Periodic Solutions Wave Equation and Energy Methods