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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2001 • Question 5

Question

Math 6420 • PDE Fall 2001 • Question 5

  1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
  2. Find an integral representation for the solution in the case that \(\alpha =0\).

Raw LaTeX

\begin{enumerate}
\item[(a)] Find all conditions on \(\alpha\), \(\beta\), \(\gamma\), and \(f(x)\) for which solutions of
\[
  u''+\alpha^2u=f(x),\qquad u(0)=\beta,\qquad \alpha u'(0)-u'(1)=\gamma
\]
exist.
\item[(b)] Find an integral representation for the solution in the case that \(\alpha=0\).
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Linear Programming and Optimization