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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2003 • Question 2

Question

Math 6420 • PDE Fall 2003 • Question 2

  1. Determine the conditions on \(f(x)\), \(\alpha \) and \(\beta \) for which there solutions of \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u'(1)-u(1)=\beta . \]
  2. Find the “best” solution of \[ \frac {d^2u}{dx^2}=1,\qquad u(0)=1,\qquad u'(1)-u(1)=1. \]

Raw LaTeX

\begin{enumerate}
\item[(a)] Determine the conditions on \(f(x)\), \(\alpha\) and \(\beta\) for which there
solutions of
\[
  \frac{d^2u}{dx^2}=f(x),\qquad u(0)=\alpha,\qquad u'(1)-u(1)=\beta.
\]
\item[(b)] Find the ``best'' solution of
\[
  \frac{d^2u}{dx^2}=1,\qquad u(0)=1,\qquad u'(1)-u(1)=1.
\]
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Fundamental Solutions Linear Programming and Optimization