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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2008 • Question 5

Question

Math 6420 • PDE Fall 2008 • Question 5

  1. Using Green’s functions, find the solution to the equation \[ u''=f(x),\qquad u(0)=0,\qquad u(1)=1. \]
  2. Find the necessary and sufficient condition on \(f(x)\) for a solution to \[ u''+m^2\pi ^2u=f(x),\qquad u(0)=u(1)=0 \] to exist.

Raw LaTeX

\begin{enumerate}
\item[(a)] Using Green's functions, find the solution to the equation
\[
  u''=f(x),\qquad u(0)=0,\qquad u(1)=1.
\]

\item[(b)] Find the necessary and sufficient condition on \(f(x)\) for a solution to
\[
  u''+m^2\pi^2u=f(x),\qquad u(0)=u(1)=0
\]
to exist.
\end{enumerate}

Topic Tags

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