Math 6420 • PDE

This problem examines features of the differential operator
\[
  Lu=u''
\]
subject to boundary conditions \(u(-1)+u(1)=0\) and \(u'(-1)=u'(1)\).
\begin{enumerate}
\item[(a)] What is the adjoint of this operator? Is it self-adjoint?

\item[(b)] What is the nullspace of the operator and what is the nullspace of the adjoint
operator?

\item[(c)] Under what conditions on \(f\), \(a\), and \(b\) does the boundary value problem
\[
  u''=f,\qquad u(-1)+u(1)=a,\qquad u'(-1)-u'(1)=b
\]
have a solution?

\item[(d)] Find the smallest least squares solution for the boundary value problem
\[
  u''=1,\qquad u(-1)+u(1)=1,\qquad u'(-1)-u'(1)=1.
\]
What equation does this least squares solution satisfy?
\end{enumerate}