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Question

Math 6410 • ODE

Fall 2016 • Question 11

Question

Math 6410 • ODE Fall 2016 • Question 11

Under what conditions on \(\lambda \), if any, does a solution \(u(r,\theta )\) of \begin{equation} \nabla ^2u=1, \tag {5} \end{equation} on the annulus \(1<r<2\), with boundary conditions \(u_r=\cos \theta \) at \(r=1\), and \(u_r=\lambda \cos ^2\theta \) at \(r=2\)? Find a solution when it exists. Is it unique?

Raw LaTeX

Under what conditions on \(\lambda\), if any, does a solution \(u(r,\theta)\) of
\begin{equation}
  \nabla^2u=1,
  \tag{5}
\end{equation}
on the annulus \(1<r<2\), with boundary conditions \(u_r=\cos\theta\) at \(r=1\), and
\(u_r=\lambda\cos^2\theta\) at \(r=2\)? Find a solution when it exists. Is it unique?

Topic Tags

Boundary Value Problems Boundary Value Problems and Sturm-Liouville Theory Elliptic Equations Poincare Maps and Periodic Solutions