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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2002 • Question 3

Question

Math 6420 • PDE Fall 2002 • Question 3

Find the eigenvalues and eigenfunctions of the integral operator \[ (Ku)(x)=\int _0^1 k(x,y)u(y)\,dy \] where \[ k(x,y)= \begin {cases} x(1-y), & 0\le x<y\le 1,\\ y(1-x), & 0\le y<x\le 1. \end {cases} \] Solve the corresponding integral equation \[ u(x)-\lambda \int _0^1 k(x,y)u(y)\,dy=f(x) \] using the eigenfunctions of \(K\) assuming that \(\lambda \ne n^2\pi ^2\) for any integer \(n\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =m^2\pi ^2\) for some integer \(m\).

Raw LaTeX

Find the eigenvalues and eigenfunctions of the integral operator
\[
  (Ku)(x)=\int_0^1 k(x,y)u(y)\,dy
\]
where
\[
k(x,y)=
\begin{cases}
x(1-y), & 0\le x<y\le 1,\\
y(1-x), & 0\le y<x\le 1.
\end{cases}
\]
Solve the corresponding integral equation
\[
  u(x)-\lambda\int_0^1 k(x,y)u(y)\,dy=f(x)
\]
using the eigenfunctions of \(K\) assuming that \(\lambda\ne n^2\pi^2\) for any integer \(n\)
(i.e. determine the resolvent operator). Write down the Fredholm alternative condition for
\(f\) when \(\lambda=m^2\pi^2\) for some integer \(m\).

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Fredholm Alternative Linear Programming and Optimization Spectral Theory of Operators