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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2004 • Question 3

Question

Math 6420 • PDE Fall 2004 • Question 3

Find the eigenvalues and eigenfunctions of the integral operator \[ (Ku)(x)=\int _0^1 k(x,y)u(y)\,dy \] where \[ k(x,y)=\min \{x,y\}. \] 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 \(\lambda \ne 1/a_n^2\) for any integer \(n\), where \(a_n=(2n+1)\pi /2\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =1/a_n^2\).

Raw LaTeX

Find the eigenvalues and eigenfunctions of the integral operator
\[
  (Ku)(x)=\int_0^1 k(x,y)u(y)\,dy
\]
where
\[
  k(x,y)=\min\{x,y\}.
\]
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 \(\lambda\ne 1/a_n^2\) for any integer \(n\),
where \(a_n=(2n+1)\pi/2\) (i.e. determine the resolvent operator). Write down the Fredholm
alternative condition for \(f\) when \(\lambda=1/a_n^2\).

Topic Tags

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