Math 641-600 Fall 2020
Current Assignment
Assignment 9 - Due Monday, November 30, 2020.
- Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and
my notes on
example problems for distributions.
- Do the following problems.
- Section 3.5: 2(b)
- Section 4.1: 4, 7
- Section 4.2: 1, 3
- Section 4.3: 3
- Let $Kf(x)=\int_0^1 k(x,y)f(y)dy$, where $k(x,y)$ is defined by $
k(x,y) = \left\{
\begin{array}{cl}
y, & 0 \le y \le x\le 1, \\
x, & x \le y \le 1.
\end{array}
\right.$
- Show that $0$ is not an eigenvalue of $K$.
- Show that $Kf(0)=0$ and $(Kf)'(1)=0$.
- Find the eigenvalues and eigenvectors of $K$. Explain why the
(normalized) eigenvectors of $K$ are a complete orthonormal basis for
$L^2[0,1]$. (Hint: Show that $K$ is a Green's function for $Lu=-u''$,
$u(0)=0$, $u'(1)=0$.)
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a self-adjoint
Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
- Use (b) and the spectral theory of compact operators to show the
orthonormal set of eigenfunctions for $L$ form a complete set in
$L^2[0,1]$.
Updated 11/19/2020.