**Assignment 12** - Due Wednesday, December 5, 2018.

- 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 3.4: 2(d) (You may use problem 4 from Assignment 11.)
- Section 4.1: 4, 7
- Section 4.2: 1, 3, 4
- Section 4.3: 3
- Let $Ku(x)=\int_0^1 k(x,y)u(y)dy$, 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 $Ku(0)=0$ and $(Ku)'(1)=0$.
- Find the eigenvalues and eigenvectors of $K$. Explain why the (normalized) eigenvectors of $K$ are a complete orthonormal basis for $L^2[0,1]$.

- 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 $L^2[0,1]$.

- Section 3.4: 2(d) (You may use problem 4 from Assignment 11.)

Updated 11/26/2018.