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\def\Rl{\mathbb{R}}
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\centerline{{\sc Problem Set 7}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Mar. 2}



\bigskip\noindent
{\bf 1.} (a) Explain why the total birth rate $B(t)$ of a population with constant reproductive rate $\lambda$ per individual
and exponential survival rate $e^{-\beta t}$ over time $t$ satisfies the renewal equation
\[
B(t) = N_0\lambda e^{-\beta t} + \lambda \int_0^t e^{-\beta (t-s)} B(s)\, ds
\]
where $N_0$ is the total initial population at $t=0$.
What are the dimensions of $N_0$, $\lambda$, $\beta$, and $B(t)$?

\smallskip\noindent
(b) Solve this integral equation and discuss the behavior of $B(t)$ as $t\to \infty$. Does your answer make sense?

\smallskip\noindent
\textsc{Hint.} One approach is to show that $B(t)$ satisfies the IVP
\[
\dot{B} = (\lambda-\beta) B, \qquad B(0) = N_0\lambda.
\]



\bigskip\noindent
{\bf 2.} (a) Consider the following Fredholm equation of the second kind
\[
u(x) - \lambda \int_0^1 x y u(y)\, dy = f(x),\qquad 0\le x \le 1
\]
where $\lambda\in \Cx$ is a constant and $f$ is a given (continuous) function.
If $\lambda\ne 3$, show that this equation has a unique (continuous) solution for $u(x)$ and find
the solution. If $\lambda = 3$, determine for what functions $f$ a solution exists and find the solutions
in that case.

\smallskip\noindent
(b) For what functions $f$ is the following Fredholm equation of the first kind
\[
\int_0^1 x y u(y)\, dy = f(x),\qquad 0\le x \le 1
\]
solvable for $u(x)$? Describe the solutions in that case.

\smallskip\noindent
\textsc{Hint.} Note that these Fredholm equations are degenerate.


\newpage\noindent
{\bf 3.} Show that the following IVP for a second-order scalar ODE for $u(t)$
\begin{align*}
&\ddot{u}(t) = f\left(t,u(t)\right),
\\
&u(0) = u_0,\quad \dot u(0) = v_0
\end{align*}
is equivalent to the Volterra integral equation
\[
u(t) = \int_0^t (t-s) f\left(s,u(s)\right)\, ds +u_0 + v_0 t.
\]

\bigskip\noindent
{\bf 4.} Consider the following BVP for $u(x)$ in $0<x<1$:
\begin{align}
\begin{split}
&-u'' = k^2 \left[1 + \epsilon q(x)\right] u,
\\
&u(0) =0,\qquad u(1) = 0.
\end{split}
\label{scat}
\end{align}
Here $k > 0$ is a constant, $\epsilon$ is a small parameter, and $f(x)$, $q(x)$ are given (continuous) functions.
Assume that $k\ne n\pi$ for any integer $n\in \Nl$, so that $k^2$ is not an eigenvalue of $-d^2/dx^2$
with Dirichlet BCs.

\smallskip\noindent
(a) Find the Green's function $G(x,\xi)$ for (\ref{scat}) with $\epsilon=0$, which satisfies
\begin{align*}
&-\frac{d^2 G}{d^2 x} = k^2 G + \delta(x-\xi)\qquad \mbox{in $0<x<1$},
\\
&G(0,\xi) =0,\qquad G(1,\xi) = 0.
\end{align*}
(b) Use the Green's function from (a) to reformulate (\ref{scat}) as a Fredholm integral equation for $u(x)$ of the form
\[
u(x) = \epsilon \int_0^1 K(x,\xi) u(\xi)\, d\xi + g(x).
\]
(c) Write out the first few terms in the Neumann series (or Born approximation) for $u$ as integrals involving $g(x)$, $q(x)$,
and $G(x,\xi)$. 


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