% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/12/5

\documentclass[a4paper,12pt]{article}

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\setcounter{homework}{10}

\begin{document}

\begin{homework}
[Optional extra questions.]

\question{perturbation}

The function $\phi$ satisfies
\begin{equation} \label{Lphi}
L \phi = 0
\end{equation}
and inhomogeneous boundary conditions.
The linear operator $L$ has the form
\begin{equation} \label{L}
L = L_0 - L_1
\end{equation}
and the function $\phi_\mathrm{b}$ satisfying
\begin{equation} \label{L0phib}
L_0 \phi_\mathrm{b} = 0
\end{equation}
and the inhomogeneous boundary conditions is known.
$L_0$ and $L_1$ are Hermitian with respect to the homogeneous boundary conditions associated with the inhomogeneous boundary conditions, and the Green's operator $G_0$ satisfying 
\begin{equation} \label{L0G0}
L_0 G_0 = 1
\end{equation}
and the homogeneous boundary conditions is known.
\begin{enumerate}
\item
Use $G_0$ to obtain an equation which can be iterated to solve \eq{Lphi} for small $L_1$.
\item
Solve the Green's operator equation
\begin{equation} \label{LG}
L G = 1
\end{equation}
and use $G$ to solve \eq{Lphi} for small $L_1$.
\end{enumerate}
Check that your answers are consistent and reexpress your answer in component form in the case that $L_0$ and $L_1$ are differential operators.


\question{neumann}

Consider the differential operator
\begin{equation}
L_x = \frac{d^2}{dx^2}
\end{equation}
acting on functions $\phi : [0,\pi] \to \R$ with boundary condition
\begin{equation}
\fn{\frac{d\phi}{dx}}{0} = \fn{\frac{d\phi}{dx}}{\pi} = 0
\end{equation}
\begin{enumerate}
\item
Show that the Green's function equation
\begin{equation}
L_x \fn{G}{x,y} = \fn{\delta}{x-y}
\end{equation}
has no solution satisfying the boundary condition
\begin{equation}
\fn{\frac{\partial G}{\partial x}}{0,y} = \fn{\frac{\partial G}{\partial x}}{\pi,y} = 0
\end{equation}
Explain why not.
\item
Express the Green's function for $L_x$ in terms of the eigenvectors of $L_x$.
\item
Hence solve
\begin{equation}
L_x \fn{\phi}{x} = \fn{\rho}{x}
\end{equation}
in general, and for
\begin{equation}
\fn{\rho}{x} = \rho_0 \cos(n x)
\end{equation}
with $n \in \N$ in particular.
What happens if $n=0$?
Explain.
\end{enumerate}


\question{force}

Use the Green's function method to solve
\begin{equation} \label{ddxf}
\ddot{x} = \fn{f}{t}
\end{equation}
with initial conditions
\begin{eqnarray}
\fn{x}{0} & = & 0
\\
\fn{\dot x}{0} & = & v
\end{eqnarray}
Compare your solution with direct integration of \eq{ddxf}.
Give a physical interpretation of the Green's function and the Green's function solution.

Hence solve
\begin{equation}
\ddot{x} + \fn{\lambda^2}{t} x = 0
\end{equation}
with the same initial conditions, to leading order in $\fn{\lambda}{t} \ll t^{-1}$.
Check your approximate solution using the exact solution in the case $\lambda = \textrm{constant}$.

\end{homework}

\end{document}

