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\begin{center}
UNM Dept. of Mathematics and Statistics \\
Math. 570, Asymptotics and Singular Perturbations \\
Problem Set IV-a \\[.2in]
\end{center}
Spring 2011 \\[.2in]
\begin{enumerate}

\item Consider the BVP (Friedrichs, 1942):
\[ \epsilon\frac{d^2 u}{dx^2} + \frac{du}{dx} - a - 2bx = 0\ ,\  
u(0) = 0\ ,\ u(1) = 1 \ .
\]
\begin{enumerate}
\item Show that the exact solution is
\[ u(x,\epsilon) = (1 - a - b + 2\epsilon b)
\left( \frac{1 - e^{-x/\epsilon}}{1 - e^{-1/\epsilon}}\right)
+ ax + bx^2 - 2 \epsilon b x = \tilde{u}(x,\epsilon) +{\cal O}
\left( e^{-1/\epsilon}\right) \ . \]
Where $\tilde{u}(x,\epsilon)$ is obtained by neglecting the
exponentially small term $e^{-1/\epsilon}$ in the denominator of $u$.
\item Using singular perturbation techniques, construct inner-outer
expansions in powers of $\epsilon$. Show that the result is $\tilde{u}$.\\
({\it Hint:} in this case the singular perturbation solution is identically
$= 0$ for ${\cal O}(\epsilon^2)$ and higher, so you only need to compute the
solution to ${\cal O}(\epsilon)$).
\item The solution found in part (ii) satisfies the B.C. at $x=0$ exactly
but is off by an exponentially small amount at $x=1$. Let
\[ V(x,\epsilon) = u(x,\epsilon) - \tilde{u}(x,\epsilon) \ .\]
Find the equation and B.C. satisfied by $V(x,\epsilon)$ and invent a 
perturbation expansion for $V$.\\
({\it Hint:} it does not involve only powers of $\epsilon$).
\end{enumerate}

\item Consider the B.V.P.
\[ \epsilon \frac{d^2 y}{dx^2} + x^{1/3}\frac{dy}{dx} - y = 0\ ,\ 
y(0) = 0\ ,\ y(1) = e^{3/2} \ .\]
Carry out an inner-outer-matching construction like the one given in class for
the case $k = 1/2$ and give a uniformly valid approximation of the solution
correct to $\cal{O}(\epsilon)$ (i.e. not including terms of $o (\epsilon)$).\\
({\it Hint :} you need to take the first 3 terms of the inner expansion to match
with the first 2 terms of the outer. Present case is simpler except for the
logarithmic term special to $k = 1/3$, as mentioned in class).

\item Following the discussion in class, (4/26/2011), devise the adiabatic
invariant for a charged particle in a slowly varying magnetic field. That is,
consider a particle with initial velocity ${\bf v} = (v_x, 0, v_z)$ in
a magnetic field ${\bf B} = B(t) {\bf e}_z$. The particle is subject to the
Lorentz force: ${\bf F} = q {\bf v}\times{\bf B} + q {\bf E}$
where the electric field ${\bf E}$ due to the slowly varying magnetic field is found as
\[ \nabla\times{\bf E} = -\frac{\partial {\bf B}}{\partial t} \rightarrow
{\bf E} = \frac{1}{2}\left(\frac{\partial {\bf B}}{\partial t}\times {\bf r}
\right) \rightarrow {\bf E} = \left( -\frac{y}{2}\dot{B},\frac{x}{2}\dot{B}
\right) \ .\]
Then, Newton's law becomes
\begin{eqnarray*}
m\frac{d^2x}{dt^2} & = & q B \frac{dy}{dt} + q E_x \\ 
m\frac{d^2y}{dt^2} & = &-q B \frac{dx}{dt} + q E_y \\ 
m\frac{d^2z}{dt^2} & = & 0  
\end{eqnarray*}

\begin{enumerate}
\item
Show that the $x$ and $y$
coordinates of the motion can be combined into $u = x + iy$ so that $u$
satisfies the (complex) ODE:
\[ \frac{d^2 u}{dt^2} + i \omega \frac{du}{dt} +\frac{i}{2} \frac{d\omega}{dt}
u = 0\ ,\ \omega = \frac{q B(t)}{m} \ ,\]
where $m, q$ are the mass and charge of the particle. 
\item
Introducing the polar form
\[ u = \phi(t) e^{-\frac{i}{2}\int_0^t\omega(s) ds} \]
show that the amplitude $\phi(t)$ satisfies the ODE
\[ \frac{d^2\phi}{dt^2} + \frac{\omega(t)^2}{4}\phi = 0 \ .\]
Assume that the frequency is a slowly varying function of time,
\[ \Omega(\tau := \epsilon t) := \omega(t)/2 \]

\item
Use the class discussion to find the adiabatic invariant $J=E/\Omega$ and
express it in terms of the original variables.
\end{enumerate}


\end{enumerate}
\end{document}
