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\title{Perturbation Methods\\
MATH 6620--1 -- Spring 2010 \\
Homework 4}
\date{Due Monday, May 17 at 5:00 PM}
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\begin{document}
\maketitle
This homework has 225 points plus 20 bonus points available but, as always, homeworks are graded out of 100 points. Full credit will generally be awarded for a solution only if it is both correctly and efficiently presented using the techniques covered in the lecture and readings, and if the reasoning is properly explained. If you used software or simulations in solving a problem, be sure to include your code, simulation results, and/or worksheets documenting your work. If you score more
than 100 points, the extra points do count toward your homework total. You are now welcome to use
computational tools to automate calculations of solutions provided that:
\begin{itemize}
\item You explain the logic of what you are doing
\item You provide the worksheet and/or code showing how you programmed the calculations
\item You write out the solution in comprehensible form. You will lose considerable credit if you
submit an excessively complicated expression for the solution which a software package produced
\end{itemize}
The fundamental principle is that your solution should show you understand how to solve the problem.
In matched asymptotic expansions, for full credit, you should explain carefully why you are hypothesizing layers in certain locations -- it is not sufficient to simply guess and check that it works.
\section{Variable Coefficient Boundary Value Problem (25 points)}
Obtain a leading order approximation to the boundary value problem:
\begin{align}
\epsilon^2 \frac{\difd^2 y}{\difd x^2} + (x^2 -1) y = 0, \\
y(x=0) = 0, \qquad \left.\frac{\difd y}{\difd x}\right|_{x=2} = 1.
\end{align}
with $ 0 < \epsilon \ll 1 $ which is uniformly valid over the whole interval $ [0,2]$.
\section{Oscillator with Weak Piecewise Linear Damping (25 points)}
Obtain a leading order approximation to the solution of the initial value problem
\begin{align*}
\frac{\difd^2 y}{\difd t^2} - \epsilon \sgn(1-|y|)\frac{\difd y}{\difd t} + y =0, \\
y(t=0) = \alpha, \qquad \left.\frac{\difd y}{\difd t}\right|_{t=0} = \beta.
\end{align*}
with small parameter $ 0 < \epsilon \ll 1 $ which is valid over time scales long enough for the weak nonlinearity to have a substantial effect.
\section{Oscillator with Frequency Integrating the Amplitude (30 points)}
Obtain a leading order approximation to the system of equations:
\begin{align*}
\frac{\difd^2 y}{\difd t^2} + \kappa y &= 0, \\
\frac{\difd \kappa}{\difd t} &= \epsilon \sqrt{\kappa} y^2
\end{align*}
with initial conditions
\begin{equation*}
y (t=0) = 1, \qquad \left. \frac{\difd y}{\difd t}\right|_{t=0} = 0, \qquad \kappa (t=0) = \kappa_0,
\end{equation*}
where $ \kappa_0 $ is a positive constant and $ 0 < \epsilon \ll 1 $ is also constant. Your approximation should be valid over a sufficiently long time interval so that the change in $ \kappa $ has a leading order effect.
\section{Weakly Damped and Driven Phase Oscillator (40 points)}
Consider the initial value problem
\begin{equation}
\frac{\difd^2 \phi}{\difd t^2} + \epsilon ( 1 + \gamma \cos \phi) \frac{\difd \phi}{\difd t}
+ \sin \phi = f \cos \omega t
\end{equation}
with initial conditions $ \phi (t=0) = \left. \frac{\difd \phi}{\difd t} \right|_{t=0} = 0 $
and positive constants $ \gamma $, $ \omega $, and $ 0 < \epsilon, f \ll 1 $.
For each of these cases:
\begin{enumerate}
\item $ \omega = 2 $,
\item $ \omega = 1 $,
\item $ \omega = 1 + \epsilon \Omega $ with $ \Omega \sim \ord(1) $
\end{enumerate}
do the following:
\begin{itemize}
\item (\textbf{20 points}) Determine an exponent $ \beta $ so that if we relate the strength of the weak forcing $ f $
to the strength of the weak damping $ \epsilon $ by $ f = \alpha \epsilon^{\beta} $, with $ \alpha \sim \ord(1) $, then the forcing term (involving $ f $) affects the solution on the same long time scale as the damping term (involving $ \epsilon $) does.
\item (\textbf{20 points}) Obtain a leading order approximation which is valid for long enough time that the damping and forcing have a significant, order unity effect.
\end{itemize}
\section{Weak Periodic Damping and Driving of Elastic String (30 points)}
Consider the partial differential equation:
\begin{equation}
\frac{\partial^2 u}{\partial t^2} + \epsilon \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
+ \epsilon (2 \sin 2 \pi x + 3 \sin 3 \pi x) \sin 2 \pi t
\end{equation}
on the domain $ 0 < x < 1, t > 0 $
where $ 0 < \epsilon \ll 1 $, with initial conditions
\begin{equation}
u(x,t=0) = \sin \pi x + \frac{1}{2} \sin 2 \pi x, \qquad \left. \frac{\partial u (x,t)}{\partial t} \right|_{t=0} = 0
\end{equation}
and boundary conditions
\begin{equation}
u(x=0,t) = u(x=1,t) = 0.
\end{equation}
Obtain a leading order approximation to the solution of this problem which is valid over time scales long enough for the weak periodic forcing to have a substantial effect.
\section{Slow Spatial Variation in Coefficient (25 points plus 20 bonus points)}
Consider the partial differential equation
\begin{equation}
\frac{\partial^2 u}{\partial t^2} = \cosh (\epsilon x) \frac{\partial}{\partial x} \left(
\sech (\epsilon x) \frac{\partial u}{\partial x}\right)
\end{equation}
on the domain $ x \in \BBR, t > 0 $
with initial condition
\begin{equation}
u(x,t=0) = 1-\tanh x, \qquad \left. \frac{\partial u (x,t)}{\partial t} \right|_{t=0} = \sech^2 x.
\end{equation}
and small parameter $0 < \epsilon \ll 1 $.
For technical reasons, assume the solution $ u(x,t) $ remains bounded along with its derivative.
\begin{enumerate}
\item (\textbf{25 points}) Obtain a leading order approximation to the solution over order unity time which is uniformly valid over the whole infinite spatial domain.
\item (\textbf{20 bonus points}) Obtain a leading order approximation to the solution which is valid uniformly over all space and for times up to order $ \epsilon^{-1} $.
\end{enumerate}
\section{Homogenization of Compressible Velocity Field (50 points)}
The homogenization of the advection-diffusion equation will be performed
in class on Monday, May 10 for the case of an incompressible velocity field. An analogous
homogenization can be pursued for the case of mean zero, compressible,
periodic velocity fields, but some variation on the approach is required. To keep the
calculations as clear as possible, consider the case of a
\emph{steady}
one-dimensional compressible velocity field $ v(\xp) $ which has
spatial period $ 1 $ along each coordinate axis:
\begin{eqaligntwo*}
v(\xp+1) &= v(\xp)
\end{eqaligntwo*}
and mean zero:
\begin{displaymath}
\int_{0}^{1} v(\xp)
\, \difd \xp = 0.
\end{displaymath}
The nondimensionalized
advection-diffusion equation reads
\begin{displaymath}
\begin{eqalign}
\frac{\partial T (x,t)}{\partial t}
+ \frac{\partial}{\partial x} \left(v(x) T(x,t)\right)
&= \Pecloc^{-1} \Delta T (x,t),\\
T(x,t=0) &= \Tinit (\delta x)
\end{eqalign}
\end{displaymath}
where $0< \delta \ll 1 $ is the ratio of the spatial period of the
velocity field to the spatial scale of variation of the initial concentration field.
\begin{enumerate}
\item (\textbf{15 points})
Pursue a formal asymptotic approximation to the large-scale
evolution of the concentration field $ T(x,t) $ when $ 0 < \delta \ll 1 $ by
an appropriate rescaling of the advection-diffusion equation and the
development of a multiple scales analysis. You will again
repeatedly encounter equations of the form $ \Diffopo g = f $ with
periodic boundary conditions,
where the differential operator $ \Diffopo $ will be somewhat
different than what it was
for the incompressible case.
\item (\textbf{15 points})
To proceed, you will need to deduce a solvability condition for
$ \Diffopo $. Assume that you can restrict attention to steady,
spatially periodic solutions (since the velocity field is steady and
spatially periodic).
You will then be able to determine the conditions for existence and
uniqueness of steady, spatially periodic solutions to $ \Diffopo g = f $ (for
time-independent $f$) by explicit calculation. This approach will not
generalize completely to multiple dimensions, but there are other more
sophisticated ways to do this.
\item (\textbf{20 points})
Complete the multiple scales calculation to obtain an effective
large scale equation for the concentration field, with an
auxiliary cell problem to determine the effective parameters in the
large-scale equation. Be sure to include appropriate side conditions
for each equation, but you may not wish to bother calculating the
initial transient response.
\end{enumerate}
\end{document}