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\title{Perturbation Methods\\
MATH 6620--1 -- Spring 2010 \\
Homework 3}
\date{Due Thursday, April 29 at 2:00 PM}
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\begin{document}
\maketitle
This homework has 170 plus 45 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. Also the matching procedure should be clearly and systematically explained unless otherwise indicated.
\section{Autonomous Nonlinear Boundary Value Problem (25 points)}
Consider the boundary value problem
\begin{equation*}
\epsilon \frac{\difd^2 y}{\difd x^2} + k \frac{\difd y}{\difd x} + y^3 = 0
\end{equation*}
where $ k $ is a constant, $ 0 < \epsilon \ll 1 $ is a small constant, and the
boundary conditions are $ y(0) = 1 $ and $ y(1) = 1/2 $.
\begin{enumerate}
\item (\textbf{15 points}) Obtain a uniform leading order approximation to the solution for $k=1$.
\item (\textbf{5 points})
Describe qualitatively how the solution behaves for larger and larger positive values of $ k $.
\item (\textbf{5 points}) How does the solution change qualitatively for negative $k $?
\end{enumerate}
\section{Nonautonomous Nonlinear Boundary Value Problem (30 points)}
Consider the boundary value problem
\begin{equation*}
2 \epsilon x y \frac{\difd^2 y}{\difd x^2} + 2 \epsilon x \left(\frac{\difd y}{\difd x}\right)^2
+ 3 \epsilon y \frac{\difd y}{\difd x} - 2 x \frac{\difd y}{\difd x} - 3 y
= 0
\end{equation*}
with $ y(1) = 1 $ and $ y(2) = 2 $, and $ 0 < \epsilon \ll 1 $. Obtain a uniform leading order approximation to the solution.
\section{Nonlinear Initial Value Problem (40 points plus 45 bonus points)}
Consider the differential equation
\begin{equation*}
\epsilon \frac{\difd^2 y}{\difd t^2} - \sgn(1-|y|)\frac{\difd y}{\difd t} + y =0
\end{equation*}
with initial data
\begin{equation*}
y(t=0) = \alpha, \qquad \frac{\difd y}{\difd t} (t=0) = \beta.
\end{equation*}
where
\begin{equation*}
\sgn (x) \equiv \begin{cases} 1 & for $ x > 0 $, \\
0 & for $ x = 0 $, \\
-1 & for $ x < 0 $.
\end{cases}
\end{equation*}
is the signum function and $ 0 < \epsilon \ll 1 $.
\begin{enumerate}
\item (\textbf{25 points})
Determine the leading order behavior of the solution and explain your reasoning. For this part, it will suffice to use crude patching of pieces of your solution rather than matching through intermediate regions of common validity. It's fine also to express your solution just for a particular choice of initial data $ (\alpha,\beta) $.
\item (\textbf{30 bonus points}) Improve your approximation to be valid through first order in $ \epsilon $ (so all errors are higher order than $ \epsilon $) and use systematic matching procedures so that every piece of your solution has a domain of validity that overlaps with the domain of validity of its neighboring piece of the solution.
\label{part:careful}
\item (\textbf{15 points plus 15 bonus points}) Compare your asymptotic approximation with a numerical solution of the differential equation. The bonus credit is showing, if you did Part~\ref{part:careful}, that your correction terms are also in agreement with the numerical solution.
\end{enumerate}
\section{Linear Partial Differential Equation (35 points)}
Consider the partial differential equation on the two-dimensional triangular domain $ D $ defined by the vertices $ (1,0) $, $(3,0) $, and $ (2, 1) $:
\begin{equation*}
\epsilon \left(\frac{\partial^2 u (x,y)}{\partial x^2} + \frac{\partial^2 u (x,y)}{\partial y^2}\right) + x \frac{\partial u(x,y)}{\partial y} = x+y \text{ for } (x,y) \in D
\end{equation*}
with boundary condition $ u(x,y) = 0 $ for $ (x,y) \in \partial D $.
Find a leading order approximation for $ u(x,y) $ when $ 0 < \epsilon \ll 1 $ which is valid everywhere except maybe the corners.
\section{Nonlinear Partial Differential Equation (40 points)}
Consider the partial differential equation
\begin{equation*}
- \sum_{j=1}^3 \frac{\partial^2 \phi (\bx)}{\partial x_j^2} + q \exp (\phi (\bx) /\epsilon) = \rho (\bx) \text{ for } \bx \in D \equiv [0,1]^3
\end{equation*}
on the unit cube where $ \rho (\bx) $ is a smooth function satisfying $ \lim_{\bx \rightarrow \partial D} \rho (\bx) = \bzero $, $ q $ is a positive constant, and $ 0 < \epsilon \ll 1 $. The boundary conditions are
\begin{equation*}
\han \dotprod \grad \phi = \sigma \text{ for } \bx \in \partial D
\end{equation*}
where $ \han $ is the unit outward normal vector, and $ \sigma $ is another constant.
Obtain a leading order approximation for $ \phi $ which is valid everywhere except maybe the corners.
\end{document}