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\noindent\hspace*{1cm}{\it Math 6720}
\hfill{\bf Appl. Complex Var. \& Asymptc Mthds}\hfill
{\it Alexander Balk}


\bigskip\bigskip

\noindent\hspace*{1cm}{\bf HW11}\hfill{\bf\large NAME:
.....................................................................}
\hfill {\bf due Wednesday, 4/13/2016}

\bigskip\bigskip

\begin{enumerate}
\item {\bf Orszag's 2/3 Rule.}
Suppose we represent unknown functions by Fourier expansions
\begin{eqnarray*}
u(x)=\sum_{k=-M}^{M} u_k \; e^{i \,k\,\frac{2\pi}{T}x},\quad\quad
v(x)=\sum_{k=-M}^{M} v_k \; e^{i \,k\,\frac{2\pi}{T}x}
\end{eqnarray*}
with $2M+1$ Fourier coefficients.
When our equation is nonlinear, we need to compute convolutions
\begin{eqnarray*}
w_k=\sum_{p+q=k} u_p v_q,
\end{eqnarray*}
where the sum is over all integers $p,q\in[-M,M]$, such that $p+q=k$. We can
do this computationally fast, \\via the FFT:
\begin{enumerate}
\item[(i)] Choose sufficiently big $N$ being a power of 2 (later we will see how
big it should be) \\
and pad Fourier coefficients vectors with zeros, so that the
resulting vectors have length $N$
\begin{eqnarray*}
\hspace*{-1cm}
{\bf F}=\left[\begin{array}{c} 0\\ \vdots\\
               0\\u_{-M}\\ \vdots\\ u_M \\ 0\\ \vdots\\ 0
              \end{array}\right]
\quad F_k=\left\{\begin{array}{cc} 0, & -\frac{N}{2}\le k\le -M-1,\\
                                     u_k, & -M\le k\le M,\\
                                     0, & M+1\le k \le\frac{N}{2}-1;
                \end{array}\right.\quad\quad
{\bf G}=\left[\begin{array}{c} 0\\ \vdots\\
               0\\v_{-M}\\ \vdots\\ v_M \\ 0\\ \vdots\\ 0
              \end{array}\right]
\quad G_k=\left\{\begin{array}{cc}  0, & -\frac{N}{2}\le k\le -M-1,\\
                                     v_k, & -M\le k\le M,\\
                                     0, & M+1\le k \le\frac{N}{2}-1;
           \end{array}\right.
\end{eqnarray*}
then make two IFFTs
\begin{eqnarray*}
f_n=\frac{1}{N}\sum_{k=-\frac{N}{2}}^{\frac{N}{2}-1} F_k \,
e^{i\,k\,\frac{2\pi}{N}\,n}\,,\quad\quad
g_n=\frac{1}{N}\sum_{k=-\frac{N}{2}}^{\frac{N}{2}-1} G_k \, 
e^{i \,k\,\frac{2\pi}{N}\;n}\,,\quad\quad n=0,1,\ldots,N-1.
\end{eqnarray*}
\item[(ii)] Calculate the point-wise product $\quad h_n=f_n\, g_n $ .
\item[(iii)] Make the FFT
\begin{eqnarray*}
H_k=\sum_{k=0}^{N-1} h_n \, e^{-i \,k\,\frac{2\pi}{N}n}.
\end{eqnarray*}
\end{enumerate}
{\sc \underline{QUESTIONS:}}
\begin{enumerate}
\item Show that the calculation (i)-(ii)-(iii) results in
\begin{eqnarray*}
H_k=\frac{1}{N}\sum_{p+q\equiv k(\mbox{\tiny mod}N)} F_p G_q.
\end{eqnarray*}
so that the sum is not just (linear) convolution, but includes all 
{\it aliases}. The idea is to choose $N$ sufficiently large so that all aliases
vanish.
\item Suppose we want to find all $w_k$, $k=-2M,-2M+1.\ldots,2M$. How
big should we choose $N$? \\What is the relation between $w_k$ and $H_k$ ?\\
{\footnotesize {\it Suggestion:} We want to elliminate all aliases 
$p+q-k\equiv 0(\mbox{mod}N)$, besides the main term, with $p+q-k=0$.\\
$p,q\in [-M,M],\; k\in [-2M,2M]$.}
\item Suppose we want to find only Fourier coefficients $w_k$ of the same
order: $k=-M,-M+1,\ldots,M$, \\as in the expansions for $u$ and $v$. How big
should we choose $N$? \\
{\footnotesize {\it Suggestion:} Now all three $p,q,k\in [-M,M]$.\\
(Aliasing would still occur for modes with $k>|M|$, but they are usually absent
from our approximation anyway.)}
\end{enumerate}

\newpage\item 
Solve Laplace's equation 
$$\phi_{xx}+\phi_{yy}=0 \quad\mbox{ in the domain } \quad 
D=\{z:\; |z-1|>1\;\&\; |z-2|<2\}$$
subject to the boundary condition 
$$\phi(x,y)=a \mbox{ when } |z-1|=1
\quad\mbox{ and } \quad
\phi(x,y)=b \mbox{ when } |z-2|=2$$
($a$ and $b$ are real parameters).

{\footnotesize{\it Suggestion:} 
Conformally transform the domain between the circles onto a domain
between two parallel straight lines and use new symmetry to reduce the PDE to an
ODE.}

\newpage\item Solve Laplace's equation 
$$\phi_{xx}+\phi_{yy}=0 \quad\mbox{ in the domain } \quad
D=\{z:\; |z|<1\; \&\; |z-i|<\sqrt{2}\}$$
subject to the boundary condition 
$$\phi(x,y)=a\; \mbox{ on the ark of the circle } |z|=1
\quad\mbox{ and } \;\quad
\phi(x,y)=b\; \mbox{ on the ark of the circle } |z-i|=\sqrt{2}$$
($a$ and $b$ are real parameters).

{\footnotesize{\it Suggestion:} Explore the maps $w=\frac{z-1}{z+1}$ and
$\zeta=\ln w$.}


\newpage\item
\begin{enumerate}
\item A function $f(x)$ has two {\bf asymptotic expansions} as 
$x\rightarrow x_0$ 
over the same asymptotic sequence $\{\phi_n(x)\} \\ (n=0,1,2,\ldots)$
\begin{eqnarray*}
f(x)\sim \sum_{n=0}^\infty a_n\;\phi_n(x),\quad\quad\quad
f(x)\sim \sum_{n=0}^\infty b_n\;\phi_n(x)
\end{eqnarray*}
($x$ is a real variable, $a_n$ and $b_n$ are real constant
coefficients). 
Is it true that $a_n=b_n$?
\item Two functions $f(x)$ and $g(x)$ have the same 
{\bf asymptotic expansion} as $x\rightarrow x_0$ 
over the same asymptotic sequence $\{\phi_n(x)\} \; (n=0,1,2,\ldots)$
\begin{eqnarray*}
f(x)\sim \sum_{n=0}^\infty a_n\;\phi_n(x),\quad\quad\quad
g(x)\sim \sum_{n=0}^\infty a_n\;\phi_n(x)
\end{eqnarray*}
($x$ is a real variable, $a_n$ are real constant coefficients). 
Is it true that $f(x)=g(x)$ in some neighborhood of $x_0$ ?
\end{enumerate}

\newpage\item 
Consider the exponential integral
\begin{eqnarray*}
Ei(x)=\int_x^\infty\frac{e^{-t}}{t} dt.
\end{eqnarray*}
\begin{enumerate}
\item Find its asymptotic expansion as $x\rightarrow\infty$, 
using {\bf integration by parts}.\\
{\footnotesize{\it Suggestion:} 
Using a few integrations by parts guess the formula}
\begin{eqnarray*}
Ei(x)=\frac{e^{-x}}{x}\left\{1-\frac{1}{x}+\frac{2}{x^2}-\ldots+
\frac{(-1)^{N-1} (N-1)!}{x^{N-1}}\right\}+R_N(x) .
\end{eqnarray*}
{\footnotesize with the reminder}
\begin{eqnarray*}
R_N(x)=(-1)^{N} N!\int_x^\infty\frac{e^{-t}}{t^{N+1}} dt.
\end{eqnarray*}
{\footnotesize Prove this formula integrating by parts one more time to find
the same formula with $N$ replaced by $N+1$. \\
Then prove that your representation is an asymptotic expansion.}

\item What is the asymptotic sequence in this case?

\item Is the expansion convergent?

\item Use table 10.1 in Keener's textbook (page 441) 
to find the optimal number of terms
$\bar N (x)$ and the corresponding accuracies $\bar\epsilon(x)$
for $x=2,\;3,\;5$.
\end{enumerate}


\end{enumerate}
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