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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 HW10}\hfill{\bf\large NAME:
.....................................................................}
\hfill {\bf due Wednesday, 4/6/2016}

\bigskip\bigskip

\begin{enumerate}
\item When applying a spectral method to a two-dimensional problem, we
represent the unknown as a linear combination of basis functions, e.g.
$$f(x,y)=\sum_{k=0}^{M-1}\sum_{l=0}^{N-1} F_{k,l}\; \phi_k(x)\,\psi_l(y).$$
which we consider on a two-dimensional grid $(x_m,y_n) \quad
(m=0,1,\ldots, M-1;\; n=0,1,\ldots,N-1).$ \\
{\footnotesize [Here we suppose that our domain is rectangle; $\phi(x)$ and
$\psi(y)$ are basis functions in $x$ and $y$ directions respectively.]}
\begin{enumerate}
\item How many operations (additions and multiplications) will the direct
computation of the double sum
$$f(x_m,y_n)=\sum_{k=0}^{M-1}\sum_{l=0}^{N-1} F_{k,l}\; \phi_k(x_m)\,
\psi_l(y_n) \quad\quad [m=0,1,\ldots, M-1;\; n=0,1,\ldots,N-1]$$
require?
\item One can significantly reduce the computational costs by
rearranging the double sum into two consecutive sums
$$f(x_m,y_n)=\sum_{k=0}^{M-1}\phi_k(x_m) \left\{\sum_{l=0}^{N-1} F_{k,l}\;
\psi_l(y_n)\right\} \quad\quad [m=0,1,\ldots, M-1;\; n=0,1,\ldots,N-1].$$
How many operations will this computation require?

{\footnotesize{\it Note:} The values of the basis functions $\phi_k(x),\,
\psi_l(y)$ at the grid points $x_m,y_n$ can be pre-computed, while the sums are
computed many times (at each time-step); so, the computation of $\phi_k(x_m),\,
\psi_l(y_n)$ should be excluded from the computational cost in (a) and (b).}
\item Consider the two dimensional FFT
$$F_{k,l}=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} f_{m,n}\;
e^{i(k\frac{2\pi}{M}m+l\frac{2\pi}{N}n)}$$
acting on the $N\times M$ matrix $[f_{n,m}]$ of samples $f_{m,n}=f(x_m,y_n)$. It
is usually realized by computing the one-dimensional FFTs of each column, then
of each row of the result. \\
How many operations will this computation require?
\end{enumerate}

\newpage\item{\bf Computer simulation of non-linear PDEs.} \\
As an example, consider PDE
$$\frac{\partial }{\partial t}(\Delta\psi-\psi) +
\psi_x+\psi_x\Delta\psi_y-\psi_y\Delta\psi_x=0.$$
{\footnotesize Several essentially
different physical problems --- the dynamics of atmosphere and
ocean, nuclear fusion with magnetic confinement, magnetohydrodynamics of
Earth's liquid iron core --- lead to this equation.}

To simulate this equation numerically, we consider it in a box 
$(0\le x\le L_x,\, 0\le y\le L_y$ with periodic boundary conditions),
represent the unknown function as a superposition of Fourier harmonics
$$\psi(x,y,t)=\sum_{k=-M_x}^{M_x}\sum_{l=-M_y}^{M_y}
\psi_{k,l}(t)\;e^{i(k\frac{2\pi}{L_x}x+l\frac{2\pi}{L_y}y)},$$
and reduce our PDE to a system of ODEs for the Fourier coefficients
$\psi_{k,l}(t)$. We then integrate this system in time $t$ using
the standard ODE solver, e.g. Runge-Kutta method.

\begin{enumerate}
 \item Write out the ODE system.
 \item How would you compute the nonlinear terms?
       The goal is speed, avoiding ``aliasing''.
\end{enumerate}


\newpage\item Prove {\bf the Schwartz Lemma:}
Let $f(z)$ be 
\begin{itemize}
 \item analytic in $U=\{z:\; |z|<1\}$,
 \item $|f(z)|\le 1$ in $U$,
 \item $f(0)=0$.
\end{itemize}
Then $|f(z)|\le |z|$ in $U$.

{\footnotesize{\it Suggestion:} Consider function $h(z)=\frac{f(z)}{z}$. 
Show that it is analytic in $U$  (the singularity at $z=0$ can be removed). \\
Take an arbitrary $\rho\in(0,1)$. 
Apply the maximum principle to $h(z)$ to show that $|h(z)|\le 1/\rho$ 
when $z$ is in the disc $U_\rho=\{z:\; |z|\le\rho\}$. \\
Finally, take the limit as $\rho\rightarrow1$. 
Why do you need to consider a smaller disc $U_\rho\subset U\;$?}

\newpage\item 
Show that {\bf a 1-to-1 conformal map of a disc onto a disc is necessarily
bilinear}.

{\footnotesize{\it Suggestion:} Consider an arbitrary disc in the $\zeta$-plane 
and another arbitrary disc in the $\omega$-plane.
Riemann says that there is a 1-to-1 conformal transformation of the first disc
onto the second one; you need to show that this conformal map is bilinear.
Follow these steps: 

\begin{enumerate}

\item 
Some point $\zeta_0$ of the first disc is taken to some $\omega_0$ of the second
disc. 

There is a bilinear transformation $\zeta\rightarrow\tilde\zeta$ of the
$\zeta$-plane disc onto the unit disc $\{|\tilde\zeta|<1\}$ (HW9).     
Herewith $\zeta_0$ is taken to some $\tilde\zeta_0$.

Similar, 
there is a bilinear transformation $\omega\rightarrow\tilde\omega$ of the $\omega$-plane disc onto 
the unit disc $\{|\tilde\omega|<1\}$.
      Herewith $\omega_0$ is taken to some $\tilde\omega_0$.

\item Find the bilinear transformation that maps $\{|\tilde\zeta|<1\}$ 
onto the unit disc $U^z=\{|z|<1\}$ and takes $\tilde\zeta_0$ to the origin.

Similar, 
find the bilinear transformation that maps $\{|\tilde\omega|<1\}$ onto $U^w=\{|w|<1\}$
and takes $\tilde\omega_0$ to the origin.

\item Use the Schwartz lemma to show that if a conformal transformation $w=f(z)$ maps $U^z$ onto $U^w$ 
so that the origin of $z$-plane is taken to the origin of the $w$-plane, 
then $f(z)=e^{i\beta}z$ ($\beta$ is a real number).

Indeed, according to the Schwartz lemma, $|w|\le|z|$.
The Schwartz lemma for the inverse transformation $z=g(w)$ says $|z|\le|w|$.\\
So, $\left|\frac{f(z)}{z}\right|=1$ for all $z\in U^z$. 
If the absolute value of an analytic function is constant, 
then the function is a constant (HW4).

\item Use the group property (HW9) to show that the original transformation 
of the $\zeta$-plane disc onto the $\omega$-plane disc is {\it bilinear}.

\end{enumerate}}

\newpage\item 
Solve Laplace's equation 
$$\phi_{xx}+\phi_{yy}=0 \quad\mbox{ in the domain between two circles } \quad
x^2+y^2=1 \quad \& \quad (x-1)^2+y^2=\left(\frac{5}{2}\right)^2$$
subject to the boundary condition 
$$\phi(x,y)=a \mbox{ when } x^2+y^2=1
\quad\mbox{ and } \quad
\phi(x,y)=b \mbox{ when } (x-1)^2+y^2=\left(\frac{5}{2}\right)^2$$
($a$ and $b$ are real parameters).

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

Such transformation is necessarily bilinear; the map 
$$\zeta=e^{i\beta}\frac{z-m}{1-\bar m z}$$ maps the unit disc to itself.
We can use the three parameters $m$ and $\beta$ to map the external circle into
some circle with its center at the origin and some radius $R$ (undetermined
constant). 
We can take $\beta=0$ and real $m$, so that we have a single real parameter; 
we find $m$ solving a certain quadratic equation.}



\end{enumerate}
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\newpage\vspace*{0.5cm}\item 
Let the function $w=f(z)$ conformally map a domain $D$ in the $z$-plane onto                                          a domain $\Delta$ in the $w$-plane 
($z=x+iy,\; w=u+iv$). Show that
\begin{enumerate}
 \item The Jacobian of the transformation $(x,y)\rightarrow(u,v)$ at point $z_0$ is $|f'(z_0)|^2$.
 \item The area of $\Delta$ is $A=\int_D |f'(z)|^2\;dx\,dy$.
\end{enumerate}

\newpage\vspace*{0.5cm}\item 
Show that the real part $u(x,y)$ and the imaginary part $v(x,y)$ of an analytic function $f(z)=u+iv$ 
are harmonic functions.

\newpage\vspace*{0.5cm}\item 
Let $u(x,y)$ be harmonic in a {\it simply connected} domain $D$. \\
Show that there exists an analytic function $f(z)\; (z\in D)$ such that $u(x,y)=\mbox{Re}[f(z)]$.

{\footnotesize{\it Suggestion:} Consider integral 
$$ v(x,y)=\int_{z_0}^z(-u_y dx + u_x dy)$$
($z_0$ is an arbitrary fixed point, $z=x+iy$ is a variable point in $D$) 
and show that this integral is independent of integration path. 
Then show that $u$ and $v$ satisfy the Cauchy-Riemann conditions.}
