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\Dauthor{�����N�l}
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\begin{document}
\chapter{�Δ����������̐��l��@�̊�b2-���ԍ����@-}
\setcounter{section}{2}
\section{���C�������ւ̉��p}
���C�������Ƃ͈ȉ��̂悤�ȕ������n�ł���.
\begin{eqnarray}\Deqlab{masatu}
\DD{U}{t} = - \kappa U \nonumber \\ 
U(0) = U_0
\end{eqnarray}
��̗�Ƃ��Ă�, �n�\�ʕt�߂̕��x�N�g���ɑ΂���n�ʖ��C�̌��ʂ�, �g�U�������̕ό`�ł������\footnote{
�g�U��������
\begin{equation}
\DD{u}{t} = \sigma \frac{\partial ^2 u}{\partial x^2} \nonumber
\end{equation}
�ł���킳���.
������t�[���G�����W�J�����, 
\begin{equation}
u = Re[U(t) e^{ikx}] \nonumber
\end{equation}
��������̕������ɑ�������, 
\begin{equation}
\DP{U}{t} = - \sigma k^2 U \nonumber
\end{equation}
�ƂȂ�, ���C�������Ɠ����`�ɂȂ�.
}.
\Deqref{masatu}�̌�������
\begin{equation}
U = U_0 e^{- \kappa t}
\end{equation}
�ł���, e-folding time �� $\frac{1}{\kappa}$�ł���.

\subsection{�������Ȃ�2�i�K�X�L�[��}
���C������������������ƈȉ��̂悤�ɂȂ�.
\begin{eqnarray}
\DD{U}{t} = - \kappa U \nonumber \\
\frac{U^{n+1} - U^n}{\Delta t} = - \kappa (\alpha U^n + \beta U^{n+1}) \nonumber \\
U^{n+1} = U^n - \kappa \Delta t (\alpha U^n + \beta U^{n+1}) \nonumber \\
= U^n - K (\alpha U^n + \beta U^{n+1}) \nonumber \\
= \frac{1-K \alpha }{1+K \beta } U^n
\end{eqnarray}
������$K = \kappa \Delta t$�ł���.
�����W��$\lambda$��
\begin{equation}
\lambda = \frac{1-K \alpha }{1+K \beta }
\end{equation}
�ƂȂ�.�܂����������
\begin{equation}
\lambda \leq 1
\end{equation}
�ł���.

\begin{enumerate}
\item{{\bf{$\alpha = 1, \beta = 0$(�I�C���[�X�L�[��)�̏ꍇ}}}
\begin{equation}
\lambda = 1-K
\end{equation}
���̃X�L�[���̈��������
\begin{eqnarray}
\mid 1-K \mid \leq  1 \nonumber \\
0 < K \leq 2
\end{eqnarray}
�ƂȂ�.
������$K > 1$�̂Ƃ�, ����1�X�e�b�v���ɕ������ς���Ă��܂�\footnote{
$\lambda \equiv \frac{U^{n+1}}{U^n}$�ł��邽��, $\lambda < 0$�ƂȂ��, $U^{n+1}$��$U^n$�̓X�e�b�v���ɕ������t�]���邱�ƂɂȂ�
}
����, �����I�v������$K < 1$�Ƃ���K�v������.

\item{{\bf{$\alpha = 0, \beta = 1$(��ރX�L�[��)�̏ꍇ}}}
\begin{equation}
\lambda = \frac{1}{1+K}
\end{equation}
���̃X�L�[���̈��������
\begin{equation}
\frac{1}{1+K} \leq 1
\end{equation}
�ƂȂ�, �S�Ă�$K$�ň���ł���.
���̃X�L�[���ł͏��$0 < \lambda $�Ȃ̂ŉ���1�X�e�b�v���ɕ������ς���Ă��܂����Ƃ�����.

\item{{\bf{$\alpha = \beta = \frac{1}{2}$(��`�X�L�[��)�̏ꍇ}}}
\begin{equation}
\lambda = \frac{2-K}{2+K}
\end{equation}
���̃X�L�[���̈��������
\begin{equation}
\frac{2-K}{2+K} \leq 1
\end{equation}
�ƂȂ�, �S�Ă�$K$�ň���ł���.
������$2 \leq K$�̂Ƃ�, ����1�X�e�b�v���ɕ������ς���Ă��܂�����, �����I�v������$K < 2$�Ƃ���K�v������.
\end{enumerate}

\subsection{��������2�i�K�X�L�[��}
�O�߂Ɠ���������������ƈȉ��̂悤�Ȍ`�ɂȂ�.
\begin{eqnarray}
U^{(n+1)*} = U^{(n)} - KU^n \nonumber \\
U^{n+1} = U^n - K (\alpha U^n + \beta U^{(n+1)*})
\end{eqnarray}

\begin{enumerate}
\item{{\bf{$\alpha = 0, \beta = 1$(����X�L�[��)�̏ꍇ}}}
\begin{eqnarray}
U^{(n+1)*} = U^{(n)} - KU^n \nonumber \\
U^{n+1} = U^n - K U^{(n+1)*} \nonumber \\
= (1-K+K^2) U^n \nonumber \\
\lambda = 1-K+K^2
\end{eqnarray}
���̃X�L�[���̈��������
\begin{eqnarray}
\lambda = 1-K+K^2 \nonumber \\
= \left( k- \frac{1}{2} \right) ^2 + \frac{3}{4} \leq 1
\end{eqnarray}
���, $0 < K < 1$�̂Ƃ������, $K=\frac{1}{2}$�̂Ƃ��ł���������.

\item{{\bf{$\alpha = \beta = \frac{1}{2}$(�z�C���X�L�[��)�̏ꍇ}}}\\
\begin{eqnarray}
U^{(n+1)*} = U^{(n)} - KU^n \nonumber \\
U^{n+1} = U^n - K\left( \frac{1}{2} U^n + \frac{1}{2} \left( U^n - K U^n \right) \right) \nonumber \\
= \left( 1 - K + \frac{1}{2} K^2 \right) U^n \nonumber \\
\lambda = 1 - K + \frac{1}{2} K^2
\end{eqnarray}
���̃X�L�[���̈��������
\begin{eqnarray}
\lambda = 1-K+K^2 \nonumber \\
= \frac{1}{2} (K-1)^2 + \frac{1}{2} \leq 1
\end{eqnarray}
���, $0 < K < 2$�̂Ƃ������, $K=1$�̂Ƃ��ł���������.
\end{enumerate}

\subsection{3�i�K�X�L�[��}
\begin{enumerate}
\item{{\bf{���[�v�t���b�O�X�L�[��}}}
\begin{equation}\Deqlab{lfs}
U^{n+1} = U^{n-1} - 2 K U^n
\end{equation}
$\lambda = \frac{U^n}{U^{n-1}}$��\Deqref{lfs}�ɑ������ƈȉ��̂悤�ɓ��$\lambda$��������.
\begin{eqnarray}
U^{n+1} = \frac{U^n}{\lambda} - 2KU^n \nonumber \\
\lambda \frac{U^{n+1}}{U^n} + 2K\lambda - 1 = 0 \nonumber \\
\lambda ^2 + 2K\lambda - 1 = 0 \nonumber \\
\lambda _1 = -K + \sqrt{K^2 + 1} \nonumber \\
\lambda _2 = -K - \sqrt{K^2 + 1} \nonumber 
\end{eqnarray}
������$K \rightarrow 0$�̋Ɍ����l�����
\begin{eqnarray}
\lambda _1 \rightarrow 1 \nonumber \\
\lambda _2 \rightarrow -1 \nonumber
\end{eqnarray}
����$\lambda _1$�ɑΉ�������𕨗����[�h, $\lambda _2$�ɑΉ���������v�Z���[�h�Ƃ���.
$K > 0$�Ȃ̂�, $\mid \lambda _2 \mid > 1$�ł��蕉�̐��ł���.
�����, �������[�h�͈���Ȃ��̂�, �v�Z���[�h�͕s���肩�X�e�b�v���ɕ������ς��.
���̒�����v�Z���[�h�����S�Ɏ�菜���͕̂s�\�ł��邽�߁A���[�v�t���b�O�X�L�[���͖��C�������������ɂ͕s�����ł���.

\item{{\bf{�A�_���X�o�b�V���t�H�[�X�X�L�[��}}}
\begin{eqnarray}
U^{n+1} = U^n - K \left( \frac{3}{2} U^n - \frac{1}{2} U^{n-1} \right) \nonumber \\
= \left( 1-\frac{3}{2} K \right) U^n + \frac{1}{2} KU^{n-1} \nonumber \\
\lambda ^2 - \left( 1- \frac{3}{2} K \right) \lambda - \frac{1}{2} K = 0 \nonumber \\
\lambda _1 = \frac{1}{2} \left( 1 - \frac{3}{2} K + \sqrt{1-K+\frac{9}{4} K^2} \right) \nonumber \\
\lambda _2 = \frac{1}{2} \left( 1 - \frac{3}{2} K - \sqrt{1-K+\frac{9}{4} K^2} \right) \nonumber 
\end{eqnarray}
������$K \rightarrow 0$�̋Ɍ����l�����
\begin{eqnarray}
\lambda _1 \rightarrow 1 \nonumber \\
\lambda _2 \rightarrow 0 \nonumber
\end{eqnarray}
����$\lambda _1$�ɑΉ���������������[�h, $\lambda _2$�ɑΉ���������v�Z���[�h�ł���.
$\mid K \mid < 1$ �̎�, �ȉ��̂悤�ɋߎ��ł���.
\begin{eqnarray}
\lambda _1 {\sim} \frac{1}{2} \left( 1 - \frac{3}{2} K + \left( 1 - \left( K-\frac{9}{4} K^2 \right) \frac{1}{2} \right) \right) \nonumber \\
= \frac{1}{2} \left( 2-2K+\frac{9}{8} K^2 \right) \nonumber \\
= 1-K+\frac{9}{16}K^2\nonumber \\
\lambda _2 {\sim} \frac{1}{2} \left( 1 - \frac{3}{2} K - \left( 1 - \left( K-\frac{9}{4} K^2 \right) \frac{1}{2} \right) \right) \nonumber \\
= \frac{1}{2} \left( -K-\frac{9}{8} K^2 \right) \nonumber \\
= -\frac{K}{2}-\frac{9}{16}K^2\nonumber
\end{eqnarray}
$K \ll 1$�̂Ƃ�, $\mid \lambda _1 \mid < 1, \mid \lambda _2 \mid \ll \mid \lambda _1 \mid $�ƂȂ�.
���̏ꍇ, �v�Z���[�h�̓X�e�b�v���ɕ������ω����邪, �������[�h�Əd�˂�ƑS�̂ł͕����͕ω����Ȃ��Ȃ�(�}�Q��).

\begin{figure}[hbtp]
 \begin{center}
  \Depsf[][]{./images/ab_stability.eps}
 \end{center}
 \caption{
  ab�X�L�[���ɂ����镨�����[�h�ƌv�Z���[�h���O���t�ɂ�������.$K \ll 1$�͈̔͂�$\lambda _1$��$\lambda _2$�𑫂���, ��ɐ��ɂȂ邱�Ƃ�������.�c��$\lambda$, ����$K$.
 }
 \Dfiglab{�}} % ���ȏ��̐}�̔ԍ�������,
                   % table �̐}�̏ꍇ�� tab0101
\end{figure}

�����, ���C�������ɑ΂�$\mid K \mid \ll 1$�ł���΃A�_���X�o�b�V���t�H�[�X�X�L�[���͈���ł���.
\end{enumerate}

\newpage

\subsection{�قȂ�X�L�[����g�ݍ��킹�Ďg���ꍇ}
��Ƃ��Ĉȉ��̂悤�Ȏ��������ꍇ���l����.
\begin{equation}\Deqlab{d_shindou}
\DD{U}{t} = i \omega U - kU
\end{equation}
����͌����U���̎��ł���, �U���������Ɩ��C��������g�ݍ��킹�����̂ł���.
�U���������ɑ΂��Ă̓��[�v�t���b�O�X�L�[�����K���邪, ���X�L�[���͖��C�������ɑ΂��Ă͕s�����ł���.
���̂悤�ȏꍇ�ɂ�, �E�ӂ̍����ƂɈقȂ�X�L�[����p����΂悢.
\Deqref{d_shindou}�̗�ł�, �E�ӑ�ꍀ�Ƀ��[�v�t���b�O�X�L�[��, ��񍀂ɃI�C���[�X�L�[����p����Ƃ������@������.
\begin{eqnarray}
\frac{U^{n+1} - U^{n-1}}{2 \Delta t} = i \omega U^n - k U^{n-1} \nonumber \\
U^{n+1} = U^{n-1} + 2 \Delta t \left( i \omega U^n - k U^{n-1} \right)
\end{eqnarray}
\end{document}
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%%%%%%%%            ���� (���x���t��)           %%%%%%%%
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%\begin{eqnarray}
% \Deqlab{1.1} % ���ȏ��ł̎��ԍ�������.
%  \DP{\rho}{t} + \Ddiv (\rho \Dvect{V}) = 0.
%\end{eqnarray}
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%%%%%%%%         �Q�l���� (�{���ɏ����ꍇ)      %%%%%%%%
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%{\bfseries �Q�l����}
%\vspace{-7mm}
%\begin{description}
% \item	���Җ�, 2000:
%	���Ж�, (��, ��).
%	�o�Ŏ�, 319pp.
%\end{description}
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%%%%%%%%             �}�̓\����                 %%%%%%%%
%
%\begin{figure}[hbtp]
% \begin{center}
%  \Depsf[][]{./SEC01/images/fig0101.eps}
% \end{center}
% \caption{
%  ���o��
% }
% \Dfiglab{fig0101} % ���ȏ��̐}�̔ԍ�������,
%                   % table �̐}�̏ꍇ�� tab0101
%\end{figure}
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