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\begin{document}
\chapter{������������̏����l���}
�Â��͒e���v�Z�ɂ͂��܂�,�ߍ��ł̓��P�b�g�̋O���v�Z�܂�,������������̏����l���Ƃ��Ē莮���������ۖ��͂���߂đ���.
�����Ă����̖��̂قƂ�ǂ͉�͓I�ɉ������Ƃ��s�\���邢�͍���Ȃ���,���l��@��p������𓾂Ȃ�.

��q�̒ʂ�Â�������v�����������Ƃ�����,�{�i�I�Ȑ��l��@�̌����͈ꐢ�I�ȏ�̗��j������.
�����ɔ������̋Z�p�����x�ɕ������Ă��邽��,�S�e��m��̂͗e�Ղł͂Ȃ�.

���̈����,�Ƃ肠�����I�C���[�@\footnote{
�I�C���[�@��1�����x�ł���.���Ȃ킿�e�C���[�W�J�̍ۂ�1���̍��ȍ~��؂�̂ĂĂ���}
������Ώ\���ł���Ƃ̍l������,�Ƃ肠���������Q�E�N�b�^�@\footnote{
�����Q�E�N�b�^�@��4�����x�ł���.}
���g���΍����x����������Ƃ����l����,����ɂ͌ÓT�I�ȃ~�����@�����ς���Ȃ��܂ܗp�����Ă�����Ƃ������ʂȌ����������Ƃ����̂��܂������ł���.

������������̏����l���̐��l��@��,���l�ϕ��@�̐e�ʂ݂����Ȃ��̂ł���,���l�ϕ��@����������7�͂ŏq�ׂ�����\footnote{
�������𑝂₵������Ɛςݎc�����o�邱�ƂȂ�.��7�͂͏ڂ������グ�Ă��Ȃ��̂ł���ȏ�̒��ӓ_�͂悭�킩��Ȃ�.
}�͂����ł��ʗp����.

\section{������������̏����l���}
����������
\begin{equation}\label{if1}
\frac{d}{dt} x = f (x,t) ������ (a\leq t\leq b)
\end{equation}
����я�������
\begin{equation}\label{if2}
x(a) = x_0
\end{equation}
��^���Ċ֐�$x(t)$ ($a\leq t\leq b$) �����߂�Ƃ����悤�Ȗ�肪������������̏����l���ł���.

�֐�$f(x,t)$���ςȊ֐�\footnote{
$\delta $�֐����̕s�A���֐�}
�Ŗ����Ȃ��,(\ref{if1})��(\ref{if2})�𖞂����悤�Ȋ֐�$x(t)$����ӂɒ�܂�Ƃ����̂�,������������_�̊�{�I�Ȓ藝�ł���.

�A��\footnote{
�֐�$x$����������悤�Ȕ���������}
������������̏ꍇ����{�I�ȍl�����͈ꏏ�ł���,
\begin{equation}\label{if1_m}
\frac{d}{dt} x^i = f^i (x^1 , x^2 \dots x^m,t) ������ (a\leq t\leq b) (i=1,\dots , m)
\end{equation}
����я�������
\begin{equation}\label{if2_m}
x^i(a) = x^i_0
\end{equation}
��^���Ċ֐�$x^i(t)$ ($a\leq t\leq b$) �����߂�Ƃ����悤�Ȗ��Ƃ��Ă���,������������̏����l���Ƃ��ċA���ł���.

����ɍ��K�̔���������\footnote{
�����K�����̍����܂ޔ���������
}
��,�����K�����̍����̂��̂�ϐ��Ƃ��Ēu�������Ă��܂���,������������̏����l���Ƃ��ċA���ł���.

�����^����ꂽ�������������ō��K�̓��֐��Ɋւ��ĉ������`�ɂȂ��Ă��Ȃ��Ƃ���,���̍ō��K�̓��֐��Ɋւ��āu���l�I�ɉ����v�T�u���[�`����p�ӂ���΂悢\footnote{
$g(x^1 ,x^2 , \dots ,x^m ,)=0$���ō��K�̓��֐��Ɋւ��ĉ����Ȃ��悤�ȗ�Ƃ���,�ȉ��̏ꍇ���l����.
\begin{equation}
\log \left( \frac{dx}{dt} \sin \left(  \frac{d^2 x}{dt^2} x \right) \right) - \frac{d^2 x}{dt^2} - k = 0
\end{equation}
������ō��K(���̗�ł͓�K�̍�)����K�Ƃ���,���ꖢ���̊K��(���̗�ł͈�K�̍�)�𑼂�x�Ɠ��������̕ϐ��Ƃ��Ĉ�����,�ȉ��̂悤�Ɉ�K�̏�����������ɋA���ł���.
\begin{equation}
\log \left( x^2 \sin \left( \frac{dx^2}{dt} x^1 \right) \right) - \frac{dx^2}{dt} - k = 0
������
(x =x^1)
(\frac{dx}{dt} = x^2)
�ƒu���Ă���.
\end{equation}
}
.

\section{Euler �@}
�I�C���[�@�Ƃ͏�����������̏����l�������������Ƃ����n�I�ȉ�@�ł���.
���ݕ��ƌĂ΂���$h$�����ɒu��,�Ɨ��ϐ�$t$��
\begin{equation}
t_n = a + nh (������ n=0,1,2,\dots )
\end{equation}
�Ƃ����悤�ɍ��ݕ����Ƃɂ��炵�Ă������Ƃ���,���̓s�x�e���m�֐��̒l$x^i (t_n)$�̋ߎ��l$x^i _n$��
\begin{eqnarray}\label{euler}
x^i _{n+1} &=& x^i _n + hf^i _n \nonumber \\
f^i _n &\equiv & f^i (x^1 _n , \dots ,x^m _n , t_n )
\end{eqnarray}
�Ƃ��Ď��X�ƒ�߂Ă������@�ł���.
���̕��@�͏�������$x^i (0)$���^�����Ă��邽�߂ɉ\�ł���.

(\ref{euler})��$t=t_n$�ɂ��������������(\ref{if1_m})�̍��ӂ�
\begin{equation}
\frac{dx^i}{dt} \approx \frac{x^i _{n+1} - x^i _n}{t_{n+1} - t_n} = \frac{x^i _{n+1} - x^i _n}{h}
\end{equation}
�Œu�����������̂ł���.

�����ŋ�̗�Ƃ��Ĉȉ��̖����������ꍇ�ɂ��čl����.
\begin{equation}
\frac{dx}{dt} = 1-x^2  , x(0) = 0 , (0 \leq t \leq 1.6)
\end{equation}
���ۂɖ��������Ƃ��ɍ���̂�,���ݕ�$h$�̒l��,����$h$��p���ē���ꂽ���l�����ǂ��܂ŗǂ��ߎ����Ă��邩�Ƃ����_�ł���.
�����ɂ��Ă͊e�_���邪,���ǂ̂Ƃ���$f^i$���ǂ��m��ꂽ�֐��łȂ�����͂悭�킩��Ȃ�.
���H�I�ɂ�,���$[a,b]$�ŉ��_���Ƃ������x�̂��Ȃ�傫�߂�$h$��ݒ肵�Ă���,$h$�̕������X�ɔ����ɂ��Ă����Ȃ���J��Ԃ��������������̂��ǂ�.
.
����������Ă݂��,�ŏ���$h$��$h/2$�̐��l�������Ȃ�H���Ⴄ��,���X�ɂ���l�ɋ߂Â��Ă����l�q��������.

���l���̌덷��,$h$�𔼕��ɂ���Ɩ񔼕��ɂȂ�Ƃ����X��������.
����𓥂܂����,�����_�ȉ�2���܂Ő�������΂悢�ꍇ�͂���Ɠ����I�[�_�[(�����_�ȉ�2��)�̍��ݕ���p�����ꍇ�̉����̗p����΂悢\footnote{
���̍��ݕ���p�������̉��̌덷��,���̎��̉��Ƃ��̑O�̍��ݕ�(�����_�ȉ�1��)�ɂ����̍����x�ƂȂ�}
.
�܂�,���̐����̓I�C���[�@���������@�\���Ă��邩�ǂ����̎w�W�ɂ��Ȃ�.

���̂悤�ɑ����̍��ݕ��ɂ��v�Z�����݂Ȃ���΂Ȃ炸,�������ŏI�I�ȉ������߂�v�Z�ȊO�͎������ʂł���.
�������Ȃ��炱�́u���ʁv�̗ʂ����ς����Ă݂��,���ǂ͂��������u�L���Ȏ�ԁv�Ɠ������x�ł���\footnote{
$ ���鍏�ݕ��ŉ������߂鎞�̌v�Z�� = 2 \times (���̔����̍��ݕ��ŉ������߂�v�Z) $������ł���.
}.
���̂悤��,��������2�{�̌v�Z�ʂŌ��ʂɑ΂��鎩�M�ƕۏ؂Ƃ�������ƍl�����,���قǖ��ʂ͑傫���Ȃ��Ƒ����邱�Ƃ��ł���.
�����������̐��l���Ɏ��X�����u�s���茻�ہv\footnote{
���̔��U�̂��Ƃł���}
��u���e���v\footnote{
3��t��p���Čv�Z����Əo�Ă���v�Z��̉��̂��Ƃł���}
�Ƃ���,���̂悤�ɒ��߂�΂����ɔ����ł���.

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% \Deqlab{1.1} % ���ȏ��ł̎��ԍ�������.
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%%%%%%%%         �Q�l���� (�{���ɏ����ꍇ)      %%%%%%%%
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%{\bfseries �Q�l����}
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% \item	���Җ�, 2000:
%	���Ж�, (��, ��).
%	�o�Ŏ�, 319pp.
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