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<H1><A NAME="SECTION00040000000000000000">4. Метод Рунге-Кутта первого порядка аппроксимации
               (метод Эйлера) </A></H1>
<P>
При <I>m</I>=1 имеем метод Эйлера 
<BR><A NAME="41"></A><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation317" SRC="img117.gif"><BR>
<BR><IMG WIDTH=406 HEIGHT=22 ALIGN=BOTTOM ALT="displaymath1875" SRC="img118.gif"><BR> 
Оценим невязку 
<BR><A NAME="42"></A><IMG WIDTH=500 HEIGHT=36 ALIGN=BOTTOM ALT="equation323" SRC="img119.gif"><BR>
Предположим, что выполнено
<P>
<I>Условие 4.1</I>. Решение <I>x</I>(<I>t</I>) системы (<A HREF="node1.html#11">1.1</A>) дифференцируемо 
на <IMG WIDTH=85 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1879" SRC="img120.gif">.
<P>
Тогда 
<BR><A NAME="43"></A><IMG WIDTH=500 HEIGHT=20 ALIGN=BOTTOM ALT="equation332" SRC="img121.gif"><BR>
причем 
<IMG WIDTH=114 HEIGHT=57 ALIGN=MIDDLE ALT="tex2html_wrap_inline1881" SRC="img122.gif">.
<P>
Подставляя (<A HREF="node4.html#43">4.3</A>) в (<A HREF="node4.html#42">4.2</A>) получаем
<BR><A NAME="44"></A><IMG WIDTH=500 HEIGHT=42 ALIGN=BOTTOM ALT="equation342" SRC="img123.gif"><BR>
откуда следует сходимость метода Эйлера.
<P>
Условие 4.1 выполняется, если выполняется одно из следующих условий.
<P>
<I>Условие 4.1.A</I>. Отображение <IMG WIDTH=92 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1673" SRC="img19.gif"> в правой части системы 
(<A HREF="node1.html#11">1.1</A>) <I>инвариантно непрерывно</I> (или <I>коинвариантно непрерывно</I>) 
в области определения [<A HREF="node13.html#Kim1">10</A>, c. 43,].
<P>
Порядок аппроксимации метода Эйлера можно уточнить, если выполнено
<P>
<I>Условие 4.2</I>. Решение <I>x</I>(<I>t</I>) системы (<A HREF="node1.html#11">1.1</A>) имеет ограниченную 
вторую производную на <IMG WIDTH=85 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1879" SRC="img120.gif">, т.е. существует константа 
<I>C</I> &gt; 0 для которой <IMG WIDTH=109 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1891" SRC="img124.gif"> при <IMG WIDTH=117 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1893" SRC="img125.gif">.
<P>
В этом случае разложение (<A HREF="node4.html#43">4.3</A>) имеет вид 
<BR><IMG WIDTH=496 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath1895" SRC="img126.gif"><BR>
откуда следует оценка <IMG WIDTH=86 HEIGHT=35 ALIGN=MIDDLE ALT="tex2html_wrap_inline1897" SRC="img127.gif">, 
т.е. метод Эйлера имеет первый порядок аппроксимации.
<P>
Условие 4.2 выполняется, если выполняется одно из следующих условий
<P>
<I>Условие 4.2.A</I>. Отображение <IMG WIDTH=92 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1673" SRC="img19.gif"> <BR> 
a) <I>инвариантно дифференцируемо</I> [<A HREF="node13.html#Kim1">10</A>, c. 43,], причем производные 
   <IMG WIDTH=24 HEIGHT=55 ALIGN=MIDDLE ALT="tex2html_wrap_inline1901" SRC="img128.gif">,<I>grad</I><I>f</I>, <IMG WIDTH=26 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1905" SRC="img129.gif"> 
   ограничены в области определения; <BR> 
b) <I>коинвариантно дифференцируемо</I> [<A HREF="node13.html#Kim1">10</A>, c. 45,], причем производные 
   <IMG WIDTH=24 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline1907" SRC="img130.gif"> и <I>grad</I><I>f</I> ограничены в области определения.
<P>
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