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		<div>Розглянемо задачу:<br />
<br />
[0, 1]  розіб'ємо на n частин,  ,<br />
<br />
Розглянемо розклади<br />
<br />
Із (3),(4) одержимо<br />
<br />
(5),(6),(7) – різницеві співвідношення, які апроксимують 1,2 похідну. Використовуючи різницеві співвідношення (5),(6),(7) апроксимуючи оператори L, l одержимо задачу:<br />
<br />
Щоб порівнювати фунції u(h) i u,  f(h) i  f,   введем норму<br />
<br />
Означення 1:<br />
<br />
Оператор Lh (l) апроксимує на функції u оператор L (l) з порядком апроксимації К, якщо  , що має місце:<br />
<br />
Означення 2:<br />
<br />
fh ( ) апроксимує f ( ) з порядком апроксимації К, якщо  , що має місце:<br />
<br />
Означення 3:<br />
<br />
Різницева схема (8), (9) апроксимує крайову задачу (1), (2) з порядком апроксимації к, якщо виконуються умови (10-13).<br />
<br />
Розглянемо апроксимацію оператора крайових або початкових умов.<br />
<br />
Відзначимо, що розв'язок задачі (14-15) задовільняє і ряд тривіальних умов. Наприклад:<br />
<br />
Оператор апроксимуєм різницевим оператором<br />
<br />
Враховуючи, що  , одержимо:<br />
<br />
,  тобто<br />
<br />
Щоб одержати апроксимацію вищого порядку, треба використовувати тривіальні умови<br />
<br />
Використаємо тривіальні умови для визначення  ; із рівняння<br />
<br />
Замість задачі (14), (15) можем розглядати задачу<br />
<br />
тому, що розв'язок задачі (14-15) однозначно визначається умовами (14), (16).<br />
<br />
Для підвищення порядку апроксимації можна користуватись ще формулами:<br />
<br />
Метод прогонки<br />
<br />
Виберемо сітку  х0, х1…хN ,  x0=a,  xn=b,  h=(b-a)/n<br />
<br />
Різницеву схему (3), (4) перетворимо до вигляду:<br />
<br />
Розглянутий метод називається різницевим методом прогонки.<br />
<br />
Якщо різницева задача (5),(6) має вигляд:<br />
<br />
То проводячи аналогічні викладки одержимометод правої прогонки, якщо<br />
<br />
то метод буде стійким до похибки округлень<br />
<br />
Якщо виконуються умови:<br />
<br />
то можна застосовувати метод лівої прогонки, який буде стійкий до похибок заокруглень<br />
<br />
Приклад:   знайти розв'язок задачі<br />
<br />
в точках   хn=0, 1n,   n=0,1,…10<br />
<br />
Розв'язок:<br />
<br />
Практична задача апроксимується на сітці   хn=0, 1n,   n=0,1,…10  різницевою  схемою<br />
<br />
Для розв'язку використовуємо метод правої прогонки<br />
<br />
1)	Знаходимо прогоночні коефіцієнти:<br />
<br />
2)  Значення уn обчислюємо по співвідношенням<br />
<br />
Результати обчислень зручно записувати в таблицю:<br />
<br />
n	xn	zn	yn<br />
<br />
Індивідуальне завдання з книжки “Сборник задач по методах вычислений” под ред. Монас-тырского, ст. 217, 204<br />
<br />
точний розв'язок – y(x) = (1-x)2<br />
<br />
0	-	-	1<br />
<br />
1	0	1	0.81<br />
<br />
2	0.5	0.49	0.64<br />
<br />
3	0.66667	0.31333	0.49<br />
<br />
4	0.75	0.22	0.36<br />
<br />
5	0.8	0.16	0.25<br />
<br />
6	0.83333	0.11667	0.16<br />
<br />
7	0.68714	0.08286	0.09<br />
<br />
8	0.87500	0.05500	0.04<br />
<br />
9	0.88889	0.03111	0.01<br />
<br />
10	0.9	0.0100	0<br />
<br />
До питання про існування розв'язку різницевої схеми<br />
<br />
На сітці  х0, х1…хN ,  x0=a,  xn=b,  h=(b-a)/n  апроксимуємо різницевою схемою<br />
<br />
Різницева схема (3-4) має єдиний розв'язок, якщо відмінний від 0 її детермінант<br />
<br />
Ця умова (7) не зручна для перевірки, тому існування розв'язку доводять використовуючи теорему:<br />
<br />
Теорема (дискретний принцип максимуму)<br />
<br />
Нехай  1) p(x), g(x), f(x) – достатньо гладкі функції;<br />
<br />
2) g(x) 0   на  [a, b]<br />
<br />
3) h настільки мале, що<br />
<br />
Тоді, якщо у внутрішніх точках проміжку [a,b] виконується умова  , то ф-ція uh  не може приймати max додатнього (min від'ємного) значення у внутрішніх точках [a, b], за винятком випадку, коли  u(h)  стала на [a, b].<br />
<br />
Оскільки (3,4) є системою лінійних р-нь, і якщо відповідна тривіальна система має лише тривіальний розв'язок, то різницева схема (3,4) має єдиний розв'язок. Те що однорідна система має лише тривіальний розв'язок доводять від супротивного використовуючи попередню теорему.<br />
<br />
Схему (3,4) можна записати у вигляді<br />
<br />
і її можна розв'язувати ітераційним методом.<br />
<br />
Якщо g(x)<0, то із (6)  , тобто ітераційний процес (8) збіжний.<br />
<br />
Збіжність і стійкість<br />
<br />
апроксимуємо її різницевою<br />
<br />
Означення:<br />
<br />
(3,4) апроксимує (1,2) з порядком апроксимації К, якщо  , що виконуються співвідношення:<br />
<br />
Означення:<br />
<br />
Різницева схема (3-4) стійка, якщо   що<br />
<br />
Теорема.   Якщо різницева схема (3,4) стійка і апроксимує крайову задачу (1),(2) з порядком апроксимації К, то розв'язок різницевої схеми збігається  до розв'язку крайової задачі при  , причому порядок збіжності рівний К.<br />
<br />
Завдання №<br />
<br />
Методом сіток з використанням методу прогонки знайти розв'язки крайових задач в точках  xk=kh,    h=0.1,     k=0,1,…,10</div>
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	width: 55%;
}

/* Категорії */

td.first .x {
	padding-left: 5px;
}

td.last .x {
	margin: 0 5px;
}

td.cat a:link, td.cat a:visited, td.cat a:hover, td.cat a:active	{ 
	color: #3366ff; 
	font-size: 80%;
	display: block;
	margin-left: 10px;
}
td.cat a:link	    { color: #3366ff; }
td.cat a:visited   { color: #3366ff; }
td.cat a:hover     { color: #3333ff;  }
td.cat a:active    { color: #ff6600;  }


td.cat {
	border-right: 1px solid #ccc;
}



/* Новини */

p.nws {
	margin-top: 10px;
}

.nws {
	font-size: 60%;
	
}

/* Сторінки */

div.main {
	margin: 10px;
	font-size: 80%;
}
.t-lt {
	border-right: 1px solid #DEDEDE;
}

.t-lt p, .t-lt ul {
	margin: 0 10px;
}


div.main div.rt {
	float: right;
	width: 240px;
	text-align:center;
}





ul.links  { 
margin-left: 1em; 
margin-top: 0.2em; 
margin-bottom: 0.2em; 
}
li.sq     { 
list-style-type: square; 
margin-bottom: 1em; 
color: #c6c6c6; 
font-size: 15px; 
line-height: 10px;
}
li.sq-new { 
list-style-type: square; 
margin-bottom: 1em; 
color: #ff9900; 
font-size: 15px ;
line-height: 10px;
}

ul.links a:link, ul.links a:visited, ul.links a:hover, ul.links a:active {
	font-size: 10px;
}

.stepnum, .stepnum-ok {
	font-size:250%; 
	font-weight:bold;
}
.stepnum {
	color:#ff9900; 
}
.stepnum-ok {
	color:#66CC00;
}

.error {
	color: red;
}


dl.tbl {
width: 100%;
height: 12em;
padding: 0;
}

dl.tbl dt {

width: 15em;
float: left;
margin: 0 0 0 10%;
padding: .5em;
}
dl.tbl dd {
float: left;
width: 24em;
margin: 0 0 0 0;
padding: .5em;
}

/* Підвал */

div.banbot {
	width: 100%;
	height: 200px;
	margin-top: 10px;
	background-color: #FCFCFC;
	border-top: 1px solid #F0F0F0;
	border-bottom: 1px solid #F0F0F0;
}

div.banbot .x {
	margin:10px 20px 10px 227px;
	width:760px;
	height:60px;	
}



div#foot {

	width: 100%;
}

#foot .lt {
	width: 18%;
	height: 50px;
	text-align: left;
	margin-left: 20px;
	float: left;
}

.cop, .cop a:link, .cop a:visited, .cop a:hover, .cop a:active { color: Gray; }
.cop a:hover, .cop a:active { color: #2C2C2C; }
.cop  { font-size: 60%;  }

.ulinks, .ulinks a:link, .ulinks a:visited, .ulinks a:active { color: Gray; }
.ulinks a:hover {	color: #2C2C2C; }
.ulinks { font-size: 60%; }

#foot .cntr {
	width: 58%;
	height: 50px;
	text-align: center;
	float: left;
}
	
#foot .rt {
	float: right;
	margin-right: 10px;
	text-align: right;
	width: 18%;
	height: 50px;
}

a.m			{ color: #3366ff }
a.m:link	{ color: #3366ff }
a.m:visited { color: #3366ff }
a.m:hover   { color: #3333ff }
a.m:active  { color: #ff9900 }

a.act			{ color: #ff9900 }
a.act:link	    { color: #ff9900 }
a.act:visited   { color: #ff9900 }
a.act:hover     { color: #ffcc00 }
a.act:active    { color: #ff9900 }


a.ref			{ color: #3366ff; text-transform: uppercase; font-weight: bold }
a.ref:link	    { color: #3366ff }
a.ref:visited   { color: #3366ff }
a.ref:hover     { color: #3333ff }
a.ref:active    { color: #ff6600 }

a.sm			{ font-size: 10px }
a.sm:link	    { font-size: 10px }
a.sm:visited    { font-size: 10px }
a.sm:hover      { font-size: 10px }
a.sm:active     { font-size: 10px }
.cntr		{ font-size: 10px; color: #999;}
a.zl		{ color: #999; text-decoration: none;}
span.t-cntr  	{  position:absolute; margin-left: 10px;  }

small   { font-size: 60% }

.inp  { font-size: 12px; font-family: courier, sans-serif}
.but  { font-size: 13px }
.sel  { font-size: 10px }

.sm   { font-size: 8pt }
.ref  { font-size: 80% }
.lim  { font-size: 100%; color: gray; font-weight: bold; 
        font-family: Verdana, Geneva, Arial, Helvetica, sans-serif }
.n    { font-size: 80%; color: gray; font-weight: bold }

.new   { color: #ff9900; font-size: 10px;  text-decoration: none;  }

/* LEX@ */
p.pager a{
	padding-left: 2px;
	padding-right: 2px;
	margin-left: 1px;
	margin-right: 1px;
}
p.pager a.active{
	color: #CCCCCC;
}
div.main div.referats{
	margin-top: -1em;
}
div.main div.referats a, div.main div.referats .ref {
	font-size:100%;
}
div.main div.referats p.cop{
	font-size:80%;
}
.w100p{
	width:99%;
}
div#login ul.errors{
	margin-top:5px;
	padding-left:15px;
}

tr.lcolor{
	background-color: #FFFFFF;
}
tr.gcolor{
	background-color: #FCFCFC;
}
tr.dcolor{
	background-color: #F9F9F9;
}
/*rating*/
#raiting {position:relative; height:16px; cursor:pointer; width:83px; float:left} /* Блок рейтинга*/
#raiting_blank, #raiting_votes, #raiting_hover {height:16px; position:absolute}
#raiting_blank { background:url(/images/ratings.png); width:83px; } /* "Чистые" звездочки */
#raiting_votes {background:url(/images/ratings.png) 0 -16px} /*  Закрашенные звездочки */ 
#raiting_hover {background:url(/images/ratings.png) 0 -32px; display:none}  /*  звездочки при голосовании */ 
#raiting_info {margin-left:100px}





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/*
 * jQuery JavaScript Library v1.3.2
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 * Copyright (c) 2009 John Resig
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 * Date: 2009-02-19 17:34:21 -0500 (Thu, 19 Feb 2009)
 * Revision: 6246
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