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Ðàçíîñòíàÿàïïðîêñèìàöèÿíà÷àëüíî-êðàåâîéçàäà÷è
äëÿóðàâíåíèÿòåïëîïðîâîäíîñòè.Ïîíÿòèåÿâíîéè
íåÿâíîéñõåìû.
1Ðàçíîñòíàÿàïïðîêñèìàöèÿóðàâíåíèÿòåïëîïðîâîä-
íîñòè
Ðàññìîòðèìðàçëè÷íûåâàðèàíòûðàçíîñòíîéàïïðîêñèìàöèèëèíåéíîãîîäíîìåðíîãî
ïîïðîñòðàíñòâóóðàâíåíèÿòåïëîïðîâîäíîñòè:
@u
@t
=
@
2
u
@x
2
+
f
(
x;t
)
;x
2
(0
;l
)
;t
2
(0
;T
]
;
(1.1)
ãäå
T�
0
íåêîòîðàÿêîíñòàíòà.
Ââåäåìâîáëàñòè
D
=
f
0
6
x
6
l;
0
6
t
6
T
g
ðàâíîìåðíóþñåòêóñøàãîì
h
ïîêîîð-
äèíàòåèøàãîì

ïîâðåìåíè:
x
i
=
ih;i
=0
;
1
;:::;N;hN
=
l
;
t
j
=
j;j
=0
;
1
;:::;M;M
=
T:
Óðàâíåíèå(1.1)ñîäåðæèòêàêïðîèçâîäíûåïîïðîñòðàíñòâåííîéïåðåìåííîé
x
,òàêè
ïîâðåìåíè
t
,ïîýòîìóäëÿïîñòðîåíèÿåãîðàçíîñòíîéàïïðîêñèìàöèèïðèäåòñÿèñïîëü-
çîâàòüóçëûñåòêè,ñîîòâåòñòâóþùèåðàçëè÷íûì
j
.Âñåóçëûñåòêè,îòâå÷àþùèåôèêñè-
ðîâàííîìó
j
,íàçûâàþò
j
-ìâðåìåííûìñëîåì
.Ñâîéñòâàðàçíîñòíûõñõåìäëÿóðàâíåíèÿ
(1.1)çàâèñÿòîòòîãî,íàêàêîìñëîå
j
ïîâðåìåíèàïïðîêñèìèðóåòñÿâûðàæåíèå
@
2
u
@x
2
.Ðàñ-
ñìîòðèìâîçìîæíûåâàðèàíòû.
Âàðèàíò1:ÿâíàÿñõåìà.
Äëÿàïïðîêñèìàöèèîïåðàòîðà
L
=
@
@t

@
2
@x
2
âóðàâíåíèè(1.1)èñïîëüçóåìøàáëîí,
ïðèâåäåííûéíàðèñ.1.
1
Ðèñ.1:Øàáëîíÿâíîéñõåìûäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè.
Ñîîòâåòñòâóþùèéðàçíîñòíûéîïåðàòîð
L
(0)
h
u
èìååòâèä:
L
(0)
h
u
=
u
(
x;t
+

)

u
(
x;t
)


u
(
x
+
h;t
)

2
u
(
x;t
)+
u
(
x

h;t
)
h
2
:
Äàëååäëÿêðàòêîñòèáóäåìèñïîëüçîâàòüñëåäóþùèåñòàíäàðòíûåîáîçíà÷åíèÿ:
u
=
u
(
x;t
);^
u
=
u
(
x;t
+

)
:
Òîãäà:
u
t
=
^
u

u

;L
(0)
h
u
=
u
t

u

xx
:
Íàéäåìïîãðåøíîñòüàïïðîêñèìàöèèðàçíîñòíûìîïåðàòîðîì
L
(0)
h
èñõîäíîãîäèôôå-
ðåíöèàëüíîãîîïåðàòîðà
L
âòî÷êå
(
x;t
)
.Âñëó÷àåäîñòàòî÷íîãëàäêîéôóíêöèè
u
(
x;t
)
ïðèäîñòàòî÷íîìàëûõøàãàõ
h
è

èìååì:
u
t
=
u
(
x;t
+

)

u
(
x;t
)

=
@u
(
x;t
)
@t
+
O
(

)
;
(1.2)
u

xx
=
@
2
u
(
x;t
)
@x
2
+
O
(
h
2
)
:
(1.3)
Ñëåäîâàòåëüíî,ðàçíîñòíûéîïåðàòîð
L
(0)
h
àïïðîêñèìèðóåòäèôôåðåíöèàëüíûéîïåðà-
òîð
L
ñïîãðåøíîñòüþ
O
(

+
h
2
)
âòî÷êå
(
x;t
)
:
L
(0)
h
u
=
@u
(
x;t
)
@t

@
2
u
(
x;t
)
@x
2
|
{z
}
L
[
u
(
x;t
)]
+
O
(

+
h
2
)
:
Ââåäåìñåòî÷íóþôóíêöèþ
'
=
'
(
x
i
;t
j
)
,àïïðîêñèìèðóþùóþïðàâóþ÷àñòü
f
(
x;t
)
óðàâíåíèÿ(1.1)íàâñåõâíóòðåííèõóçëàõ
(
x
i
;t
j
)
ñåòêèñïîãðåøíîñòüþ
O
(

+
h
2
)
.合֌-
ñòâå
'
ìîæíîâçÿòü,íàïðèìåð
'
(
x
i
;t
j
)=
f
(
x
i
;t
j
)
.Òîãäàðàçíîñòíîåóðàâíåíèå
L
(0)
h
y
=
'
áóäåòàïïðîêñèìèðîâàòüèñõîäíîåäèôôåðåíöèàëüíîåóðàâíåíèåòåïëîïðîâîäíîñòè(1.1)ñ
ïåðâûìïîðÿäêîìïîãðåøíîñòèïî

èâòîðûìïî
h
.
2
Âàðèàíò2.×èñòîíåÿâíàÿñõåìà.
Èñïîëüçóåìäëÿàïïðîêñèìàöèèîïåðàòîðà
L
=
@
@t

@
2
@x
2
âóðàâíåíèè(1.1)øàáëîí,
ïðèâåäåííûéíàðèñ.2.
Ðèñ.2:Øàáëîííåÿâíîéñõåìûäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè.
Òîãäàðàçíîñòíàÿàïïðîêñèìàöèÿîïåðàòîðà
L
óðàâíåíèÿòåïëîïðîâîäíîñòèáóäåòâû-
ãëÿäåòüñëåäóþùèìîáðàçîì:
L
(1)
h
u
=
u
(
x;t
+

)

u
(
x;t
)


u
(
x
+
h;t
+

)

2
u
(
x;t
+

)+
u
(
x

h;t
+

)
h
2
=
u
t

^
u

xx
:
Ðàññìîòðèìïîãðåøíîñòüàïïðîêñèìàöèèðàçíîñòíûìîïåðàòîðîì
L
(1)
h
èñõîäíîãîäèô-
ôåðåíöèàëüíîãîîïåðàòîðà
L
âòî÷êàõ
(
x;t
)
,
(
x;t
+

)
.Òàêêàêäëÿäîñòàòî÷íîãëàäêîé
ôóíêöèè
u
(
x;t
)
ñïðàâåäëèâûðàâåíñòâà
^
u

xx
=
@
2
u
(
x;t
+

)
@x
2
+
O
(
h
2
)=
@
2
u
(
x;t
)
@x
2
+
O
(

+
h
2
)
;
(1.4)
òîñó÷åòîì(1.2)ïîëó÷àåì,÷òîîïåðàòîð
L
(1)
h
àïïðîêñèìèðóåòäèôôåðåíöèàëüíûéîïåðà-
òîð
L
âóðàâíåíèè(1.1)ñïîãðåøíîñòüþ
O
(

+
h
2
)
âòî÷êàõ
(
x;t
)
è
(
x;t
+

)
:
L
(1)
h
u
=
@u
(
x;t
)
@t
+
O
(

)

@
2
u
(
x;t
)
@x
2
+
O
(

+
h
2
)=
@u
(
x;t
)
@t

@
2
u
(
x;t
)
@x
2
|
{z
}
L
[
u
(
x;t
)]
+
O
(

+
h
2
)
;
L
(1)
h
u
=
@u
(
x;t
+

)
@t
+
O
(

)

@
2
u
(
x;t
+

)
@x
2
+
O
(
h
2
)=
@u
(
x;t
+

)
@t

@
2
u
(
x;t
+

)
@x
2
|
{z
}
L
[
u
(
x;t
+

)]
+
O
(

+
h
2
)
:
Áåðÿâêà÷åñòâåñåòî÷íîéàïïðîêñèìàöèèïðàâîé÷àñòèóðàâíåíèÿ(1.1),íàïðèìåð,
ôóíêöèþ
'
(
x
i
;t
j
)=
f
(
x
i
;t
j
+1
)
,ïîëó÷èìðàçíîñòíîåóðàâíåíèå
L
(1)
h
y
=
';
àïïðîêñèìèðóþùåå(1.1)ñïîãðåøíîñòüþ
O
(

+
h
2
)
.
3
Âàðèàíò3.Íåÿâíàÿñõåìàñâåñàìè.
Èñïîëüçóåìøàáëîí,ïðèâåäåííûéíàðèñ.3,èëèíåéíóþêîìáèíàöèþîïåðàòîðîâ
L
(0)
h
è
L
(1)
h
äëÿàïïðîêñèìàöèèäèôôåðåíöèàëüíîãîîïåðàòîðà
L
:
L
(

)
h
u
=
L
(1)
h
u
+(1


)
L
(0)
h
u
=
u
t


^
u

xx
+(1


)
u
t

(1


)
u

xx
=
u
t

(

^
u

xx
+(1


)
u

xx
)
;
ãäå

2
(0
;
1)
.
Ðèñ.3:Øàáëîííåÿâíîéñõåìûñâåñàìèäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè.
Ïîëüçóÿñüðàâåíñòâàìè(1.2),(1.3)è(1.4),ïîëó÷àåì,÷òîîïåðàòîð
L
(

)
h
àïïðîêñèìèðóåò
èñõîäíûéäèôôåðåíöèàëüíûéîïåðàòîð
L
ñïîãðåøíîñòüþ
O
(

+
h
2
)
âòî÷êàõ
(
x;t
)
,
(
x;t
+

)
ïðèëþáîì

.
Ïîîïðåäåëåíèþïîãðåøíîñòü

(
x;t
)=
L
(

)
h
u

Lu
(1.5)
àïïðîêñèìàöèèâûðàæåíèÿ
Lu
ðàçíîñòíûìâûðàæåíèåì
L
(

)
h
u
ìîæåòâû÷èñëÿòüñÿâëþáîé
òî÷êå
(
x;t
)
,àíåîáÿçàòåëüíîâêàêîì-ëèáîóçëåñåòêè,òàêêàêâñîîòíîøåíèè(1.5)ôóíêöèÿ
u
(
x;t
)
ýòîïðîèçâîëüíàÿäîñòàòî÷íîãëàäêàÿôóíêöèÿíåïðåðûâíûõàðãóìåíòîâ
x
è
t
.
Ïîýòîìóðàññìîòðèìïîãðåøíîñòüàïïðîêñèìàöèèîïåðàòîðîì
L
(

)
h
äèôôåðåíöèàëüíîãî
îïåðàòîðà
L
âöåíòðàëüíîéòî÷êå
(
x;t
+0
:
5

)
øàáëîíà,ïðèâåäåííîãîíàðèñ.3.Ïîëüçóÿñü
äëÿäîñòàòî÷íîãëàäêîéôóíêöèè
u
(
x;t
)
ðàçëîæåíèåìâðÿäÒåéëîðàâîêðåñòíîñòèòî÷êè
(
x;t
+0
:
5

)
,ïðèìàëûõ

è
h
ïîëó÷àåì:
u
t
=
u
(
x;t
+

)

u
(
x;t
)

=
@u
@t




(
x;t
+0
:
5

)
+
O
(

2
)
;
^
u

xx
=
@
2
u
@x
2




(
x;t
+

)
+
O
(
h
2
)=
@
2
u
@x
2




(
x;t
+0
:
5

)
+

2
@
3
u
@[email protected]
2




(
x;t
+0
:
5

)
+
O
(

2
+
h
2
)
;
u

xx
=
@
2
u
@x
2




(
x;t
)
+
O
(
h
2
)=
@
2
u
@x
2




(
x;t
+0
:
5

)


2
@
3
u
@[email protected]
2




(
x;t
+0
:
5

)
+
O
(

2
+
h
2
)
:
4
Ñëåäîâàòåëüíî,ïðè

=0
:
5
âòî÷êå
(
x;t
+0
:
5

)
îïåðàòîð
L
(0
:
5)
h
âñèëóñâîåéñèììåòðèè
àïïðîêñèìèðóåò
L
ñîâòîðûìïîðÿäêîìïîãðåøíîñòèàïïðîêñèìàöèèïî

è
h
:
L
(

)
h
u
=
@u
(
x;t
+

2
)
@t

@
2
u
(
x;t
+

2
)
@x
2
|
{z
}
L
[
u
(
x;t
+

2
)]


2
(2


1)
|
{z
}
0
ïðè

=0
:
5
@
3
u
(
x;t
+

2
)
@x
2
@t
+
O
(

2
+
h
2
)
:
Äëÿòîãî,÷òîáûïîëó÷èòüðàçíîñòíîåóðàâíåíèå,àïïðîêñèìèðóþùååäèôôåðåíöèàëü-
íîåóðàâíåíèå
@u
@t
=
@
2
u
@x
2
+
f
(
x;t
)
ñïîãðåøíîñòüþ
O
(

2
+
h
2
)
âòî÷êå
(
x;t
+

2
)
,äîñòàòî÷íîâçÿòüâêà÷åñòâåñåòî÷íîéàï-
ïðîêñèìàöèèïðàâîé÷àñòè
f
(
x;t
)
ýòîãîóðàâíåíèÿôóíêöèþ
'
(
x
i
;t
j
)=
f
(
x
i
;t
j
+0
:
5

)
.
Èòàê,ðàçíîñòíîåóðàâíåíèå
L
(0
:
5)
h
y
=
';
ãäå
'
(
x
i
;t
j
)=
f
(
x
i
;t
j
+0
:
5

)
,àïïðîêñèìèðóåòóðàâíåíèå(1.1)ñîâòîðûìïîðÿäêîìïî-
ãðåøíîñòèàïïðîêñèìàöèèïî

è
h
.
2Ðåàëèçàöèÿÿâíîé,íåÿâíîéèñèììåòðè÷íîéðàçíîñò-
íûõñõåìäëÿíà÷àëüíî-êðàåâîéçàäà÷èäëÿóðàâíåíèÿ
òåïëîïðîâîäíîñòèíàîòðåçêå.
Ïðèìåð2.1.
Ïîñòðîéòåÿâíóþðàçíîñòíóþñõåìóäëÿñëåäóþùåéíà÷àëüíî-êðàåâîéçà-
äà÷èíàîòðåçêå
x
2
[0
;
1]
:
8















:
@u
@t
=
@
2
u
@x
2
+
x;
0
x
1
;
0
t
6
1
;
u
(
x;
0)=sin

3
x
2

;
u
(0
;t
)=0
;
@u
@x




x
=1
=
t:
(2.1)
Ñðàâíèòå÷èñëåííîåðåøåíèåñàíàëèòè÷åñêèìèèññëåäóéòåçàâèñèìîñòüïîãðåøíîñòè
îòøàãîâñåòêè.Ïðè÷èñëåííîìðåøåíèèñîáëþäàéòåóñëîâèåóñòîé÷èâîñòèÿâíîéñõå-
ìû:

6
h
2
=
2
.
Ðåøåíèå.
Ïðåæäåâñåãî,íàéäåìàíàëèòè÷åñêîåðåøåíèåçàäà÷è(2.1).Âñèëóååëèíåé-
íîñòèðåøåíèåìîæíîèñêàòüââèäå
u
(
x;t
)=
v
(
x;t
)+
xt
,ãäåôóíêöèÿ
v
(
x;t
)
óäîâëåòâîðÿåò
5
çàäà÷åñîäíîðîäíûìèãðàíè÷íûìèóñëîâèÿìè:
8















:
@v
@t
=
@
2
v
@x
2
;
0
x
1
;
0
t
6
1
;
v
(
x;
0)=sin

3
x
2

;
v
(0
;t
)=0
;
@v
@x




x
=1
=0
:
Èñïîëüçóÿìåòîäðàçäåëåíèÿïåðåìåííûõ,ïîëó÷àåì:
v
=
e

(3
=
2)
2
t
sin(3
x=
2)
.Ñëåäîâà-
òåëüíî,àíàëèòè÷åñêîåðåøåíèåçàäà÷è(2.1)èìååòâèä:
u
(
x;t
)=
xt
+
e

(
3

2
)
2
t

sin

3
x
2

:
Äëÿòîãî,÷òîáûïîëó÷èòü÷èñëåííîåðåøåíèå,ââåäåìâðàñ÷åòíîéîáëàñòèðàâíîìåð-
íóþñåòêó:
x
i
=
ih;i
=0
;
1
;:::;N;hN
=1
;t
j
=
j;j
=0
;
1
;:::;M;M
=1
;
èáóäåìäëÿêðàòêîñòèèñïîëüçîâàòüîáîçíà÷åíèÿ
u
j
i
=
u
(
x
i
;t
j
)
,
y
j
i
=
y
(
x
i
;t
j
)
.
Ïîñòðîèìðàçíîñòíóþàïïðîêñèìàöèþóðàâíåíèÿâñîîòâåòñòâèèñÿâíîéñõåìîé:
y
j
+1
i

y
j
i

=
y
j
i

1

2
y
j
i
+
y
j
i
+1
h
2
+
x
i
;i
=1
;
2
;:::;N

1
;j
=0
;
1
;:::;M

1
:
(2.2)
Ýòîðàçíîñòíîåóðàâíåíèåíåîáõîäèìîäîïîëíèòüñîîòâåòñòâóþùèìèíà÷àëüíûìèè
ãðàíè÷íûìèóñëîâèÿìèíàñåòêå.Íà÷àëüíîåóñëîâèåèãðàíè÷íîåóñëîâèåÄèðèõëåïðè
x
=0
àïïðîêñèìèðóþòñÿòî÷íî:
y
0
i
=sin

3
x
i
2

;i
=0
;
1
;:::;N;
y
j
0
=0
;j
=0
;
1
;:::;M:
Ãðàíè÷íîåóñëîâèåïðè
x
=1
ñîäåðæèòïðîèçâîäíóþ
@u
@x
.Åñëèååïðîñòîçàìåíèòü
îäíîñòîðîííåéðàçíîñòíîéïðîèçâîäíîé,òîóðàâíåíèå
y
j
N

y
j
N

1
h
=
t
j
;j
=0
;
1
;:::;M;
(2.3)
áóäåòàïïðîêñèìèðîâàòüñîîòâåòñòâóþùååãðàíè÷íîåóñëîâèåñïåðâûìïîðÿäêîìïîãðåø-
íîñòèàïïðîêñèìàöèè.Ýòîîçíà÷àåò,÷òîèäëÿâñåéðàçíîñòíîéñõåìûïîðÿäîêïîãðåø-
íîñòèàïïðîêñèìàöèèïî
h
áóäåòïåðâûì.Íàïîìíèì,÷òîðàçíîñòíîåóðàâíåíèå(2.2)àï-
ïðîêñèìèðîâàëîäèôôåðåíöèàëüíîåóðàâíåíèåâçàäà÷å(2.1)ñïîãðåøíîñòüþ
O
(

+
h
2
)
.
6
Èòàê,ïåðâûéâàðèàíòÿâíîéðàçíîñòíîéñõåìûäëÿçàäà÷è(2.1),îáëàäàþùåéïîãðåø-
íîñòüþàïïðîêñèìàöèè
O
(

+
h
)
,èìååòâèä:
8



















:
y
j
+1
i

y
j
i

=
y
j
i

1

2
y
j
i
+
y
j
i
+1
h
2
+
x
i
;i
=1
;
2
;:::;N

1
;j
=0
;
1
;:::;M

1
;
y
0
i
=sin

3
x
i
2

;i
=0
;
1
;:::;N;
y
j
0
=0
;
y
j
N

y
j
N

1
h
=
t
j
;j
=0
;
1
;:::;M:
(2.4)
Ðàññìîòðèìàëãîðèòìðåøåíèÿñèñòåìû(2.4).Ïðè
j
=0
çíà÷åíèÿ
y
0
i
èçâåñòíûèç
íà÷àëüíîãîóñëîâèÿ,àçíà÷åíèÿ
y
1
i
íåèçâåñòíûèäîëæíûáûòüíàéäåíûäëÿâñåõ
i
=
0
;
1
;:::;N
.Êîãäàíàéäåíûâñåçíà÷åíèÿ
y
1
i
,íóæíîíàéòè
y
2
i
èò.ä.Ñëåäîâàòåëüíî,ïðèêàæ-
äîìôèêñèðîâàííîì
j
=0
;
1
;:::;M

1
íåèçâåñòíûìèÿâëÿþòñÿçíà÷åíèÿ
y
j
+1
i
.Íàéòèèõ
ìîæíîñëåäóþùèìîáðàçîì:
1)ïðè
i
=1
;
2
;:::;N

1
èçïåðâîãîóðàâíåíèÿñèñòåìû(2.4)íàõîäèì
y
j
+1
i
=
y
j
i
+

h
2

y
j
i
+1

2
y
j
i
+
y
j
i

1

+
x
i
;
2)ïðè
i
=0
è
i
=
N
ïîëüçóåìñÿãðàíè÷íûìèóñëîâèÿìè,ó÷èòûâàÿ,÷òî
y
j
+1
1
è
y
j
+1
N

1
óæå
èçâåñòíû:
y
j
+1
0
=0
;y
j
+1
N
=
y
j
+1
N

1
+
h

t
j
+1
;
3)ïåðåõîäèìíàíîâûéñëîéïîâðåìåíè,óâåëè÷èâàÿ
j
íàåäèíèöóèïîâòîðÿåìäåéñòâèÿ
1)è2).
Íàðèñ.4-6ïðèâåäåíûàíàëèòè÷åñêîåðåøåíèåçàäà÷è2.1,ðåçóëüòàòûåå÷èñëåííîãî
ðåøåíèÿïîñõåìå(2.4)äëÿ
N
=50
è
M
=10+2
N
2
(÷èñëî
M
ïîäîáðàíîòàê,÷òîáû
óñëîâèåóñòîé÷èâîñòèÿâíîéñõåìûâûïîëíÿëîñü)èïîãðåøíîñòü÷èñëåííîãîðåøåíèÿ.
Ðèñ.4:Àíàëèòè÷åñêîåðåøåíèåçàäà÷è(2.1).
7
Ðèñ.5:×èñëåííîåðåøåíèåçàäà÷è(2.1)ñïîìîùüþÿâíîéñõåìû(2.4).
Ðèñ.6:Ïîãðåøíîñòü÷èñëåííîãîðåøåíèÿçàäà÷è(2.1)ñïîìîùüþÿâíîéñõåìû(2.4).
Åñëèìûõîòèì,÷òîáûÿâíàÿñõåìààïïðîêñèìèðîâàëàèñõîäíóþçàäà÷óñïîãðåøíî-
ñòüþ
O
(

+
h
2
)
,òîìîæíîèñïîëüçîâàòüòîòæåïðèåì,êîòîðûéïðèìåíÿëñÿðàíååäëÿàï-
ïðîêñèìàöèèãðàíè÷íîãîóñëîâèÿ,ñîäåðæàùåãîïðîèçâîäíóþ,âêðàåâîéçàäà÷åäëÿîáûê-
íîâåííîãîäèôôåðåíöèàëüíîãîóðàâíåíèÿíàîòðåçêå.Ïóñòü
u
(
x;t
)
ðåøåíèåçàäà÷è(2.1).
Ðàññìîòðèìâûðàæåíèå:
u

x
=
u
(
x;t
)

u
(
x

h;t
)
h
=
@u
(
x;t
)
@x

h
2
@
2
u
(
x;t
)
@x
2
+
O
(
h
2
)=
=
@u
(
x;t
)
@x

h
2

@u
(
x;t
)
@t

x

+
O
(
h
2
)
:
Çàìåíÿÿâíåìïðîèçâîäíóþ
@u
@t
êîíå÷íîéðàçíîñòüþèó÷èòûâàÿ,÷òî
@u
(
x;t
)
@t
=
u
(
x;t
)

u
(
x;t


)

+
O
(

)
;
ïîëó÷èì
u
(
x;t
)

u
(
x

h;t
)
h
=
@u
(
x;t
)
@x

h
2

u
(
x;t
)

u
(
x;t


)


x

+
O
(

+
h
2
)
:
8
Ïåðåõîäÿâïîëó÷åííîìðàâåíñòâåêïðåäåëóïðè
x
!
1
èó÷èòûâàÿ,÷òîïîóñëîâèþ
@u
@x




x
=1
=
t;
íàõîäèì,÷òîïðè
t
=
t
j
+1
èìååòìåñòîðàâåíñòâî:
u
j
+1
N

u
j
+1
N

1
h
=
t
j
+1

h
2

u
j
+1
N

u
j
N


1
!
+
O
(

+
h
2
)
:
Ñëåäîâàòåëüíî,ðàçíîñòíîåóðàâíåíèå
y
j
+1
N

y
j
+1
N

1
h
=
t
j
+1

h
2

y
j
+1
N

y
j
N


1
!
(2.5)
àïïðîêñèìèðóåòãðàíè÷íîåóñëîâèåÍåéìàíàïðè
x
=1
ñïîãðåøíîñòüþ
O
(

+
h
2
)
.Òàêèì
îáðàçîì,ìåíÿÿâñõåìå(2.4)óðàâíåíèå(2.3)íà(2.5),ìûïîëó÷èìñõåìó,àïïðîêñèìèðóþ-
ùóþèñõîäíóþçàäà÷óíàååðåøåíèèñïîãðåøíîñòüþ
O
(

+
h
2
)
.
Óðàâíåíèå(2.5)óäîáíîïåðåïèñàòüââèäå:
y
j
+1
N
=

1+
h
2
2



1

y
j
+1
N

1
+
ht
j
+1
+
h
2
2

1+
y
j
N

!!
;j
=0
;
1
;:::;M

1
;
èèñïîëüçîâàòüïðèóæåíàéäåííûõ
y
j
+1
N

1
,
y
j
N
äëÿçàâåðøåíèÿïåðåõîäàíàñëîé
j
+1
.
Ðåçóëüòàòûðàñ÷åòîâïîñîîòâåòñòâóþùåéÿâíîéñõåìåíàòîéæåñåòêå,÷òîèâïðåäû-
äóùåìñëó÷àå,ïðèâåäåíûíàðèñ.7-8.
Ðèñ.7:×èñëåííîåðåøåíèåçàäà÷èñïîìîùüþÿâíîéñõåìûñãðàíè÷íûìóñëîâèåì(2.5).
Òàêæåäëÿïîëó÷åíèÿñõåìû,èìåþùåéïîãðåøíîñòüàïïðîêñèìàöèè
O
(

+
h
2
)
,ìîæíî
àïïðîêñèìèðîâàòüãðàíè÷íîåóñëîâèåÍåéìàíàïðè
x
=1
ñïîìîùüþòðåõòî÷å÷íîéïåðâîé
ðàçíîñòíîéïðîèçâîäíîé:
3
y
j
+1
N

4
y
j
+1
N

1
+
y
j
+1
N

2
2
h
=
t
j
+1
;j
=0
;
1
;:::;M

1
:
9
Ðèñ.8:Ïîãðåøíîñòüðåøåíèÿçàäà÷èñïîìîùüþÿâíîéñõåìûñãðàíè÷íûìóñëîâèåì(2.5).
Ïåðåïèñûâàÿýòîóðàâíåíèåââèäå
y
j
+1
N
=
4
3
y
j
+1
N

1

1
3
y
j
+1
N

2
+
2
ht
j
+1
3
;
(2.6)
ìûìîæåìèñïîëüçîâàòüåãîäëÿçàâåðøåíèÿïåðåõîäàíàñëîé
j
+1
ïðèóæåíàéäåííûõ
y
j
+1
N

1
è
y
j
+1
N

2
.
Ïîãðåøíîñòüâû÷èñëåíèéïîÿâíîéñõåìåñóñëîâèåì(2.6)ïðèâåäåíàíàðèñ.9.
Ðèñ.9:Ïîãðåøíîñòüðåøåíèÿçàäà÷èñïîìîùüþÿâíîéñõåìûñãðàíè÷íûìóñëîâèåì(2.6).
Ïðèìåð2.2.
Ïîñòðîéòå÷èñòîíåÿâíóþðàçíîñòíóþñõåìóäëÿíà÷àëüíî-êðàåâîéçà-
äà÷è(2.1).Ñðàâíèòå÷èñëåííîåðåøåíèåñàíàëèòè÷åñêèìèèññëåäóéòåçàâèñèìîñòü
ïîãðåøíîñòèîòøàãîâñåòêè.
Ðåøåíèå.
Èñïîëüçóåìòóæåñåòêó,÷òîèâïðåäûäóùåìïðèìåðåñòîéëèøüðàçíè-
öåé,÷òîñîîòíîøåíèåøàãîâ

è
h
òåïåðüìîæåòáûòüëþáûì.Ðàçíîñòíàÿàïïðîêñèìàöèÿ
óðàâíåíèÿâñîîòâåòñòâèèñíåÿâíîéñõåìîéèìååòâèä:
y
j
+1
i

y
j
i

=
y
j
+1
i

1

2
y
j
+1
i
+
y
j
+1
i
+1
h
2
+
x
i
;i
=1
;
2
;:::;N

1
;j
=0
;
1
;:::;M

1
:
(2.7)
10
Äîïîëíèìðàçíîñòíîåóðàâíåíèå(2.7)íà÷àëüíûìèèãðàíè÷íûìèóñëîâèÿìèíàñåòêå.
Êàêèâñëó÷àåÿâíîéñõåìû,íà÷àëüíîåóñëîâèåèãðàíè÷íîåóñëîâèåÄèðèõëåïðè
x
=0
àïïðîêñèìèðóþòñÿòî÷íî:
y
0
i
=sin

3
x
i
2

;i
=0
;
1
;:::;N
;
y
j
+1
0
=0
;j
=1
;:::;M

1
:
Äëÿàïïðîêñèìàöèèãðàíè÷íîãîóñëîâèÿïðè
x
=1
èñïîëüçóåìòåæåòðèñïîñîáà,÷òî
èâñëó÷àåÿâíîéñõåìû,ðàçîáðàííîéâïðåäûäóùåìïðèìåðå.
Ïåðâûéâàðèàíòàïïðîêñèìàöèèãðàíè÷íîãîóñëîâèÿÍåéìàíàïðè
x
=1
:
y
j
+1
N

y
j
+1
N

1
h
=
t
j
+1
;j
=1
;:::;M

1
:
Ïîëó÷àþùàÿñÿïðèýòîìíåÿâíàÿðàçíîñòíàÿñõåìà
8



















:
y
0
i
=sin

3
x
i
2

;i
=0
;
1
;:::;N;
y
j
+1
0
=0
;j
=0
;
1
;:::;M

1
;
y
j
+1
i

y
j
i

=
y
j
+1
i

1

2
y
j
+1
i
+
y
j
+1
i
+1
h
2
+
x
i
;i
=1
;
2
;:::;N

1
;j
=0
;
1
;:::;M

1
;
y
j
+1
N

y
j
+1
N

1
h
=
t
j
+1
;j
=0
;
1
;:::;M

1
(2.8)
èìååòïîãðåøíîñòüàïïðîêñèìàöèè
O
(

+
h
)
.Çíà÷åíèÿñåòî÷íîéôóíêöèè
y
j
i
íàíóëåâîì
ñëîåïîâðåìåíèèçâåñòíûèçíà÷àëüíîãîóñëîâèÿ,ïîýòîìó,êàêèâñëó÷àåÿâíîéñõåìû,
ïðèêàæäîìôèêñèðîâàííîì
j
=0
;
1
;:::;M

1
íåèçâåñòíûìèÿâëÿþòñÿ
y
j
+1
i
.Ñèñòåìà
óðàâíåíèé,êîòîðûìîíèóäîâëåòâîðÿþò,èìååòâèä:
8







:
y
j
+1
0
=0
;

h
2
y
j
+1
i

1


1+
2

h
2

y
j
+1
i
+

h
2
y
j
+1
i
+1
=


y
j
i
+
x
i

;i
=1
;
2
;:::;N

1
;
y
j
+1
N
=
y
j
+1
N

1
+
ht
j
+1
;
(2.9)
òîåñòüÿâëÿåòñÿñèñòåìîéñòðåõäèàãîíàëüíîéìàòðèöåé:
8





:
y
j
+1
0
=
{
1
y
j
+1
1
+

1
;
A
i
y
j
+1
i

1

C
i
y
j
+1
i
+
B
i
y
j
+1
i
+1
=

F
i
;i
=1
;
2
;:::;N

1
;
y
j
+1
N
=
{
2
y
j
+1
N

1
+

2
;
(2.10)
ãäå
{
1
=0
,

1
=0
,
A
i
=
B
i
=

h
2
,
C
i
=1+
2

h
2
,
F
i
=
y
j
i
+
x
i
,
{
2
=1
,

2
=
ht
j
+1
.Î÷åâèäíî,
÷òîäîñòàòî÷íûåóñëîâèÿóñòîé÷èâîñòèïðîãîíêè:
A
i

0
;B
i

0
;C
i

A
i
+
B
i
;C
i
6
A
i
+
B
i
;i
=1
;
2
;:::;N

1
;
0
6
{
p
6
1
;p
=1
;
2
11
äëÿñèñòåìû(2.9)âûïîëíåíû.
Ðåøàÿñèñòåìó(2.9)ìåòîäîìïðîãîíêèèïîñëåäîâàòåëüíîóâåëè÷èâàÿçíà÷åíèÿ
j
íà
åäèíèöó,ìûïîëíîñòüþðåøèìñèñòåìó(2.8).Ðåçóëüòàòûâû÷èñëåíèéïîíåÿâíîéñõåìå
(2.8)âñëó÷àå
N
=
M
=50
ïðèâåäåíûíàðèñ.10-11.
Ðèñ.10:×èñëåííîåðåøåíèåçàäà÷è(2.1)ñïîìîùüþíåÿâíîéñõåìû(2.8).
Ðèñ.11:Ïîãðåøíîñòü÷èñëåííîãîðåøåíèÿçàäà÷è(2.1)ñïîìîùüþíåÿâíîéñõåìû(2.8).
Âòîðîéâàðèàíòàïïðîêñèìàöèèãðàíè÷íîãîóñëîâèÿÍåéìàíàïðè
x
=1
:
y
j
+1
N
=

1+
h
2
2



1

y
j
+1
N

1
+
ht
j
+1
+
h
2
2

1+
y
j
N

!!
;j
=0
;
1
;:::;M

1
:
(2.11)
Âýòîìñëó÷àåäëÿíåèçâåñòíûõ
y
j
+1
i
ïðèêàæäîìôèêñèðîâàííîì
j
ïîëó÷àåìòðåõäèà-
ãîíàëüíóþñèñòåìóâèäà(2.10),ãäå
{
2
=

1+
h
2
2



1
;
2
=
{
2


ht
j
+1
+
h
2
2

1+
y
j
N

!!
:
Ïîãðåøíîñòüðàñ÷åòîâïîñîîòâåòñòâóþùåéíåÿâíîéñõåìåâñëó÷àå
N
=
M
=50
ïðè-
âåäåíàíàðèñ.12.
12
Ðèñ.12:Ïîãðåøíîñòüðåøåíèÿçàäà÷è(2.1)ñïîìîùüþíåÿâíîéñõåìûñãðàíè÷íûìóñëî-
âèåì(2.11).
Òðåòèéâàðèàíòàïïðîêñèìàöèèãðàíè÷íîãîóñëîâèÿÍåéìàíàïðè
x
=1
:
y
j
+1
N
=
4
3
y
j
+1
N

1

1
3
y
j
+1
N

2
+
2
ht
j
+1
3
(2.12)
Äëÿòîãî,÷òîáûïîëó÷èòüäëÿíåèçâåñòíûõ
y
j
+1
i
ñèñòåìóñòðåõäèàãîíàëüíîéìàòðèöåé
ïðèêàæäîìôèêñèðîâàííîì
j
,èñêëþ÷èìèçóðàâíåíèÿ(2.12)íåèçâåñòíîå
y
j
+1
N

2
.Äëÿýòîãî
âîñïîëüçóåìñÿóðàâíåíèåì(2.7)ïðè
i
=
N

1
:

h
2
y
j
+1
N

2

(1+
2

h
2
)
y
j
+1
N

1
+

h
2
y
j
+1
N
=

F
N

1
:
Ñëåäîâàòåëüíî,
y
j
+1
N

2
=

h
2

+2

y
j
+1
N

1

y
j
+1
N

h
2

F
N

1
;
èóðàâíåíèå(2.12)ïðèíèìàåòâèä:
y
j
+1
N
=

1

h
2
2


y
j
+1
N

1
+
h
2
2

F
N

1
+
ht
j
+1
:
Âðåçóëüòàòåäëÿíåèçâåñòíûõ
y
j
+1
i
ïðèõîäèìêñèñòåìåñòðåõäèàãîíàëüíîéìàòðèöåéâèäà
(2.10),ãäå
{
2
=1

h
2
2

;
2
=
h
2
2

F
N

1
+
ht
j
+1
:
Ïîãðåøíîñòüðàñ÷åòîâïîñîîòâåòñòâóþùåéñõåìåâñëó÷àå
N
=
M
=50
ïðèâåäåíàíà
ðèñ.13.
Ïðèìåð2.3.
Ïîñòðîéòåñèììåòðè÷íóþðàçíîñòíóþñõåìó(ñõåìóñâåñîì

=0
:
5
)äëÿ
íà÷àëüíî-êðàåâîéçàäà÷è(2.1).Ñðàâíèòå÷èñëåííîåðåøåíèåñàíàëèòè÷åñêèìèèññëå-
äóéòåçàâèñèìîñòüïîãðåøíîñòèîòøàãîâñåòêè.
Ðåøåíèå.
Àïïðîêñèìàöèÿóðàâíåíèÿ
@u
@t
=
@
2
u
@x
2
+
x
13
Ðèñ.13:Ïîãðåøíîñòü÷èñëåííîãîðåøåíèÿçàäà÷è(2.1)ñïîìîùüþíåÿâíîéñõåìûñãðà-
íè÷íûìóñëîâèåì(2.12).
âñîîòâåòñòâèèññèììåòðè÷íîéðàçíîñòíîéñõåìîéèìååòâèä:
y
j
+1
i

y
j
i

=
1
2

y
j
+1
i

1

2
y
j
+1
i
+
y
j
+1
i
+1
h
2
+
y
j
i

1

2
y
j
i
+
y
j
i
+1
h
2
!
+
x
i
;
(2.13)
ãäå
i
=1
;
2
;:::;N

1
;j
=0
;
1
;:::;M

1
.Ðàçíîñòíîåóðàâíåíèå(2.13)àïïðîêñèìèðóåò
èñõîäíîåäèôôåðåíöèàëüíîåóðàâíåíèåòåïëîïðîâîäíîñòèñïîãðåøíîñòüþ
O
(

2
+
h
2
)
â
òî÷êàõ
(
x
i
;t
j
+0
:
5

)
äëÿâñåõâíóòðåííèõóçëîâ
x
i
ïðè
j
=0
;
1
;:::;M

1
.
Íà÷àëüíîåóñëîâèåèóñëîâèåÄèðèõëåïðè
x
=0
àïïðîêñèìèðóþòñÿòàêæå,êàêèâ
äâóõðàññìîòðåííûõðàíååñëó÷àÿõ.Ãðàíè÷íîåóñëîâèåÍåéìàíàïðè
x
=1
ìîæíîàïïðîê-
ñèìèðîâàòüêàêñïåðâûì,òàêèñîâòîðûìïîðÿäêîìïî
h
.
Åñëèâêà÷åñòâåàïïðîêñèìàöèèóñëîâèÿïðè
x
=1
áåðåòñÿðàçíîñòíîåóðàâíåíèå
y
j
+1
N

y
j
+1
N

1
h
=
t
j
+1
;j
=1
;:::;M

1
;
òîñõåìàáóäåòèìåòüïîãðåøíîñòüàïïðîêñèìàöèè
O
(

2
+
h
)
.Ñîîòâåòñòâóþùàÿñèñòåìà
äëÿíåèçâåñòíûõ
y
j
+1
i
áóäåòòðåõäèàãîíàëüíîé:
8





:
y
j
+1
0
=0
;
A
i
y
j
+1
i

1

C
i
y
j
+1
i
+
B
i
y
j
+1
i
+1
=

F
i
;i
=1
;
2
;:::;N

1
;
y
j
+1
N
=
y
j
+1
N

1
+
ht
j
+1
;
(2.14)
ãäå
A
i
=
B
i
=

2
h
2
;C
i
=1+2
A
i
;F
i
=
y
j
i
+
x
i
+

2
h
2
(
y
j
i

1

2
y
j
i
+
y
j
i
+1
)
:
Äîñòàòî÷íûåóñëîâèÿóñòîé÷èâîñòèïðîãîíêèäëÿñèñòåìû(2.14)âûïîëíåíû.Ïîãðåø-
íîñòüðåøåíèÿçàäà÷èïîñõåìå(2.14)äëÿ
N
=
M
=50
ïðèâåäåíàíàðèñ.14.
14
Ðèñ.14:Ïîãðåøíîñòü÷èñëåííîãîðåøåíèÿçàäà÷è(2.1)ñïîìîùüþñèììåòðè÷íîéñõåìû.
Ïîñòðîèìàïïðîêñèìàöèþãðàíè÷íîãîóñëîâèÿÍåéìàíàïðè
x
=1
ñïîãðåøíîñòüþ
O
(

2
+
h
2
)
.Ðàññìîòðèìðàâåíñòâî:
u
(
x;t
)

u
(
x

h;t
)
h
=
@u
(
x;t
)
@x

h
2

@u
(
x;t
)
@t

x

+
O
(
h
2
)
;
(2.15)
ãäå
u
(
x;t
)
ðåøåíèåèñõîäíîéçàäà÷è(2.1).Ïîëîæèìâðàâåíñòâå(2.15)
t
=
t
j
+0
:
5

.Òàê
êàê
u
(
x;t
j
+0
:
5

)

u
(
x

h;t
j
+0
:
5

)
h
=
=
1
2

u
(
x;t
j
)

u
(
x

h;t
j
)
h
+
u
(
x;t
j
+1
)

u
(
x

h;t
j
+1
)
h

+
O
(

2
)
è
@u
(
x;t
)
@t




t
=
t
j
+0
:
5

=
u
(
x;t
j
+1
)

u
(
x;t
j
)

+
O
(

2
)
;
ïîëó÷àåì:
1
2

u
j
i

u
j
i

1
h
+
u
j
+1
i

u
j
+1
i

1
h
!
=
@u
(
x;t
)
@x




(
x
i
;t
j
+0
:
5

)

h
2

u
j
+1
i

u
j
i


x
i
!
+
O
(

2
+
h
2
)
:
Ïåðåéäåìâïîëó÷åííîìðàâåíñòâåêïðåäåëóïðè
x
i
!
1
(òîåñòüïðè
i
!
N
),ó÷èòûâàÿ
ãðàíè÷íûåóñëîâèÿçàäà÷è:
1
2

u
j
N

u
j
N

1
h
+
u
j
+1
N

u
j
+1
N

1
h
!
=(
t
j
+0
:
5

)

h
2

u
j
+1
N

u
j
N


1
!
+
O
(

2
+
h
2
)
:
Ñëåäîâàòåëüíî,ðàçíîñòíîåóðàâíåíèå
1
2

y
j
N

y
j
N

1
h
+
y
j
+1
N

y
j
+1
N

1
h
!
=(
t
j
+0
:
5

)

h
2

y
j
+1
N

y
j
N


1
!
;j
=0
;
1
;:::;M

1
(2.16)
áóäåòàïïðîêñèìèðîâàòüóñëîâèå
@u
@x




x
=1
=
t
15
ñïîãðåøíîñòüþ
O
(

2
+
h
2
)
.Ñîîòâåòñòâóþùàÿñèñòåìàäëÿ
y
j
+1
i
ïðèôèêñèðîâàííîì
j
èìååò
âèä:
8





:
y
j
+1
0
=0
;
A
i
y
j
+1
i

1

C
i
y
j
+1
i
+
B
i
y
j
+1
i
+1
=

F
i
;i
=1
;
2
;:::;N

1
;
y
j
+1
N
=
{
2
y
j
+1
N

1
+

2
;
(2.17)
ãäå
{
2
=

1+
h
2



1
;
2
=
{
2


h
2

1+
y
j
N

!
+2
h

t
j
+

2


y
j
N
+
y
j
N

1
!
:
Ïîãðåøíîñòü,ïîëó÷àåìàÿïðè÷èñëåííîìðåøåíèèçàäà÷èñèñïîëüçîâàíèåìãðàíè÷íî-
ãîóñëîâèÿ(2.16),äëÿ
N
=
M
=50
ïðèâåäåíàíàðèñ.15.
Ðèñ.15:Ïîãðåøíîñòü÷èñëåííîãîðåøåíèÿçàäà÷è(2.1)ñïîìîùüþñèììåòðè÷íîéñõåìû
ñãðàíè÷íûìóñëîâèåì(2.16).
Òàêîéæåïîðÿäîêïîãðåøíîñòèàïïðîêñèìàöèèìîæíîïîëó÷èòü,èñïîëüçóÿãðàíè÷íîå
óñëîâèå
y
j
+1
N
=
4
3
y
j
+1
N

1

1
3
y
j
+1
N

2
+
2
ht
j
+1
3
:
(2.18)
Èñêëþ÷èìèçýòîãîóðàâíåíèÿíåèçâåñòíîå
y
j
+1
N

2
,èñïîëüçóÿóðàâíåíèå(2.13)ïðè
i
=
N

1
:

2
h
2
y
j
+1
N

2


1+

h
2

y
j
+1
N

1
+

2
h
2
y
j
+1
N
=

F
N

1
:
Òàêêàê
y
j
+1
N

2
=

2
h
2

+2

y
j
+1
N

1

y
j
+1
N

2
h
2

F
N

1
;
óðàâíåíèå(2.18)ìîæíîïåðåïèñàòüââèäå:
y
j
+1
N
=

1

h
2


y
j
+1
N

1
+
h
2

F
N

1
+
ht
j
+1
:
Âðåçóëüòàòåìûñíîâàïðèäåìêñèñòåìåñòðåõäèàãîíàëüíîéìàòðèöåéâèäà(2.17)äëÿ
íåèçâåñòíûõ
y
j
+1
i
ïðèêàæäîìôèêñèðîâàííîì
j
=0
;
1
;:::;M

1
,ãäåòåïåðü
{
2
=1

h
2

;
2
=
h
2

F
N

1
+
ht
j
+1
:
16
Ïîãðåøíîñòüðåøåíèÿïîïðåäëîæåííîéñõåìåïðè
N
=
M
=50
ïðèâåäåíàíàðèñ.16.
Ðèñ.16:Ïîãðåøíîñòüðåøåíèÿçàäà÷èñïîìîùüþñèììåòðè÷íîéñõåìûñãðàíè÷íûìóñëî-
âèåì(2.18).
3Çàäà÷èäëÿñàìîñòîÿòåëüíîãîðåøåíèÿ
Ðåøèòåàíàëèòè÷åñêèè÷èñëåííîïðèïîìîùèÿâíîé,íåÿâíîéèñèììåòðè÷íîéñõåìíà÷àëüíî-
êðàåâóþçàäà÷óäëÿóðàâíåíèÿòåïëîïðîâîäíîñòèíàîòðåçêå:
8

















:
@u
@t
=
a
2
@
2
u
@x
2
+
f
(
x;t
)
;x
2
(0
;l
)
;t
2
(0
;T
]
;
u
(
x;
0)=
u
0
(
x
)
;


0
@u
@x
+

0
u




x
=0
=
g
0
(
t
)
;

1
@u
@x
+

1
u




x
=
l
=
g
1
(
t
)
;
ãäå:
à)
a
=2
,
f
=cos

x
2


e
t
,
u
0
=


x
,

0
=1
,

0
=0
,

1
=0
,

1
=1
,
g
0
=1
,
g
1
=0
,
l
=

;
á)
a
=1
,
f
=
e
t

x
2
=
2

1

,
u
0
=1+
x
2
=
2
,

0
=1
,

0
=0
,

1
=1
,

1
=0
,
g
0
=0
,
g
1
=
e
t
,
l
=1
;
â)
a
=0
:
5
,
f
=
e
t
,
u
0
=1+sin(3
x
)
,

0
=0
,

0
=1
,

1
=1
,

1
=0
,
g
0
=
e
t
,
g
1
=0
,
l
=
=
2
;
ã)
a
=1
,
f
=0
,
u
0
=3

x
+cos

3
x
4

,

0
=1
,

0
=0
,

1
=0
,

1
=1
,
g
0
=1
,
g
1
=1
,
l
=2
;
ä)
a
=0
:
1
,
f
=0
,
u
0
=cos(
x
)+
x
2
+
x
,

0
=1
,

0
=0
,

1
=1
,

1
=0
,
g
0
=

1
,
g
1
=5
,
l
=2
.
Ñðàâíèòåðåçóëüòàòû÷èñëåííîãîðåøåíèÿïîðàçíûìñõåìàììåæäóñîáîéèñàíàëè-
òè÷åñêèìðåøåíèåìçàäà÷è.Ïðèèñïîëüçîâàíèèÿâíîéñõåìûñîáëþäàéòåóñëîâèåóñòîé-
÷èâîñòè

6
h
2
2
a
2
.
17

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