Прокудин Д.А., Глухарева Т.В., Казаченко И.В. Уравнения математической физики


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ÔÅÄÅÐÀËÜÍÎÅÃÎÑÓÄÀÐÑÒÂÅÍÍÎÅÁÞÄÆÅÒÍÎÅ
ÎÁÐÀÇÎÂÀÒÅËÜÍÎÅÓ×ÐÅÆÄÅÍÈÅ
ÂÛÑØÅÃÎÏÐÎÔÅÑÑÈÎÍÀËÜÍÎÃÎÎÁÐÀÇÎÂÀÍÈß
"ÊÅÌÅÐÎÂÑÊÈÉÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉÓÍÈÂÅÐÑÈÒÅÒ"
Ä.À.ÏÐÎÊÓÄÈÍ,Ò.Â.ÃËÓÕÀÐÅÂÀ,È.Â.ÊÀÇÀ×ÅÍÊÎ
ÓÐÀÂÍÅÍÈßÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÔÈÇÈÊÈ
Ó÷åáíîåïîñîáèå
Êåìåðîâî2014
ÁÁÊ22.311
ÓÄÊ517.95
Ï80
Ïå÷àòàåòñÿïîðåøåíèþðåäàêöèîííî-èçäàòåëüñêîãîñîâåòà
Êåìåðîâñêîãîãîñóäàðñòâåííîãîóíèâåðñèòåòà
Ðåöåíçåíòû:
äîêòîðòåõíè÷åñêèõíàóê,ïðîôåññîðÊóçÃÒÓ
Ñ.Â.×åðäàíöåâ
;
äîêòîðòåõíè÷åñêèõíàóê,ïðîôåññîðÊåìÒÈÏÏà
Ò.Â.Øåâ÷åíêî
Ïðîêóäèí,Ä.À.
Ï80Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè:ó÷åáíîåïîñî-
áèå/Ä.À.Ïðîêóäèí,Ò.Â.Ãëóõàðåâà,È.Â.Êàçà÷åíêî;
Êåìåðîâñêèéãîñóäàðñòâåííûéóíèâåðñèòåò.Êåìåðîâî,
2014.163ñ.
ISBN978-5-8353-1631-1
Ó÷åáíîåïîñîáèåðàçðàáîòàíîïîäèñöèïëèíå¾Óðàâíåíèÿìàòåìàòè÷åñêîé
ôèçèêè¿âñîîòâåòñòâèèñòðåáîâàíèÿìèÔÃÎÑÂÏÎ,ñîäåðæèòîñíîâíûå
îïðåäåëåíèÿ,ôîðìóëèðîâêèèäîêàçàòåëüñòâàòåîðåìäèñöèïëèíû¾Óðàâ-
íåíèÿìàòåìàòè÷åñêîéôèçèêè¿,ìåòîäûèññëåäîâàíèÿðàçðåøèìîñòèðàç-
ëè÷íûõçàäà÷äëÿóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîèâòîðîãî
ïîðÿäêîâ,àòàêæåâîïðîñûèçàäà÷è,ïðåäíàçíà÷åííûåäëÿñàìîñòîÿòåëü-
íîãîðåøåíèÿ.Ïðåäíàçíà÷åíîäëÿñòóäåíòîâíàïðàâëåíèÿ¾Ìàòåìàòè÷å-
ñêîåîáåñïå÷åíèåèàäìèíèñòðèðîâàíèåèíôîðìàöèîííûõñèñòåì¿.
ÁÁÊ22.311
ÓÄÊ517.95
ISBN978-5-8353-1631-1
c

ÏðîêóäèíÄ.À.,ÃëóõàðåâàÒ.Â.,
Êàçà÷åíêîÈ.Â.,2014
c

Êåìåðîâñêèéãîñóäàðñòâåííûé
óíèâåðñèòåò,2014
Ñîäåðæàíèå
Ïðåäèñëîâèå6
1.Ââåäåíèå8
1.1.Óðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà...
13
1.1.1.Ïîíÿòèåõàðàêòåðèñòèêèêâàçèëèíåéíîãîóðàâíåíèÿ
ïåðâîãîïîðÿäêà.....................
13
1.1.2.Èíòåãðèðîâàíèåëèíåéíûõóðàâíåíèéïåðâîãîïîðÿäêà
16
1.1.3.Èíòåãðèðîâàíèåêâàçèëèíåéíûõóðàâíåíèéïåðâîãî
ïîðÿäêà..........................
19
1.1.4.Çàäà÷àÊîøèäëÿêâàçèëèíåéíîãîóðàâíåíèÿïåðâîãî
ïîðÿäêà..........................
21
1.2.Óðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêà...
25
1.2.1.Ïîíÿòèåõàðàêòåðèñòè÷åñêîéôîðìûè
êëàññèôèêàöèÿëèíåéíûõóðàâíåíèéâòîðîãîïîðÿäêà
25
1.2.2.Êëàññèôèêàöèÿíåëèíåéíûõóðàâíåíèéâòîðîãîïîðÿäêà
28
1.2.3.Êëàññèôèêàöèÿñèñòåìäâóõëèíåéíûõóðàâíåíèé
ïåðâîãîïîðÿäêà.....................
30
1.2.4.Ïðèâåäåíèåêêàíîíè÷åñêîìóâèäóëèíåéíîãî
óðàâíåíèÿâòîðîãîïîðÿäêàñïîñòîÿííûìè
êîýôôèöèåíòàìè.....................
31
1.2.5.Ïðèâåäåíèåêêàíîíè÷åñêîìóâèäóëèíåéíîãî
óðàâíåíèÿâòîðîãîïîðÿäêàñäâóìÿíåçàâèñèìûìè
ïåðåìåííûìè.......................
34
1.2.6.Çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòîðîãî
ïîðÿäêàãèïåðáîëè÷åñêîãîòèïà.............
43
1.2.7.Çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòîðîãî
ïîðÿäêàñàíàëèòè÷åñêèìèäàííûìè.
ÔîðìóëèðîâêàòåîðåìûÊîøèÊîâàëåâñêîé......
53
1.2.8.Ïîíÿòèåêîððåêòíîñòèçàäà÷èìàòåìàòè÷åñêîé
ôèçèêè.ÏðèìåðÀäàìàðà................
56
1.3.Âûâîäîñíîâíûõóðàâíåíèéìàòåìàòè÷åñêîéôèçèêè.....
57
1.3.1.Óðàâíåíèåêîëåáàíèéñòðóíû..............
57
1.3.2.Óðàâíåíèåêîëåáàíèéìåìáðàíû............
63
1.3.3.Óðàâíåíèåðàñïðîñòðàíåíèÿòåïëàâèçîòðîïíîì
òâåðäîìòåëå.......................
67
3
1.3.4.Óðàâíåíèÿ,îïèñûâàþùèåñòàöèîíàðíûåïðîöåññû
ðàñïðîñòðàíåíèÿòåïëà..................
73
1.4.Âîïðîñûèçàäà÷è........................
73
2.Óðàâíåíèÿãèïåðáîëè÷åñêîãîòèïà76
2.1.Îäíîðîäíîåâîëíîâîåóðàâíåíèå................
76
2.1.1.Çàäà÷àÊîøèäëÿîäíîìåðíîãîâîëíîâîãîóðàâíåíèÿ.
ÔîðìóëàÄàëàìáåðà...................
76
2.1.2.Çàäà÷àñíà÷àëüíûìèóñëîâèÿìèäëÿâîëíîâîãî
óðàâíåíèÿñòðåìÿïðîñòðàíñòâåííûìèïåðåìåííûìè.
ÔîðìóëàÊèðõãîôà....................
79
2.1.3.Çàäà÷àÊîøèäëÿâîëíîâîãîóðàâíåíèÿñäâóìÿ
ïðîñòðàíñòâåííûìèïåðåìåííûìè.Ìåòîäñïóñêà.
ÔîðìóëàÏóàññîíà....................
85
2.1.4.Àíàëèçðåøåíèÿ(ïîíÿòèåîáëàñòèçàâèñèìîñòè,
îáëàñòèâëèÿíèÿèîáëàñòèîïðåäåëåíèÿ).......
87
2.2.Íåîäíîðîäíîåâîëíîâîåóðàâíåíèå...............
90
2.2.1.Ñëó÷àéîäíîéïðîñòðàíñòâåííîéïåðåìåííîé.....
90
2.2.2.Ñëó÷àéòðåõïðîñòðàíñòâåííûõïåðåìåííûõ.
Çàïàçäûâàþùèéïîòåíöèàë...............
91
2.2.3.Ñëó÷àéäâóõïðîñòðàíñòâåííûõïåðåìåííûõ.....
94
2.3.Êîððåêòíîïîñòàâëåííûåçàäà÷èäëÿãèïåðáîëè÷åñêèõ
óðàâíåíèé.............................
94
2.3.1.Åäèíñòâåííîñòüðåøåíèÿçàäà÷èÊîøè........
94
2.3.2.Îáùàÿïîñòàíîâêàçàäà÷èÊîøè............
96
2.3.3.Çàäà÷àÃóðñà(õàðàêòåðèñòè÷åñêàÿçàäà÷à)......
99
2.4.Âîïðîñûèçàäà÷è........................
100
3.Óðàâíåíèÿïàðàáîëè÷åñêîãîòèïà103
3.1.Ïðèíöèïìàêñèìóìà.......................
103
3.2.Ïåðâàÿêðàåâàÿçàäà÷àäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè..
105
3.3.Ïîñòàíîâêàçàäà÷èÊîøèèäîêàçàòåëüñòâîñóùåñòâîâàíèÿåå
ðåøåíèÿ..............................
108
3.4.Ãëàäêîñòüðåøåíèé........................
112
3.5.Íåîäíîðîäíîåóðàâíåíèåòåïëîïðîâîäíîñòè..........
113
3.6.Âîïðîñûèçàäà÷è........................
113
4.Óðàâíåíèÿýëëèïòè÷åñêîãîòèïà115
4.1.Îñíîâíûåñâîéñòâàãàðìîíè÷åñêèõôóíêöèé.........
115
4
4.1.1.Èíòåãðàëüíîåïðåäñòàâëåíèåãàðìîíè÷åñêèõôóíêöèé
115
4.1.2.Òåîðåìàîñðåäíåì....................
119
4.1.3.Ïðèíöèïýêñòðåìóìàèåãîñëåäñòâèÿ.........
120
4.2.ÔóíêöèÿÃðèíà.Ðåøåíèåçàäà÷èÄèðèõëåäëÿøàðàè
ïîëóïðîñòðàíñòâà........................
122
4.2.1.ÏîíÿòèåôóíêöèèÃðèíàçàäà÷èÄèðèõëå
äëÿóðàâíåíèÿËàïëàñà.................
122
4.2.2.Ðåøåíèåçàäà÷èÄèðèõëåäëÿøàðà.ÔîðìóëàÏóàññîíà
123
4.2.3.Ðåøåíèåçàäà÷èÄèðèõëåäëÿïîëóïðîñòðàíñòâà...
128
4.2.4.Íåêîòîðûåñëåäñòâèÿ,âûòåêàþùèåèçôîðìóëû
Ïóàññîíà.ÒåîðåìûËèóâèëëÿèÃàðíàêà........
130
4.3.Âîïðîñûèçàäà÷è........................
132
5.Ìåòîäðàçäåëåíèÿïåðåìåííûõ(ìåòîäÔóðüå)135
5.1.Ðåøåíèåñìåøàííûõçàäà÷äëÿóðàâíåíèéãèïåðáîëè÷åñêîãî
òèïàìåòîäîìðàçäåëåíèÿïåðåìåííûõ.............
135
5.2.Ðåøåíèåñìåøàííûõçàäà÷äëÿóðàâíåíèéïàðàáîëè÷åñêîãî
òèïàìåòîäîìðàçäåëåíèÿïåðåìåííûõ.............
141
5.3.Ðåøåíèåêðàåâûõçàäà÷äëÿóðàâíåíèéýëëèïòè÷åñêîãîòèïà
ìåòîäîìðàçäåëåíèÿïåðåìåííûõ................
146
5.3.1.Ïîñòðîåíèåðåøåíèéêðàåâûõçàäà÷âïðÿìîóãîëüíûõ
îáëàñòÿõ..........................
146
5.3.2.Ïîñòðîåíèåðåøåíèéêðàåâûõçàäà÷âêðóãîâûõ
îáëàñòÿõ..........................
152
5.4.Âîïðîñûèçàäà÷è........................
158
5
Ïðåäèñëîâèå
Êóðñ¾Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè¿íàìàòåìàòè-
÷åñêîìôàêóëüòåòåîòíîñèòñÿêáëîêóîáùèõìàòåìàòè÷åñêèõ
èåñòåñòâåííîíàó÷íûõäèñöèïëèíèèçó÷àåòñÿñòóäåíòàìèíà-
ïðàâëåíèÿ¾Ìàòåìàòè÷åñêîåîáåñïå÷åíèåèàäìèíèñòðèðîâàíèå
èíôîðìàöèîííûõñèñòåì¿âòå÷åíèåïÿòîãîñåìåñòðàíàòðå-
òüåìêóðñåîáó÷åíèÿ.
Äàííîåó÷åáíîåïîñîáèåðàçðàáîòàíîâñîîòâåòñòâèèñòðå-
áîâàíèÿìèÔÃÎÑÂÏÎèìîæåòáûòüèñïîëüçîâàíîäëÿïðî-
âåäåíèÿëåêöèîííûõçàíÿòèéñîñòóäåíòàìè.Ïîñîáèåñïîñîá-
ñòâóåòôîðìèðîâàíèþóîáó÷àþùèõñÿòàêèõêîìïåòåíöèéêàê
çíàíèåêîððåêòíûõïîñòàíîâîêêëàññè÷åñêèõçàäà÷(ÏÊ9),ïî-
íèìàíèåêîððåêòíîñòèïîñòàíîâîêçàäà÷(ÏÊ10)èâûäåëåíèå
ãëàâíûõñìûñëîâûõàñïåêòîââäîêàçàòåëüñòâàõ(ÏÊ16).
Ïîñîáèåñîñòîèòèç5ðàçäåëîâ,êàæäûéèçêîòîðûõñîäåð-
æèòâñâîþî÷åðåäüíåñêîëüêîïîäðàçäåëîâ.
Âïåðâîìðàçäåëåðàññìàòðèâàþòñÿòàêèåâîïðîñûêóðñà
¾Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè¿,êàêîñíîâíûåïîíÿòèÿ
óðàâíåíèéìàòåìàòè÷åñêîéôèçèêè,óðàâíåíèÿñ÷àñòíûìèïðî-
èçâîäíûìèïåðâîãîïîðÿäêàèìåòîäûèõðåøåíèÿ,ïîíÿòèåõà-
ðàêòåðèñòè÷åñêîéôîðìûèêëàññèôèêàöèÿóðàâíåíèéñ÷àñò-
íûìèïðîèçâîäíûìèâòîðîãîïîðÿäêà,ïðèâåäåíèåêêàíîíè-
÷åñêîìóâèäóëèíåéíîãîóðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìè
âòîðîãîïîðÿäêàñäâóìÿíåçàâèñèìûìèïåðåìåííûìè,ðåøåíèå
çàäà÷èÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòîðîãîïîðÿäêàãèïåð-
áîëè÷åñêîãîòèïà,çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòî-
ðîãîïîðÿäêàñàíàëèòè÷åñêèìèäàííûìè,ôîðìóëèðîâêàòåîðå-
ìûÊîøèÊîâàëåâñêîé,ïîíÿòèåêîððåêòíîñòèçàäà÷èìàòåìà-
òè÷åñêîéôèçèêè,âûâîäîñíîâíûõóðàâíåíèéìàòåìàòè÷åñêîé
ôèçèêè.
Âîâòîðîìðàçäåëåèçó÷àþòñÿðàçëè÷íûåçàäà÷èäëÿóðàâ-
6
íåíèéãèïåðáîëè÷åñêîãîòèïà.
Òðåòèéðàçäåëïîñâÿùåíèçó÷åíèþíà÷àëüíî-êðàåâûõçàäà÷
äëÿóðàâíåíèéïàðàáîëè÷åñêîãîòèïà.
Â÷åòâåðòîìðàçäåëåäàåòñÿýëåìåíòàðíàÿòåîðèÿãàðìîíè-
÷åñêèõôóíêöèéèðàññìàòðèâàåòñÿðåøåíèåîñíîâíûõêðàåâûõ
çàäà÷äëÿóðàâíåíèéýëëèïòè÷åñêîãîòèïà.
Âïÿòîìðàçäåëåðàññìàòðèâàåòñÿðåøåíèåñìåøàííûõçà-
äà÷äëÿóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêà
ãèïåðáîëè÷åñêîãîèïàðàáîëè÷åñêîãîòèïîâ,ðåøåíèåêðàåâûõ
çàäà÷âêðóãîâûõîáëàñòÿõäëÿóðàâíåíèéñ÷àñòíûìèïðîèç-
âîäíûõâòîðîãîïîðÿäêàýëëèïòè÷åñêîãîòèïàìåòîäîìðàçäå-
ëåíèÿïåðåìåííûõ(ìåòîäîìÔóðüå).
Âêîíöåêàæäîãîðàçäåëàïðèâåäåíûâîïðîñûèçàäà÷è,
ïðåäíàçíà÷åííûåäëÿñàìîñòîÿòåëüíîãîðåøåíèÿ.
7
1.Ââåäåíèå
Óðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìèóðàâíåíèå,ñîäåð-
æàùååíåèçâåñòíóþôóíêöèþäâóõèëèáîëååíåçàâèñèìûõïå-
ðåìåííûõèíåêîòîðûå÷àñòíûåïðîèçâîäíûåýòîéôóíêöèè.
Îáîçíà÷èì÷åðåç
D
îáëàñòü
n
-ìåðíîãîåâêëèäîâàïðî-
ñòðàíñòâà
R
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òî÷åê
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äåéñòâèòåëüíàÿ
ôóíêöèÿòî÷åê
x
îáëàñòè
D
èäåéñòâèòåëüíûõïåðåìåí-
íûõ
p
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ñíåîòðèöàòåëüíûìèöåëî÷èñëåííûìèèíäåêñàìè
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m

1
è,ïîêðàéíåéìåðå,
îäíàèçïðîèçâîäíîéêîòîðîé
@F
@p
i
1
;:::;i
n
;
n
X
j
=1
i
j
=
m
îòëè÷íàîòíóëÿ.
Îïðåäåëåíèå11.
Ðàâåíñòâîâèäà
F

x
;:::;
@
k
u
@x
i
1
1
;:::;@x
i
n
n
;:::
!
=0
;
n
X
j
=1
i
j
=
k;k
=0
;:::;m
(1.1)
íàçûâàåòñÿäèôôåðåíöèàëüíûìóðàâíåíèåìñ÷àñòíûìèïðî-
èçâîäíûìèïîðÿäêà
m
îòíîñèòåëüíîíåèçâåñòíîéôóíêöèè
u
(
x
)
,
x
2
D
.
Óðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêàìî-
æåòáûòüçàïèñàíîââèäå
F

x
;u;
@u
@x
1
;:::;
@u
@x
n

=0
:
(1.2)
1
Èíîãäàâìåñòî
x
1
;x
2
;x
3
;:::
áóäåìïèñàòü
x;y;z;:::
.
2
Ìûíåáóäåìîáñóæäàòüñòåïåíüãëàäêîñòèôóíêöèè
F
,ïîëàãàÿååíåïðåðûâíîäèôôåðåíöèðóåìîé
ñòîëüêîðàç,ñêîëüêîïîòðåáóåòñÿ.
8
Óðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêàèìå-
åòâèä
F

x
;u;
@u
@x
1
;:::;
@u
@x
n
;:::
@
2
u
@x
i
@x
j
;:::

=0
;i;j
=1
;:::;n:
(1.3)
Áóäåìãîâîðèòü,÷òîìûðåøèëèóðàâíåíèåñ÷àñòíûìèïðî-
èçâîäíûìè,åñëèíàéäåíûâñåôóíêöèè
u
,óäîâëåòâîðÿþùèå
(1.1)(âîçìîæíî,ëèøüâêëàññåôóíêöèé,óäîâëåòâîðÿþùèõ
âñïîìîãàòåëüíûìãðàíè÷íûìóñëîâèÿìíàíåêîòîðîé÷àñòè

ãðàíèöû
@D
).Êîãäàìûãîâîðèì,÷òîðåøåíèåíàéäåíî,âèäå-
àëüíîìñëó÷àåýòîîçíà÷àåò,÷òîíàéäåíûïðîñòûåÿâíûåôîð-
ìóëûäëÿðåøåíèÿèëè,åñëèòàêîåíåâîçìîæíîèëèñëèøêîì
ñëîæíî,äîêàçàíîñóùåñòâîâàíèåðåøåíèÿèóñòàíîâëåíûåãî
íåêîòîðûåñâîéñòâà.
Îïðåäåëåíèå12.
Îïðåäåëåííàÿâîáëàñòè
D
çàäàíèÿ
óðàâíåíèÿ(1.1)äåéñòâèòåëüíàÿôóíêöèÿ
u
(
x
)
,íåïðåðûâíàÿ
âìåñòåñîñâîèìè÷àñòíûìèïðîèçâîäíûìè,âõîäÿùèìèâýòî
óðàâíåíèå,èîáðàùàþùàÿåãîâòîæäåñòâî,íàçûâàåòñÿêëàñ-
ñè÷åñêèìðåøåíèåì(èëèðåãóëÿðíûìðåøåíèåì).
Îïðåäåëåíèå13.
Åñëè
u
(
x
)
ðåãóëÿðíîåðåøåíèåóðàâíå-
íèÿ(1.1),òîïîâåðõíîñòü
u
=
u
(
x
)
âïðîñòðàíñòâåïåðåìåí-
íûõ(
x
,
u
)íàçûâàåòñÿèíòåãðàëüíîéïîâåðõíîñòüþóðàâíåíèÿ
(1.1).
Îïðåäåëåíèå14.
Óðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìè
(1.1)íàçûâàåòñÿëèíåéíûì,åñëèôóíêöèÿ
F
(
x
;:::;p
i
1
;:::;i
n
;:::
)
ÿâëÿåòñÿëèíåéíîéîòíîñèòåëüíîâñåõïåðåìåííûõ
p
i
1
;:::;i
n
;
n
X
j
=1
i
j
=
k
,
k
=0
;:::;m
.
Óðàâíåíèåâèäà
n
X
i
=1
a
i
(
x
)
@u
@x
i
+
b
(
x
)
u
=
f
(
x
)
(1.4)
9
åñòüëèíåéíîåóðàâíåíèåïåðâîãîïîðÿäêà,àóðàâíåíèå
n
X
ij
=1
a
ij
(
x
)
@
2
u
@x
i
@x
j
+
n
X
i
=1
b
i
(
x
)
@u
@x
i
+
c
(
x
)
u
=
f
(
x
)
;a
ij
=
a
ji
(1.5)
ÿâëÿåòñÿëèíåéíûìóðàâíåíèåìâòîðîãîïîðÿäêàîòíîñèòåëüíî
íåèçâåñòíîéôóíêöèè
u
(
x
)
.
Îïðåäåëåíèå15.
Åñëèôóíêöèÿ
F
(
x
;:::;p
i
1
;:::;i
n
;:::
)
ëè-
íåéíàîòíîñèòåëüíîïåðåìåííûõ
p
i
1
;:::;i
n
ïðè
n
X
j
=1
i
j
=
m
,òî
óðàâíåíèå(1.1)íàçûâàåòñÿêâàçèëèíåéíûì.
Óðàâíåíèå
n
X
i
=1
a
i
(
x
;u
)
@u
@x
i
=
f
(
x
;u
)
(1.6)
åñòüêâàçèëèíåéíîåóðàâíåíèåïåðâîãîïîðÿäêà,àóðàâíåíèå
n
X
ij
=1
a
ij

x
;u;
@u
@x
1
;:::;
@u
@x
n

@
2
u
@x
i
@x
j
=
f

x
;u;
@u
@x
1
;:::;
@u
@x
n

(1.7)
êâàçèëèíåéíîåóðàâíåíèåâòîðîãîïîðÿäêàîòíîñèòåëüíî
íåèçâåñòíîéôóíêöèè
u
(
x
)
.
Ñèñòåìàóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèñîâîêóï-
íîñòüíåñêîëüêèõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèîòíî-
ñèòåëüíîíåñêîëüêèõíåèçâåñòíûõôóíêöèé.
Îïðåäåëåíèå16.
Ðàâåíñòâîâèäà
F

x
;:::;
@
k
u
@x
i
1
1
;:::;@x
i
n
n
;:::
!
=0
;
n
X
j
=1
i
j
=
k;k
=0
;:::;m;
(1.8)
ãäå
F
=(
F
1
;:::;F
s
)
çàäàíà,à
u
=(
u
1
;:::;u
s
)
íåèçâåñòíà,íàçû-
âàåòñÿñèñòåìîéóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìè
m
ãî
ïîðÿäêà.
10
Ìûáóäåìðàññìàòðèâàòüñèñòåìû,óêîòîðûõ÷èñëîóðàâ-
íåíèé
s
ñîâïàäàåòñ÷èñëîìíåèçâåñòíûõ
u
1
;:::;u
s
.Îáû÷íîðàñ-
ñìàòðèâàþòñÿèìåííîòàêèåñèñòåìû,õîòÿñëó÷àè,êîãäà÷èñëî
óðàâíåíèéìåíüøåèëèáîëüøå÷èñëàíåèçâåñòíûõ,òàêæåèñ-
ñëåäóþòñÿ.
Î÷åâèäíî,÷òîìîæíîêëàññèôèöèðîâàòüñèñòåìûïîïðè-
çíàêóëèíåéíîñòèèêâàçèëèíåéíîñòè.
Íåñóùåñòâóåòîáùåéòåîðèè,óñòàíàâëèâàþùåéðàçðåøè-
ìîñòüâñåõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìè.Âåñüìàñî-
ìíèòåëüíî,÷òîñîçäàíèåòàêîéòåîðèèâîîáùåâîçìîæíîââèäó
áîëüøîãîìíîãîîáðàçèÿôèçè÷åñêèõ,ãåîìåòðè÷åñêèõèâåðîÿò-
íîñòíûõÿâëåíèé,êîòîðûåìîäåëèðóþòñÿóðàâíåíèÿìèñ÷àñò-
íûìèïðîèçâîäíûìè.Ïîýòîìóèññëåäîâàíèÿêîíöåíòðèðóþòñÿ
âîêðóãíåêîòîðûõêîíêðåòíûõóðàâíåíèé,âàæíûõäëÿïðèëî-
æåíèé,êàêâðàìêàõñàìîéìàòåìàòèêè,òàêèäëÿñìåæíûõ
äèñöèïëèí,ñíàäåæäîé,÷òîèíòóèòèâíîåïîíèìàíèåèñòîêîâ
ýòèõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèïîäñêàæåòïóòüê
èõðåøåíèþ.
Ïåðå÷èñëèìíåêîòîðûåâàæíûåâñîâðåìåííûõèññëåäîâà-
íèÿõóðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìè,ñòåì,÷òîáûèìåòü
õîòÿáûíà÷àëüíîåïðåäñòàâëåíèå(íàçâàíèåèâèä)îáýòèõõî-
ðîøîèçâåñòíûõóðàâíåíèÿõ.Íèæåìûîáñóäèìïðîèñõîæäåíèå
äëÿóðàâíåíèéèçïðèâåäåííîãîíèæåñïèñêà.
Âñþäóäàëåå
x
2
D
,ãäå
D
îáëàñòüïðîñòðàíñòâà
R
n

t

0
âðåìÿ.
1)Ïðèèçó÷åíèèðàçëè÷íûõâèäîââîëíóïðóãèõ,çâóêî-
âûõ,ýëåêòðîìàãíèòíûõ,àòàêæåäðóãèõêîëåáàòåëüíûõÿâëå-
íèéìûïðèõîäèìêâîëíîâîìóóðàâíåíèþ
@
2
u
@t
2
=
c
2

u
+
f
(
x
;t
)
;
=
n
X
i
=1
@
2
@x
2
i
;
(1.9)
ãäå
c
ñêîðîñòüðàñïðîñòðàíåíèÿâîëíûâäàííîéñðåäå,
11
f
(
x
;t
)
âíåøíÿÿñèëà.
2)Ïðîöåññûðàñïðîñòðàíåíèÿòåïëàâîäíîðîäíîìèçîòðîï-
íîìòåëå,òàêæåêàêèÿâëåíèÿäèôôóçèè,îïèñûâàþòñÿóðàâ-
íåíèåìòåïëîïðîâîäíîñòè
@u
@t
=
a
2

u
+
g
(
x
;t
)
;
(1.10)
ãäå
a
êîýôôèöèåíòòåìïåðàòóðîïðîâîäíîñòè,
g
(
x
;t
)
ïëîò-
íîñòüòåïëîâûõèñòî÷íèêîâ.
3)Ïðèðàññìîòðåíèèóñòàíîâèâøèõñÿêîëåáàòåëüíûõÿâëå-
íèéèëèñòàöèîíàðíîãîòåïëîâîãîñîñòîÿíèÿâîäíîðîäíîìèçî-
òðîïíîìòåëåìûïðèõîäèìêóðàâíåíèþÏóàññîíà

u
=

h
(
x
)
:
(1.11)
Ïðè
h
(
x
)=0
óðàâíåíèå(1.11)ïåðåõîäèòâóðàâíåíèåËàïëàñà

u
=0
:
(1.12)
Ïîòåíöèàëûïîëÿòÿãîòåíèÿèñòàöèîíàðíîãîýëåêòðè÷åñêîãî
ïîëÿòàêæåóäîâëåòâîðÿþòóðàâíåíèþËàïëàñà,âêîòîðîìîò-
ñóòñòâóþòìàññûèñîîòâåòñòâåííîýëåêòðè÷åñêèåçàðÿäû.
Óðàâíåíèÿ(1.9)-(1.12)÷àñòîíàçûâàþòîñíîâíûìèóðàâíå-
íèÿìèìàòåìàòè÷åñêîéôèçèêè.Èõïîäðîáíîåèçó÷åíèåäàåò
âîçìîæíîñòüïîñòðîèòüòåîðèþøèðîêîãîêðóãàôèçè÷åñêèõÿâ-
ëåíèéèðåøèòüðÿäôèçè÷åñêèõèòåõíè÷åñêèõçàäà÷.
Êàæäîåèçóðàâíåíèé(1.9)-(1.12)èìååòáåñ÷èñëåííîåìíî-
æåñòâî÷àñòíûõðåøåíèé.Ïðèðåøåíèèêîíêðåòíîéôèçè÷å-
ñêîéçàäà÷èíåîáõîäèìîèçâñåõýòèõðåøåíèéâûáðàòüòî,êîòî-
ðîåóäîâëåòâîðÿåòíåêîòîðûìäîïîëíèòåëüíûìóñëîâèÿì,âû-
òåêàþùèìèçå¼ôèçè÷åñêîãîñìûñëà.Èòàê,çàäà÷èìàòåìàòè-
÷åñêîéôèçèêèñîñòîÿòâîòûñêàíèèðåøåíèéóðàâíåíèéâ÷àñò-
íûõïðîèçâîäíûõ,óäîâëåòâîðÿþùèõíåêîòîðûìäîïîëíèòåëü-
íûìóñëîâèÿì.Òàêèìèäîïîëíèòåëüíûìèóñëîâèÿìè÷àùåâñå-
ãîÿâëÿþòñÿòàêíàçûâàåìûåãðàíè÷íûåóñëîâèÿ,ò.å.óñëîâèÿ,
12
çàäàííûåíàíåêîòîðîé÷àñòèãðàíèöûðàññìàòðèâàåìîéñðåäû,
èíà÷àëüíûåóñëîâèÿ,îòíîñÿùèåñÿêîäíîìóêàêîìó-íèáóäüìî-
ìåíòóâðåìåíè,ñêîòîðîãîíà÷èíàåòñÿèçó÷åíèåäàííîãîôèçè-
÷åñêîãîÿâëåíèÿ.
1.1.Óðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà
1.1.1.Ïîíÿòèåõàðàêòåðèñòèêèêâàçèëèíåéíîãîóðàâíåíèÿïåðâîãîïîðÿäêà
Ðàññìîòðèìêâàçèëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèç-
âîäíûìèïåðâîãîïîðÿäêà
n
X
i
=1
a
i
(
x
;u
)
@u
@x
i
=
f
(
x
;u
)
;
(1.13)
ãäå
a
i
(
x
;u
)
,
i
=1
;:::;n
,
f
(
x
;u
)
èçâåñòíûåíåïðåðûâíîäèôôå-
ðåíöèðóåìûåôóíêöèèâîáëàñòè
G
ïðîñòðàíñòâàïåðåìåííûõ
x
;u
,ïðè÷åìâýòîéîáëàñòè
n
X
i
=1
a
2
i
6
=0
.
Ïóñòü
u
(
x
)
ðåãóëÿðíîåðåøåíèåóðàâíåíèÿ(1.13),îïðå-
äåëåííîåâíåêîòîðîéîáëàñòè
D
ïðîñòðàíñòâàïåðåìåííûõ
x
,
èïóñòü
x
=
x
(
s
)
ãëàäêàÿêðèâàÿëåæàùàÿâ
D
.Òîãäà,ðàñ-
ñìàòðèâàÿðåøåíèå
u
(
x
)
íàýòîéêðèâîé,ïîëó÷àåìôóíêöèþ
~
u
(
s
)=
u
(
x
(
s
))
.
Ïîäáåðåìêðèâóþ
x
(
s
)
òàê,÷òîáû
dx
i
ds
=
a
i
(
x
(
s
)
;
~
u
(
s
))
;i
=1
;:::;n:
Òîãäà,äèôôåðåíöèðóÿôóíêöèþ
~
u
(
s
)
,ïîëó÷àåì
d
~
u
ds
=
n
X
i
=1
@u
@x
i
(
x
(
s
))
dx
i
ds
=
=
n
X
i
=1
@u
@x
i
(
x
(
s
))
a
i
(
x
(
s
)
;
~
u
(
s
))=
f
(
x
(
s
)
;
~
u
(
s
))
:
13
Ñëåäîâàòåëüíî,èñêîìàÿêðèâàÿ
x
(
s
)
äîëæíàáûòüòàêîé,÷òîáû
âûðàæåíèÿ
x
=
x
(
s
)
;u
=~
u
(
s
)
îïðåäåëÿëèòðàåêòîðèþñèñòåìûîáûêíîâåííûõäèôôåðåíöè-
àëüíûõóðàâíåíèé
8





:
dx
i
ds
=
a
i
(
x
;u
)
;i
=1
;:::;n;
du
ds
=
f
(
x
;u
)
:
(1.14)
Îïðåäåëåíèå17.
Ñèñòåìàóðàâíåíèé(1.14)íàçûâàåò-
ñÿõàðàêòåðèñòè÷åñêîéñèñòåìîéóðàâíåíèÿ(1.13),àååòðà-
åêòîðèèâïðîñòðàíñòâåïåðåìåííûõ(
x
,
u
)õàðàêòåðèñòè-
êàìèóðàâíåíèÿ(1.13).
Çàìå÷àíèå11.
Ïîä÷åðêíåì,÷òîïàðàìåòð
s
íàõàðàêòå-
ðèñòèêåóðàâíåíèÿ(1.13)îïðåäåëåíëèøüñòî÷íîñòüþäîïî-
ñòîÿííîãîñëàãàåìîãî.
Çàìå÷àíèå12.
Ñîãëàñíîíàøèìïðåäïîëîæåíèÿì,ôóíê-
öèè
a
i
(
x
;u
)
,
i
=1
;:::;n
,
f
(
x
;u
)
ïðèíàäëåæàòêëàññó
C
1
(
G
)
,ïî-
ýòîìóäëÿñèñòåìûóðàâíåíèé(1.14)âûïîëíåíûóñëîâèÿòåîðå-
ìûñóùåñòâîâàíèÿ.
Òåîðåìà11.
Åñëèïîâåðõíîñòü
S
:
u
=
u
(
x
)
êëàññà
C
1
â
ïðîñòðàíñòâåïåðåìåííûõ(
x
,
u
)òàêîâà,÷òî,êàêîâàáûíè
áûëàòî÷êà
(
x
0
;u
0
)
2
S
,õàðàêòåðèñòèêàóðàâíåíèÿ(1.13),
ïðîõîäÿùàÿ÷åðåç
(
x
0
;u
0
)
,êàñàåòñÿ
S
âýòîéòî÷êå,òî
S
ÿâ-
ëÿåòñÿèíòåãðàëüíîéïîâåðõíîñòüþóðàâíåíèÿ(1.13).
Äîêàçàòåëüñòâî.
Ïóñòü
(
x
0
;u
0
)
,
u
0
=
u
(
x
0
)
ïðîèç-
âîëüíàÿòî÷êàïîâåðõíîñòè
S
.Ðàññìîòðèìõàðàêòåðèñòèêó
x
=
x
(
s
)
;u
=
~
u
(
s
)
óðàâíåíèÿ(1.13),ïðîõîäÿùóþ÷åðåçýòóòî÷êó.Òàêèìîáðàçîì,
14
ïðèíåêîòîðîì
s
=
s
0
x
(
s
0
)=
x
0
;
~
u
(
s
0
)=
u
0
:
Êàñàòåëüíàÿïëîñêîñòüêïîâåðõíîñòè
S
âòî÷êå
(
x
0
;u
0
)
èìååòóðàâíåíèå
u

u
0
=
n
X
i
=1
@u
@x
i
(
x
0
)(
x
i

x
0
i
)
:
Ïîóñëîâèþòåîðåìûõàðàêòåðèñòèêà,êàñàåòñÿïîâåðõíîñòè
S
âòî÷êå
(
x
0
;u
0
)
.Ýòîîçíà÷àåò,÷òîêàñàòåëüíûéêõàðàêòåðè-
ñòèêåâåêòîð

dx
1
ds
(
s
0
)
;:::;
dx
n
ds
(
s
0
)
;
d
~
u
ds
(
s
0
)

ëåæèòâóêàçàííîé
ïëîñêîñòè,ò.å.
n
X
i
=1
@u
@x
i
(
x
0
)
dx
i
ds
(
s
0
)=
d
~
u
ds
(
s
0
)
:
(1.15)
Òàêêàê
x
=
x
(
s
)
,
u
=
~
u
(
s
)
ðåøåíèåñèñòåìû(1.14),òî
dx
i
ds
(
s
0
)=
a
i
(
x
(
s
0
)
;
~
u
(
s
0
))=
a
i
(
x
0
;u
0
)
;
d
~
u
ds
(
s
0
)=
b
(
x
(
s
0
)
;
~
u
(
s
0
))=
f
(
x
0
;u
0
)
:
(1.16)
Âñèëó(1.16)ðàâåíñòâî(1.15)ìîæíîïåðåïèñàòüòàê:
n
X
i
=1
a
i
(
x
0
;u
0
)
@u
@x
i
(
x
0
)=
f
(
x
0
;u
0
)
;
ò.å.ôóíêöèÿ
u
=
u
(
x
)
óäîâëåòâîðÿåòóðàâíåíèþ(1.13)ïðè
x
=
x
0
:
Òàêêàê
(
x
0
;u
0
)
ïðîèçâîëüíàÿòî÷êàïîâåðõíîñòè
S
,òî
u
=
u
(
x
)
ïðåäñòàâëÿåòñîáîéðåøåíèåóðàâíåíèÿ
1
:
13
.
Òåîðåìàäîêàçàíà.
Òåîðåìà12.
Ïóñòü
u
=
u
(
x
)
ðåøåíèåóðàâíåíèÿ
(1.13),îïðåäåëåííîåâíåêîòîðîéîáëàñòè
D
,
x
=
x
(
s
)
,
u
=
~
u
(
s
)
15
(
s
)ðåøåíèåñèñòåìû(1.14).Åñëèïðèýòîìêðèâàÿ
x
=
x
(
s
)
(
s
)ëåæèòâ
D
è
~
u
(
s
0
)=
u
(
x
(
s
0
))
,òî
u
(
x
(
s
))

~
u
(
s
)
:
Äîêàçàòåëüñòâî.
Ïóñòü
v
(
s
)=
u
(
x
(
s
))
.Òîãäà,ò.ê.
x
=
x
(
s
)
,
u
=~
u
(
s
)
ðåøåíèåñèñòåìûóðàâíåíèé(1.14),òî
dv
ds
=
n
X
i
=1
@u
@x
i
(
x
(
s
))
dx
i
ds
=
n
X
i
=1
@u
@x
i
(
x
(
s
))
a
i
(
x
(
s
)
;
~
u
(
s
))
:
(1.17)
Òàêêàê
u
(
x
(
s
))
ðåøåíèåóðàâíåíèÿ(1.13),òî,âñèëó(1.14),
èçðàâåíñòâà(1.17)ïîëó÷èì
dv
ds
=
f
(
x
(
s
)
;
~
u
(
s
))=
d
~
u
ds
;
îòêóäàñëåäóåò,÷òî
v
(
s
)=~
u
(
s
)+const
:
Íî
~
u
(
s
0
)=
u
(
x
(
s
0
))
,ò.å.
v
(
s
0
)=~
u
(
s
0
)
.Ñëåäîâàòåëüíî,
const=0
è
v
(
s
)

~
u
(
s
)
.Òåîðåìàäîêàçàíà.
Çàìå÷àíèå13.
Ëþáàÿèíòåãðàëüíàÿïîâåðõíîñòüóðàâ-
íåíèÿ(1.13)îáðàçîâàíàíåêîòîðûìñåìåéñòâîìõàðàêòåðèñòèê.
1.1.2.Èíòåãðèðîâàíèåëèíåéíûõóðàâíåíèéïåðâîãîïîðÿäêà
Ëèíåéíîåíåîäíîðîäíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíû-
ìèïåðâîãîïîðÿäêàèìååòâèä
n
X
i
=1
a
i
(
x
)
@u
@x
i
=
f
(
x
)
;
(1.18)
ãäå
a
i
(
x
)
,
i
=1
;:::;n
,
f
(
x
)
èçâåñòíûåäîñòàòî÷íîãëàäêèå
ôóíêöèèâîáëàñòè
D
ïðîñòðàíñòâàïåðåìåííûõ
x
=(
x
1
;:::;x
n
)
,
ïðè÷åìâýòîéîáëàñòè
n
X
i
=1
a
2
i
(
x
)
6
=0
.
16
Åñëèâïðàâîé÷àñòèóðàâíåíèÿ(1.18)ôóíêöèÿ
f
(
x
)

0
,
ò.å.ýòîóðàâíåíèåèìååòâèä
n
X
i
=1
a
i
(
x
)
@u
@x
i
=0
;
(1.19)
òîòàêîåóðàâíåíèåíàçûâàåòñÿëèíåéíûìîäíîðîäíûìóðàâíå-
íèåìñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà.
Âûïèøåìõàðàêòåðèñòè÷åñêóþñèñòåìó,ñîîòâåòñòâóþùóþ
óðàâíåíèþ(1.19):
dx
i
ds
=
a
i
(
x
)
;i
=1
;:::;n
(1.20)
èëè
dx
1
a
1
(
x
)
=
dx
2
a
2
(
x
)
=
:::
=
dx
n
a
n
(
x
)
:
(1.21)
Ñèñòåìóîáûêíîâåííûõäèôôåðåíöèàëüíûõóðàâíåíèé(1.21)
ìîæíîïåðåïèñàòüâñëåäóþùåì(íîðìàëüíîì)âèäå(ïðåäïîëà-
ãàÿ,÷òî
a
n
(
x
)
6
=0
â
D
):
8



















:
dx
1
dx
n
=
a
1
(
x
)
a
n
(
x
)
;
dx
2
dx
n
=
a
2
(
x
)
a
n
(
x
)
;
:::
dx
n

1
dx
n
=
a
n

1
(
x
)
a
n
(
x
)
:
(1.22)
Êàêèçâåñòíîèçêóðñàîáûêíîâåííûõäèôôåðåíöèàëüíûõóðàâ-
íåíèé,îáùååðåøåíèåñèñòåìû(1.22)ìîæíîçàïèñàòüòàê:
x
i
=
x
i
(
x
n
;C
1
;C
2
;:::;C
n

1
)
;C
i
2
R
;i
=1
;:::;n

1
:
(1.23)
Òàêèìîáðàçîì(1.23)ýòîîáùèéèíòåãðàëñèñòåìû(1.22).
Âûðàæåíèå(1.23)ìîæíîïåðåïèñàòüîòíîñèòåëüíîïîñòî-
ÿííûõ
C
i
:
C
i
=
'
i
(
x
1
;x
2
;:::;x
n
)
;i
=1
;:::;n

1
:
(1.24)
17
Îïðåäåëåíèå18.
Ïåðâûìèíòåãðàëîìñèñòåìû(1.21)
íàçûâàåòñÿôóíêöèÿ
'
=
'
(
x
1
;:::;x
n
)
òîæäåñòâåííîðàâíàÿ
ïîñòîÿííîéíàèíòåãðàëüíîéêðèâîéñèñòåìû(1.21).
Äðóãèìèñëîâàìè(1.24)îïðåäåëÿåò
(
n

1)
íåçàâèñèìûõ
ïåðâûõèíòåãðàëîâñèñòåìû(1.21)(ðàíãìàòðèöû



@'
i
@x
j



ðàâåí
n

1
â
D
).
Òåîðåìà13.
Âñÿêîåðåøåíèå
u
=
u
(
x
)
óðàâíåíèÿ(1.19)
ÿâëÿåòñÿïåðâûìèíòåãðàëîìñèñòåìû(1.21).Èîáðàòíî,âñÿ-
êèéïåðâûéèíòåãðàëñèñòåìû(1.21)
'
(
x
1
;:::;x
n
)
ÿâëÿåòñÿðå-
øåíèåìóðàâíåíèÿ(1.19).
Òåîðåìà14.
Ðåøåíèå
u
=
u
(
x
)
óðàâíåíèÿ(1.19)ïðåä-
ñòàâèìîââèäå
u
=
F
(
'
1
;:::;'
n

1
)
,ãäå
F
íåêîòîðàÿ
íåïðåðûâíîäèôôåðåíöèðóåìàÿôóíêöèÿñâîèõàðãóìåíòîâ,
'
1
;:::;'
n

1
íåçàâèñèìûåïåðâûåèíòåãðàëûñèñòåìû(1.21).
Ïðèìåð11.
Ðåøèòüóðàâíåíèå
x
@u
@x
+
y
@u
@y
=0
.
Ðåøåíèå.
Ñîñòàâëÿåìõàðàêòåðèñòè÷åñêóþñèñòåìóäëÿ
äàííîãîóðàâíåíèÿ:
8



:
dx
ds
=
x;
dy
ds
=
y:
Îòñþäàñëåäóåò,÷òî
dx
dy
=
x
y
:
Ðàçäåëÿÿïåðåìåííûåèèíòåãðèðóÿ,ïîëó÷àåìïåðâûéèíòåãðàë
õàðàêòåðèñòè÷åñêîéñèñòåìû
'
(
x;y
)=
x
y
=
C;C
2
R
:
Òîãäàîáùååðåøåíèåèñõîäíîãîóðàâíåíèÿìîæíîçàïèñàòüâ
ñëåäóþùåìâèäå:
u
(
x;y
)=
F

x
y

;
18
ãäå
F
ïðîèçâîëüíàÿíåïðåðûâíîäèôôåðåíöèðóåìàÿôóíê-
öèÿ.
1.1.3.Èíòåãðèðîâàíèåêâàçèëèíåéíûõóðàâíåíèéïåðâîãîïîðÿäêà
Ðàññìîòðèìòåïåðüêâàçèëèíåéíîåóðàâíåíèåñ÷àñòíûìè
ïðîèçâîäíûìèïåðâîãîïîðÿäêà
n
X
i
=1
a
i
(
x
;u
)
@u
@x
i
=
f
(
x
;u
)
;
(1.25)
ãäå
a
i
(
x
;u
)
,
i
=1
;:::;n
,
f
(
x
;u
)
2
C
(
G
)
,ïðè÷åìâîáëàñòè
G
ïðîñòðàíñòâàïåðåìåííûõ
x
;u
èìååì
n
X
i
=1
a
2
i
6
=0
.
Íàïîìíèì,÷òîõàðàêòåðèñòè÷åñêàÿñèñòåìàóðàâíåíèÿ
(1.25)èìååòâèä
8







:
dx
i
ds
=
a
i
(
x
;u
)
;i
=1
;:::;n;
du
ds
=
f
(
x
;u
)
:
(1.26)
Áóäåìèñêàòüðåøåíèåóðàâíåíèÿ(1.25)âíåÿâíîìâèäå
v
(
x
1
;:::;x
n
;u
)=0
:
(1.27)
Ïðîäèôôåðåíöèðóåì(1.27)ïî
x
i
,ïîëó÷èì
@v
@u
@u
@x
i
+
@v
@x
i
=0
;i
=1
;:::;n:
(1.28)
Äîìíîæèâòåïåðüîáå÷àñòèóðàâíåíèÿ(1.25)íà
@v
@u
,è,èñïîëü-
çóÿðàâåíñòâà(1.28),ïðèõîäèìêëèíåéíîìóîäíîðîäíîìóóðàâ-
íåíèþñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà
n
X
i
=1
a
i
(
x
;u
)
@v
@x
i
+
f
(
x
;u
)
@v
@u
=0
(1.29)
19
îòíîñèòåëüíîíåèçâåñòíîéôóíêöèè
v
=
v
(
x
;u
)
.
Äàëååäåéñòâóåìêàêâïðåäûäóùåìðàçäåëå.Ñîñòàâëÿ-
åìõàðàêòåðèñòè÷åñêóþñèñòåìóóðàâíåíèÿ(1.29)èíàõîäèì
ååíåçàâèñèìûåïåðâûåèíòåãðàëû
'
1
(
x
;u
)
;:::;'
n
(
x
;u
)
.Òî-
ãäàîáùååðåøåíèåóðàâíåíèÿ(1.29)ìîæíîçàïèñàòüââèäå
v
=
F
(
'
1
;:::;'
n
)
,ãäå
F
íåêîòîðàÿíåïðåðûâíîäèôôåðåíöè-
ðóåìàÿôóíêöèÿñâîèõàðãóìåíòîâ.Íàêîíåö,ó÷èòûâàÿ(1.27),
ïîëó÷àåìîáùååðåøåíèåèñõîäíîãîóðàâíåíèÿ(1.25)âíåÿâíîì
âèäå
F
(
'
1
;:::;'
n
)=0
:
(1.30)
Îòìåòèì,÷òîåñëè
u
âõîäèòòîëüêîâîäèíèçïåðâûõèí-
òåãðàëîâ
'
1
;:::;'
n
,íàïðèìåð,âïîñëåäíèé,ò.å.
'
n
=
'
n
(
x
;u
)
,
'
i
=
'
i
(
x
)
,
i
=1
;:::;n

1
,òîîáùååðåøåíèåóðàâíåíèÿ(1.25)
ìîæíîçàïèñàòüâñëåäóþùåìâèäå:
'
n
(
x
;u
)=
f
(
'
1
;:::;'
n

1
)
;
(1.31)
ãäå
f
íåêîòîðàÿíåïðåðûâíîäèôôåðåíöèðóåìàÿôóíêöèÿ
ñâîèõàðãóìåíòîâÐàçðåøèâäàííîåðàâåíñòâîîòíîñèòåëüíî
u
(êîíå÷íîåñëèýòîâîçìîæíî),ïîëó÷èìðåøåíèåóðàâíåíèÿ
(1.25)âÿâíîìâèäå.
Ïðèìåð12.
Ðåøèòüóðàâíåíèå
xz
@z
@x
+
yz
@z
@y
=
x
.
Ðåøåíèå.
Áóäåìèñêàòüðåøåíèåââèäå
v
(
x;y;z
)=0
,ãäå
v
íåèçâåñòíàÿôóíêöèÿïåðåìåííûõ
x;y;z
.Äèôôåðåíöèðóÿ
ïîñëåäíååâûðàæåíèåïî
x
è
y
,ïîëó÷èìñîîòâåòñòâåííî
@v
@z
@z
@x
+
@v
@x
=0
è
@v
@z
@z
@y
+
@v
@y
=0
:
Òåïåðüóìíîæèìèñõîäíîåóðàâíåíèåíà
@v
@z
è,ó÷èòûâàÿïî-
ñëåäíèåäâàðàâåíñòâà,èìååìëèíåéíîåîäíîðîäíîåóðàâíåíèå
20
ñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà
xz
@v
@x
+
yz
@v
@y
+
x
@v
@z
=0
:
Çàïèøåìõàðàêòåðèñòè÷åñêóþñèñòåìóýòîãîóðàâíåíèÿ:
dx
xz
=
dy
yz
=
dz
x
:
Îòñþäàíàõîäèìäâàïåðâûõèíòåãðàëà
'
1
=
z
2

2
x
è
'
2
=
x
y
:
Ïîñêîëüêó
z
âõîäèòÿâíîòîëüêîâîäèíèçïåðâûõèíòåãðàëîâ
(âïåðâûé),òîîáùååðåøåíèåèñõîäíîãîóðàâíåíèÿìîæíîçà-
ïèñàòüââèäå
z
2

2
x
=
f

x
y

èëè
z
(
x;y
)=

r
f

x
y

+2
x;
ãäå
f
ïðîèçâîëüíàÿíåïðåðûâíîäèôôåðåíöèðóåìàÿôóíê-
öèÿ.
1.1.4.Çàäà÷àÊîøèäëÿêâàçèëèíåéíîãîóðàâíåíèÿïåðâîãîïîðÿäêà
Êàêìûóæåóáåäèëèñü,óðàâíåíèÿñ÷àñòíûìèïðîèçâîä-
íûìèïåðâîãîïîðÿäêàäîïóñêàþòáåñêîíå÷íîìíîãîðåøåíèé.
Äëÿîäíîçíà÷íîãîâûäåëåíèÿðåøåíèÿíåîáõîäèìîêóðàâíåíèþ
ïðèñîåäèíèòüíà÷àëüíûåóñëîâèÿ.
Ðàññìîòðèìêâàçèëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèç-
âîäíûìèïåðâîãîïîðÿäêàâñëó÷àåäâóõïðîñòðàíñòâåííûõïå-
ðåìåííûõ
x
è
y
a
1
(
x;y;u
)
@u
@x
+
a
2
(
x;y;u
)
@u
@y
=
f
(
x;y;u
)
;
(1.32)
21
ãäå
a
i
(
x;y;u
)
,
i
=1
;
2
,
f
(
x;y;u
)
èçâåñòíûåãëàäêèåôóíêöèè,
ïðè÷åì
a
2
1
+
a
2
2
6
=0
.
Ïóñòüçàäàíàäîñòàòî÷íîãëàäêàÿêðèâàÿ
l
âïàðàìåòðè÷å-
ñêîéôîðìå
8







:
x
=
x
(
t
)
;
y
=
y
(
t
)
;
u
=
u
(
t
)
;
(1.33)
ïðè÷åì
x
2
t
+
y
2
t
6
=0
.Îáîçíà÷èì÷åðåç
l
0
ïðîåêöèþêðèâîé
l
íà
ïëîñêîñòü
Oxy:
Çàäà÷àÊîøèäëÿóðàâíåíèÿ(1.32)ñòàâèòñÿòàê:âîêðåñò-
íîñòèïðîåêöèè
l
0
íàéòèèíòåãðàëüíóþïîâåðõíîñòüóðàâíåíèÿ
(1.32),ïðîõîäÿùóþ÷åðåççàäàííóþêðèâóþ
l
,ò.å.íàéòèòàêîå
ðåøåíèÿóðàâíåíèÿ(1.32),êîòîðîåïðèíèìàåòçàäàííûåçíà÷å-
íèÿâòî÷êàõêðèâîé
l
0
.
Äëÿðåøåíèÿçàäà÷èÊîøèïðîâåäåì÷åðåçêàæäóþòî÷êó
êðèâîé
l
õàðàêòåðèñòèêó,ò.å.èíòåãðàëüíóþêðèâóþñèñòåìû
8











:
dx
ds
=
a
1
(
x;y;u
)
;
dy
ds
=
a
2
(
x;y;u
)
;
du
ds
=
f
(
x;y;u
)
:
(1.34)
Çàìåòèì,÷òîâíåêîòîðîéîêðåñòíîñòèêðèâîé
l
ýòîìîæíîñäå-
ëàòüåäèíñòâåííûìîáðàçîì.Òàêèìîáðàçîììûïîëó÷èëèñå-
ìåéñòâîõàðàêòåðèñòè÷åñêèõêðèâûõ,çàâèñÿùèõåùåîòïàðà-
ìåòðà
t
:
x
=
x
(
s;t
)
;y
=
y
(
s;t
)
;u
=
u
(
s;t
)
:
(1.35)
Êðèâûå(1.35)îáðàçóþòïîâåðõíîñòü
u
=
u
(
x;y
)
,åñëèèçïåð-
âûõäâóõóðàâíåíèé(1.35)ìîæíîâûðàçèòü
s
è
t
÷åðåç
x
è
y
.
Äëÿýòîãîäîñòàòî÷íî,÷òîáûíàêðèâîé
l
íåîáðàùàëñÿâíóëü
22
ÿêîáèàí
=








dx
ds
dx
dt
dy
ds
dy
dt








=
a
1
dy
dt

a
2
dx
dt
:
(1.36)
Åñëèíà
l
âûïîëíÿåòñÿóñëîâèå

6
=0
,òî
u
ÿâëÿåòñÿôóíê-
öèåé
x
è
y
.Íåòðóäíîâèäåòü,÷òîýòàôóíêöèÿåñòüðåøåíèå
óðàâíåíèÿ(1.32).Äåéñòâèòåëüíî,ïîëüçóÿñüïðàâèëîìäèôôå-
ðåíöèðîâàíèÿñëîæíîéôóíêöèèèóðàâíåíèÿìè(1.34),ïîëó-
÷èì
du
ds
=
a
1
@u
@x
+
a
2
@u
y
:
Íî
du
ds
=
f
è,ñëåäîâàòåëüíî,
u
(
x;y
)
óäîâëåòâîðÿåòóðàâíåíèþ
(1.32).Åäèíñòâåííîñòüðåøåíèÿçàäà÷èÊîøèñëåäóåòèçòîãî,
÷òîõàðàêòåðèñòè÷åñêàÿêðèâàÿ,èìåþùàÿîäíóîáùóþòî÷êóñ
èíòåãðàëüíîéïîâåðõíîñòüþ,öåëèêîìëåæèòíàýòîéïîâåðõíî-
ñòè.Ýòîçíà÷èò,÷òîëþáàÿèíòåãðàëüíàÿïîâåðõíîñòü,ïðîõî-
äÿùàÿ÷åðåçêðèâóþ
l
,öåëèêîìñîäåðæèòñåìåéñòâîõàðàêòåðè-
ñòèê,ïðîõîäÿùèõ÷åðåç
l
è,ñëåäîâàòåëüíî,ñîâïàäàåòc
u
(
x;y
)
.
Åñëè
=0
âñþäóíàêðèâîé
l
èåñëèñóùåñòâóåòèíòåãðàëü-
íàÿïîâåðõíîñòü
u
=
u
(
x;y
)
ñíåïðåðûâíûìèïðîèçâîäíûìè
ïåðâîãîïîðÿäêà,ïðîõîäÿùàÿ÷åðåç
l
,òîýòàêðèâàÿäîëæíà
áûòüõàðàêòåðèñòèêîé.Âñàìîìäåëå,âýòîìñëó÷àåïàðàìåòð
t
íàêðèâîé
l
ìîæíîâûáðàòüòàê,÷òîâäîëüýòîéêðèâîé
a
1
=
dx
dt
,
a
2
=
dy
dt
.Äàëåå,ïîäñòàâëÿÿâ
u
(
x;y
)
âûðàæåíèÿ
x
=
x
(
t
)
,
y
=
y
(
t
)
èäèôôåðåíöèðóÿïî
t
,áóäåìèìåòü
du
dt
=
a
1
@u
@x
+
a
2
@u
@y
.
Îòñþäà,ó÷èòûâàÿ,÷òî
u
(
x;y
)
åñòüðåøåíèÿóðàâíåíèÿ(1.32),
ïîëó÷èì
du
dt
=
f
.Ñëåäîâàòåëüíî
l
ÿâëÿåòñÿõàðàêòåðèñòèêîé.
Íî,åñëè
l
õàðàêòåðèñòèêà,òî÷åðåçíååïðîõîäèòíåòîëüêî
23
îäíà,àáåñêîíå÷íîìíîãîèíòåãðàëüíûõïîâåðõíîñòåé.Äåéñòâè-
òåëüíî,ïðîâåäåì÷åðåçëþáóþòî÷êóêðèâîé
l
êðèâóþ
l
0
,êîòî-
ðàÿóæåíåÿâëÿåòñÿõàðàêòåðèñòèêîé.Èíòåãðàëüíàÿïîâåðõ-
íîñòü,ïðîõîäÿùàÿ÷åðåç
l
0
,îáÿçàòåëüíîñîäåðæèòõàðàêòåðè-
ñòèêó
l
.Òàêèìîáðàçîì,ìíîæåñòâîðåøåíèéçàäà÷èÊîøèäëÿ
õàðàêòåðèñòèêè
l
îïðåäåëÿåòñÿìíîæåñòâîìêðèâûõ
l
0
.Âñåèí-
òåãðàëüíûåïîâåðõíîñòè,ïðîõîäÿùèå÷åðåçêðèâûåýòîãîìíî-
æåñòâà,ñîäåðæàòõàðàêòåðèñòèêó
l
.Ñëåäîâàòåëüíî,õàðàêòå-
ðèñòèêèÿâëÿþòñÿëèíèÿìèïåðåñå÷åíèÿèíòåãðàëüíûõïîâåðõ-
íîñòåé(ëèíèÿìèâåòâëåíèÿ),òîãäàêàê÷åðåçíåõàðàêòåðèñòè-
÷åñêóþêðèâóþíåìîæåòïðîõîäèòüáîëååîäíîéèíòåãðàëüíîé
ïîâåðõíîñòè.
Ñôîðìóëèðóåìïîëó÷åííûåðåçóëüòàòû.
Òåîðåìà15.
Åñëè

6
=0
âñþäóíàíà÷àëüíîéêðèâîé
l
,òî
çàäà÷àÊîøèäëÿóðàâíåíèÿ(1.32)èìååòîäíîèòîëüêîîäíî
ðåøåíèå.Åñëèæå
=0
âñþäóíà
l
,òîäëÿòîãî÷òîáûçà-
äà÷àÊîøèèìåëàðåøåíèå,êðèâàÿ
l
äîëæíàáûòüõàðàêòåðè-
ñòèêîé.Âýòîìñëó÷àåçàäà÷àÊîøèèìååòáåñêîíå÷íîìíîãî
ðåøåíèé.
Ïðèìåð13.
Ðàññìîòðèìóðàâíåíèå
(
y
2

u
)
@u
@x
+
y
@u
@y
=
u:
(1.37)
Ñèñòåìà
(1
:
34)
èìååòâèä
dx
ds
=
y
2

u;
dy
ds
=
y;
du
ds
=
u
(1.38)
èååðåøåíèå,âûðàæåííîå÷åðåçíà÷àëüíûåçíà÷åíèÿ
(
x
0
;y
0
;u
0
)
ïåðåìåííûõ
(
x;y;u
)
ïðè
s
=0
,áóäåò
x
=

1
2
y
2
0
e
s

u
0

e
s
+
x
0
+

u
0

1
2
y
2
0

;y
=
y
0
e
s
;u
=
u
0
e
s
:
(1.39)
24
Ïîëîæèì,÷òîêðèâàÿ
l
,÷åðåçêîòîðóþäîëæíàïðîõîäèòüèí-
òåãðàëüíàÿïîâåðõíîñòü,çàäàíàóðàâíåíèÿìè
x
0
=1
;y
0
=
t;u
0
=
1
2
t
2
:
(1.40)
Ïîäñòàâèâ(1.40)â(1.39),ïîëó÷èì
x
=
t
2
2
(
e
s

1)
e
s
+1
;y
=
te
s
;u
=
t
2
2
e
s
:
Îïðåäåëèòåëü
=
dx
ds
dy
dt

dx
dt
dy
ds
=
t
2
2
e
2
s
;
íåîáðàùàåòñÿâíóëüïðè
s
=0
è
t
6
=0
.Èñêëþ÷àÿ
s
è
t
,ìû
ïîëó÷èìóðàâíåíèåèíòåãðàëüíîéïîâåðõíîñòè
u
=1

x
+
y
2
2
:
1.2.Óðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêà
1.2.1.Ïîíÿòèåõàðàêòåðèñòè÷åñêîéôîðìûè
êëàññèôèêàöèÿëèíåéíûõóðàâíåíèéâòîðîãîïîðÿäêà
Ðàññìîòðèìëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìè
âòîðîãîïîðÿäêà
n
X
i;j
=1
a
ij
(
x
)
@
2
u
@x
i
@x
j
+
n
X
i
=1
b
i
(
x
)
@u
@x
i
+
c
(
x
)
u
=
f
(
x
)
;
(1.41)
ãäå
a
ij
=
a
ji
,
b
i
,
c
,
f
çàäàííûåâîáëàñòè
D

R
n
äåéñòâè-
òåëüíûåãëàäêèåôóíêöèèòî÷êè
x
=(
x
1
;:::;x
n
)
.Ïðåäïîëàãà-
åòñÿ,÷òîõîòÿáûîäèíèçêîýôôèöèåíòîâ
a
ij
âòî÷êàõ
x
2
D
îòëè÷åíîòíóëÿ,èíà÷åóðàâíåíèåïåðåñòàåòáûòüóðàâíåíèåì
âòîðîãîïîðÿäêà,ò.å.âóêàçàííûõòî÷êàõïîðÿäîêóðàâíåíèÿ
(1.41)âûðîæäàåòñÿ.
25
Îïðåäåëåíèå19.
Ôîðìàïîðÿäêà
m
K
(

1
;:::;
n
)=
X
i
1

0
;:::;i
n

0
@F
@p
i
1
:::i
n

i
1
1


i
n
n
;
n
X
j
=1
i
j
=
m
(1.42)
îòíîñèòåëüíîäåéñòâèòåëüíûõïàðàìåòðîâ

1
;:::;
n
íàçûâà-
åòñÿõàðàêòåðèñòè÷åñêîéôîðìîéñîîòâåòñòâóþùåéóðàâíå-
íèþ(1.1).
Âñëó÷àåëèíåéíîãîóðàâíåíèÿâòîðîãîïîðÿäêàõàðàêòåðè-
ñòè÷åñêàÿôîðìà(1.42)ÿâëÿåòñÿêâàäðàòè÷íîé:
Q
(

1
;:::;
n
)=
n
X
i;j
=1
a
ij
(
x
)

i

j
:
(1.43)
Âêàæäîéòî÷êå
x
2
D
êâàäðàòè÷íàÿôîðìà
Q
ïðèïî-
ìîùèíåâûðîæäåííîãîàôôèííîãîïðåîáðàçîâàíèÿïåðåìåííûõ

i
=

i
(

1
;:::;
n
)
,
i
=1
;:::;n
ìîæåòáûòüïðèâåäåíàêêàíîíè-
÷åñêîìóâèäó
Q
(

1
;:::;
n
)=
n
X
i
=1
a
i

2
i
;
(1.44)
ãäå
a
i
=

1
;
0
;
1
;i
=1
;:::;n
,ïðè÷åì÷èñëîîòðèöàòåëüíûõèíó-
ëåâûõêîýôôèöèåíòîâôîðìû
Q
íåçàâèñèòîòñïîñîáàïðèâåäå-
íèÿýòîéôîðìûêêàíîíè÷åñêîìóâèäó.Íàýòîìôàêòåîñíîâàíà
êëàññèôèêàöèÿëèíåéíûõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíû-
ìèâòîðîãîïîðÿäêà.
Îïðåäåëåíèå110.
Óðàâíåíèå(1.41)ïðèíàäëåæèòýë-
ëèïòè÷åñêîìóòèïóâòî÷êå
x
2
D
,åñëèâýòîéòî÷êåêâàä-
ðàòè÷íàÿôîðìà
Q
ïîëîæèòåëüíîîïðåäåëåííàÿèëèîòðè-
öàòåëüíîîïðåäåëåííàÿ,ò.å.âñå
a
i
=1
èëèâñå
a
i
=

1
,
i
=1
;:::;n
.
Óðàâíåíèå(1.41)ïðèíàäëåæèòãèïåðáîëè÷åñêîìóòèïóâ
òî÷êå
x
2
D
,åñëèâýòîéòî÷êåâêâàäðàòè÷íîéôîðìå
Q
èìååìâñåêîýôôèöèåíòû,êðîìåîäíîãî,îïðåäåëåííîãîçíàêà,
àîñòàâøèéñÿîäèíêîýôôèöèåíòïðîòèâîïîëîæíîãîçíàêà.
26
Óðàâíåíèå(1.41)ïðèíàäëåæèòóëüòðàãèïåðáîëè÷åñêîìó
òèïóâòî÷êå
x
2
D
,åñëèâýòîéòî÷êåêâàäðàòè÷íàÿôîð-
ìà
Q
èìååòáîëüøåîäíîãîïîëîæèòåëüíîãîêîýôôèöèåíòàè
áîëååîäíîãîîòðèöàòåëüíîãî,ïðè÷åìâñåêîýôôèöèåíòûîò-
ëè÷íûîòíóëÿ.
Óðàâíåíèå(1.41)ïðèíàäëåæèòïàðàáîëè÷åñêîìóòèïóâ
òî÷êå
x
2
D
,åñëèâýòîéòî÷êåêâàäðàòè÷íàÿôîðìà
Q
èìååò
òîëüêîîäèíêîýôôèöèåíò,ðàâíûéíóëþ,âñåæåäðóãèåêîýô-
ôèöèåíòûèìåþòîäèíàêîâûåçíàêè.
Óðàâíåíèå(1.41)ïðèíàäëåæèòýëëèïòè÷åñêîìóòèïó,ñîîò-
âåòñòâåííîãèïåðáîëè÷åñêîìóòèïóèò.ä.âîáëàñòè
D
,åñëèâî
âñåõòî÷êàõýòîéîáëàñòèîíîïðèíàäëåæèòýëëèïòè÷åñêîìóòè-
ïó,ñîîòâåòñòâåííîãèïåðáîëè÷åñêîìóòèïóèò.ä.
Îïðåäåëåíèå111.
Ýëëèïòè÷åñêîåâîáëàñòè
D
óðàâíå-
íèå(1.41)íàçûâàåòñÿðàâíîìåðíîýëëèïòè÷åñêèì,åñëèñóùå-
ñòâóþòäåéñòâèòåëüíûå÷èñëà
k
1
,
k
2
,îòëè÷íûåîòíóëÿè
îäíîãîçíàêàòàêèå,÷òî
k
1
n
X
i
=1

2
i
6
Q
(

1
;:::;
n
)
6
k
2
n
X
i
=1

2
i
8
x
2
D:
Êîãäàâðàçíûõòî÷êàõîáëàñòè
D
óðàâíåíèå(1.41)ïðèíàä-
ëåæèòðàçëè÷íûìòèïàì,òîîíîÿâëÿåòñÿóðàâíåíèåìñìåøàí-
íîãîòèïàâýòîéîáëàñòè.
Åñëèêîýôôèöèåíòû
a
ij
ïîñòîÿííûå,òîïðèíàäëåæíîñòü
óðàâíåíèÿêòîìóèëèèíîìóòèïóíåçàâèñèòîòçíà÷åíèéíåçà-
âèñèìûõïåðåìåííûõ.
Ïðèìåð14.
Äëÿìíîãîìåðíîãîâîëíîâîãîóðàâíåíèÿâèäà
@
2
u
@t
2
=
n
X
i
=1
@
2
u
@x
2
i
=
u;
x
2
R
n
ñîñòàâèìêâàäðàòè÷íóþôîðìó
Q
(

1
;:::;
n
+1
)=


2
1


2
2

:::


2
n
+

2
n
+1
:
27
Âñîîòâåòñòâèèñêëàññèôèêàöèåéäàííîåóðàâíåíèåïðèíàäëå-
æèòãèïåðáîëè÷åñêîìóòèïó.
Ïðèìåð15.
ÐàññìîòðèìóðàâíåíèåËàïëàñàâìíîãîìåð-
íîìñëó÷àå:

u
=0
;
x
2
R
n
:
Êâàäðàòè÷íàÿôîðìàèìååòâèä
Q
(

1
;:::;
n
)=

2
1
+
:::
+

2
n
;
ñëåäîâàòåëüíî,èñõîäíîåóðàâíåíèåïðèíàäëåæèòýëëèïòè÷å-
ñêîìóòèïó.
Ïðèìåð16.
Äëÿóðàâíåíèÿòåïëîïðîâîäíîñòèâèäà
@u
@t
=
u;
x
2
R
n
õàðàêòåðèñòè÷åñêàÿôîðìàèìååòâèä
Q
(

1
;:::;
n
+1
)=


2
1

:::


2
n
+0


2
n
+1
;
òàêèìîáðàçîì,óðàâíåíèåòåïëîïðîâîäíîñòèÿâëÿåòñÿóðàâíå-
íèåìïàðàáîëè÷åñêîãîòèïà.
Ïðèìåð17.
ÐàññìîòðèìóðàâíåíèåÒðèêîìè:
y
@
2
u
@x
2
+
@
2
u
@y
2
=0
:
Êîýôôèöèåíòûõàðàêòåðèñòè÷åñêîéôîðìû
Q
=
y
2
1
+

2
2
çàâèñÿòîò
y
,ñëåäîâàòåëüíî,äàííîåóðàâíåíèåâðàçíûõòî÷êàõ
îáëàñòè
D
èìååòðàçíûéòèï,ò.å.óðàâíåíèåÒðèêîìèÿâëÿåòñÿ
óðàâíåíèåìñ÷àñòíûìèïðîèçâîäíûìèñìåøàííîãîòèïà.
1.2.2.Êëàññèôèêàöèÿíåëèíåéíûõóðàâíåíèéâòîðîãîïîðÿäêà
Ðàññìîòðèìíåëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíû-
ìèâòîðîãîïîðÿäêà
F

x
;u;
@u
@x
1
;:::;
@u
@x
n
;:::
@
2
u
@x
i
@x
j
;:::

=0
;i;j
=1
;:::;n;
(1.45)
28
ãäå
F
(
x
;:::;p
i
1
;:::;i
n
;:::
)
çàäàííàÿãëàäêàÿäåéñòâèòåëüíàÿ
ôóíêöèÿòî÷åê
x
îáëàñòè
D
èäåéñòâèòåëüíûõïåðåìåí-
íûõ
p
i
1
;:::;i
n
ñíåîòðèöàòåëüíûìèöåëî÷èñëåííûìèèíäåêñàìè
i
1
;:::;i
n
,
n
X
j
=1
i
j
=
k
,
k
=0
;
1
;
2
,è,ïîêðàéíåéìåðå,îäíàèç
ïðîèçâîäíîéêîòîðîé
@F
@p
i
1
;:::;i
n
;
n
X
j
=1
i
j
=2
îòëè÷íàîòíóëÿ.
Êëàññèôèêàöèÿóðàâíåíèÿ(1.45)âíåëèíåéíîìñëó÷àåïðî-
èñõîäèòàíàëîãè÷íûìîáðàçîì,÷òîèêëàññèôèêàöèÿóðàâíåíèÿ
(1.41)ïîõàðàêòåðóõàðàêòåðèñòè÷åñêîéôîðìû
Q
(

1
;:::;
n
)=
n
X
i;j
=1
@F
@u
x
i
x
j

i

j
:
(1.46)
Ïîñêîëüêóêîýôôèöèåíòûôîðìû(1.46)çàâèñÿòîòèñêîìî-
ãîðåøåíèÿèåãîïðîèçâîäíûõ,òîêëàññèôèêàöèÿèìååòñìûñë
ëèøüäëÿýòîãîðåøåíèÿ.
Ïðèìåð18.
Îïðåäåëèòüòèïóðàâíåíèÿ
u
2
xx
+(
u
xx

2)
u
xy

u
2
yy
=0
íàðåøåíèè
u
=
x
2
+
y
2
.
Ðåøåíèå.
Õàðàêòåðèñòè÷åñêàÿôîðìà
Q
(

1
;
2
)=
@F
@u
xx

2
1
+
@F
@u
xy

1

2
+
@F
@u
yy

2
2
äëÿäàííîãîóðàâíåíèÿïðèìåòâèä
Q
(

1
;
2
)=(2
u
xx
+
u
xy
)

2
1
+(
u
xx

2)

1

2

2
u
yy

2
2
:
Íàðåøåíèè
u
=
x
2
+
y
2
èìååì
Q
(

1
;
2
)=4

2
1

4

2
2
29
èëè
Q
(

1
;
2
)=

2
1


2
2
;
i
=2

i
;i
=1
;
2
:
Ñëåäîâàòåëüíî,äàííîåóðàâíåíèåïðèíàäëåæèòãèïåðáîëè÷å-
ñêîìóòèïóíàäàííîìðåøåíèè.
1.2.3.Êëàññèôèêàöèÿñèñòåìäâóõëèíåéíûõóðàâíåíèé
ïåðâîãîïîðÿäêà
Ðàññìîòðèìñèñòåìóäâóõëèíåéíûõóðàâíåíèéñ÷àñòíûìè
ïðîèçâîäíûìèïåðâîãîïîðÿäêà
8









:
n
X
i
=1
a
1
i
(
x
)
u
x
i
+
n
X
i
=1
b
1
i
(
x
)
v
x
i
+
c
1
(
x
)
u
+
d
1
(
x
)
v
=
f
1
(
x
)
;
n
X
i
=1
a
2
i
(
x
)
u
x
i
+
n
X
i
=1
b
2
i
(
x
)
v
x
i
+
c
2
(
x
)
u
+
d
2
(
x
)
v
=
f
2
(
x
)
;
(1.47)
ãäå
a
ji
,
b
ji
,
i
=1
;:::;n
,
c
j
,
d
j
,
f
j
,
j
=1
;
2
çàäàííûåâîáëà-
ñòè
D

R
n
äåéñòâèòåëüíûåãëàäêèåôóíêöèè,çàâèñÿùèåîò
x
=(
x
1
;:::;x
n
)
.
Êëàññèôèêàöèÿñèñòåìâèäà(1.47)ïðîèñõîäèòïîõàðàêòå-
ðóõàðàêòåðèñòè÷åñêîéôîðìû(õàðàêòåðèñòè÷åñêîãîäåòåðìè-
íàíòà)
Q
(

1
;:::;
n
)=










n
X
i
=1
a
1
i

i
n
X
i
=1
b
1
i

i
n
X
i
=1
a
2
i

i
n
X
i
=1
b
2
i

i










(1.48)
òî÷íîòàêæåêàêèêëàññèôèêàöèÿîäíîãîëèíåéíîãîóðàâíåíèÿ
âòîðîãîïîðÿäêà(1.41).
Ïðèìåð19.
Ðàññìîòðèìñèñòåìóóðàâíåíèé
(
5
u
x
+22
;
5
v
x
+2
u
y
+
v
y

6
u
=0
;
5
v
x
+2
u
y
+3
v
y

2
xu
=0
:
30
Âûïèøåìõàðàêòåðèñòè÷åñêóþôîðìó,ñîîòâåòñòâóþùóþäàí-
íîéñèñòåìå
Q
(

1
;
2
)=





5

1
+2

2
22
;
5

1
+

2
2

2
5

1
+3

2





èëè
Q
(

1
;
2
)=(5

1

2

2
)
2
:
Îòñþäà
Q
(

1
;
2
)=

2
1
+0


2
2
;
ãäå

1
=5

1

2

2
;
2
=

2
(

1
;
2
)
ëèíåéíî-íåçàâèñèìàÿñ

1
.
Ñëåäîâàòåëüíî,äàííàÿñèñòåìàóðàâíåíèéïðèíàäëåæèòïàðà-
áîëè÷åñêîìóòèïó.
1.2.4.Ïðèâåäåíèåêêàíîíè÷åñêîìóâèäóëèíåéíîãî
óðàâíåíèÿâòîðîãîïîðÿäêàñïîñòîÿííûìè
êîýôôèöèåíòàìè
Ðàññìîòðèìëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìè
âòîðîãîïîðÿäêàñïîñòîÿííûìèêîýôôèöèåíòàìè
n
X
i;j
=1
a
ij
@
2
u
@x
i
@x
j
+
n
X
i
=1
b
i
@u
@x
i
+
cu
=
f
(
x
1
;:::;x
n
)
:
(1.49)
Çäåñü
a
ij
=
a
ji
,
b
i
;i;j
=1
;:::;n
,
c
èçâåñòíûåïîñòîÿííûå,ïðè-
÷åìîäèíèç
a
ij
íåðàâåííóëþ,
f
çàäàííàÿâîáëàñòè
D

R
n
ãëàäêàÿäåéñòâèòåëüíàÿôóíêöèÿïåðåìåííûõ
x
1
;:::;x
n
.
Ââåäåìâìåñòî
x
1
;:::;x
n
íîâûåíåçàâèñèìûåïåðåìåííûå

1
;:::;
n
ïðèïîìîùèëèíåéíîãîïðåîáðàçîâàíèÿ

k
=
n
X
i
=1
c
ki
x
i
;k
=1
;:::;n:
(1.50)
Ìûïðåäïîëàãàåì,÷òîïðåîáðàçîâàíèå(1.50)íåâûðîæäåííîå,
ò.å.÷òîîïðåäåëèòåëü
j
c
ki
j
íåðàâåííóëþ.Ïðîèçâîäíûåïîñòà-
ðûìïåðåìåííûìïåðåñ÷èòàþòñÿ÷åðåçïðîèçâîäíûåïîíîâûì
31
ïåðåìåííûìñëåäóþùèìèôîðìóëàìè:
@u
@x
i
=
n
X
k
=1
c
ki
@v
@
k
;
@
2
u
@x
i
@x
j
=
n
X
k;l
=1
c
ki
c
lj
@
2
v
@
k
@
l
;
(1.51)
ãäå
u
(
x
1
;:::;x
n
)=
v
(

1
(
x
1
;:::;x
n
)
;:::;
n
(
x
1
;:::;x
n
))
.Ïîäñòàâèâ
(1.51)âóðàâíåíèå(1.49),ïîëó÷èì
n
X
k;l
=1

a
kl
@
2
v
@
k
@
l
+
n
X
k
=1

b
k
@v
@
k
+
cv
=
f
1
(

1
;:::;
n
)
;
(1.52)
ãäå

a
kl
=
n
X
i;j
=1
a
ij
c
ki
c
lj
;

b
k
=
n
X
i
=1
b
i
c
ki
:
(1.53)
Íåòðóäíîïðîâåðèòü,÷òîôîðìóëûïðåîáðàçîâàíèÿ(1.51)êîýô-
ôèöèåíòîâïðèâòîðûõïðîèçâîäíûõîòôóíêöèè
u
ïðèçàìåíå
íåçàâèñèìûõïåðåìåííûõïîôîðìóëàì(1.50)ñîâïàäàþòñôîð-
ìóëàìèïðåîáðàçîâàíèÿêîýôôèöèåíòîâêâàäðàòè÷íîéôîðìû
n
X
i;j
=1
a
ij
t
i
t
j
;
(1.54)
åñëèâíåéïðîèçâåñòèëèíåéíîåíåâûðîæäåííîåïðåîáðàçîâàíèå
t
k
=
n
X
i
=1
c
ik

i
;k
=1
;:::;n;
(1.55)
ïðèâîäÿùååå¼êâèäó
n
X
k;l
=1

a
kl

k

l
:
(1.56)
Âàëãåáðåäîêàçûâàåòñÿ,÷òîâñåãäàìîæíîïîäîáðàòüêîýôôè-
öèåíòû
c
ki
òàê,÷òîáûêâàäðàòè÷íàÿôîðìà(1.54)ïðèâåëàñüê
êàíîíè÷åñêîìóâèäó,ò.å.
n
X
k
=1

k

2
k
;
(1.57)
32
èëè,èíà÷åãîâîðÿ,â(1.56)

a
kl
=0
ïðè
k
6
=
l
è

a
kk
=

k
3
.
Êîýôôèöèåíòû

k
ðàâíû

1
èëè0.Çíàêèêîýôôèöèåíòîâ

k
èîïðåäåëÿþòòèïóðàâíåíèÿ(1.49).Òàêèìîáðàçîìïðåîáðàçî-
âàííîåóðàâíåíèå(1.52)ïðèíèìàåòâèä
n
X
k
=1

k
@
2
v
@
2
k
+
n
X
i
=1

b
i
@v
@
i
+
cv
=
f
1
(

1
;:::;
n
)
:
(1.58)
Ýòîòâèäóðàâíåíèÿ(1.49)íàçûâàåòñÿåãîêàíîíè÷åñêèìâèäîì.
Ïîëîæèì,÷òîâñå

k
îòëè÷íûîòíóëÿ,ò.å.÷òîóðàâíåíèå
(1.49)íåïàðàáîëè÷åñêîãîòèïàèïîêàæåì,÷òîâýòîìñëó÷àå
ïðèïîìîùèïðåîáðàçîâàíèÿôóíêöèè
v
ìîæíîîñâîáîäèòüñÿîò
ïðîèçâîäíûõïåðâîãîïîðÿäêà.Ñýòîéöåëüþâìåñòî
v
ââåäåì
íîâóþèñêîìóþôóíêöèþ
w
ïîôîðìóëå
v
=
we

1
2
n
P
k
=1

b
k

k

k
:
Ïîäñòàâèâýòîâóðàâíåíèå(1.58),ïîëó÷èì,êàêíåòðóäíîïðî-
âåðèòü,óðàâíåíèåâèäà
n
X
k
=1

k
@
2
w
@
2
k
+
c
1
w
=
f
2
(

1
;:::;
n
)
:
Äëÿóðàâíåíèÿýëëèïòè÷åñêîãîòèïàâñå

k
=1
èëè

k
=

1
,è,óìíîæàÿ,åñëèíàäî,îáå÷àñòèóðàâíåíèÿíà

1
,ìû
ìîæåìñ÷èòàòü,÷òîâñå

k
=1
.Òàêèìîáðàçîì,ñîõðàíÿÿïðåæ-
íèåîáîçíà÷åíèÿ,ìûìîæåìóòâåðæäàòü,÷òîâñÿêîåëèíåéíîå
óðàâíåíèåâòîðîãîïîðÿäêàýëëèïòè÷åñêîãîòèïàñïîñòîÿííû-
ìèêîýôôèöèåíòàìèìîæåòáûòüïðèâåäåíîêâèäó
n
X
k
=1
@
2
u
@x
2
k
+
c
2
u
=
f
3
(
x
1
;:::;x
n
)
:
(1.59)
3
Ñîãëàñíîçàêîíóèíåðöèèäëÿêâàäðàòè÷íûõôîðì÷èñëîïîëîæèòåëüíûõèîòðèöàòåëüíûõêî-
ýôôèöèåíòîâ

k
èíâàðèàíòíîîòíîñèòåëüíîëèíåéíîãîïðåîáðàçîâàíèÿ,ïðèâîäÿùåãîêâàäðàòè÷íóþ
ôîðìó(1.54)êâèäó(1.57).
33
Âñëó÷àåóðàâíåíèÿãèïåðáîëè÷åñêîãîòèïàáóäåìñ÷èòàòü,
÷òîèìååòñÿ
n
+1
íåçàâèñèìûõïåðåìåííûõ,èïîëîæèì

n
+1
=
t
.
Òîãäàâñÿêîåëèíåéíîåóðàâíåíèåãèïåðáîëè÷åñêîãîòèïàñïî-
ñòîÿííûìèêîýôôèöèåíòàìèïðèâîäèòñÿêâèäó
@
2
u
@t
2

n
X
k
=1
@
2
u
@x
2
k
+
c
3
u
=
f
4
(
x
1
;:::;x
n
;t
)
:
(1.60)
Âñëó÷àåóðàâíåíèÿ(1.49)ñïåðåìåííûìèêîýôôèöèåíòà-
ìèäëÿêàæäîéòî÷êè
(
x
0
1
;:::;x
0
n
)
îáëàñòè
D
ìîæíîóêàçàòüòà-
êîåíåâûðîæäåííîåïðåîáðàçîâàíèåíåçàâèñèìûõïåðåìåííûõ,
êîòîðîåïðèâîäèòóðàâíåíèå(1.49)êêàíîíè÷åñêîìóâèäóâ
ýòîéòî÷êå.Äëÿêàæäîéòî÷êè
(
x
0
1
;:::;x
0
n
)
èìååòñÿ,âîîáùåãîâî-
ðÿ,ñâîåïðåîáðàçîâàíèåíåçàâèñèìûõïåðåìåííûõ,ïðèâîäÿùåå
óðàâíåíèåêêàíîíè÷åñêîìóâèäó.Äèôôåðåíöèàëüíîåóðàâíå-
íèåñ÷èñëîìíåçàâèñèìûõïåðåìåííûõáîëüøåäâóõ(åñëèèñ-
êëþ÷èòüñëó÷àéïîñòîÿííûõêîýôôèöèåíòîâ),âîîáùåãîâîðÿ,
íåâîçìîæíîïðèâåñòèñïîìîùüþïðåîáðàçîâàíèÿíåçàâèñèìûõ
ïåðåìåííûõêêàíîíè÷åñêîìóâèäóäàæåâêàêóãîäíîìàëîé
îáëàñòè.Âñëó÷àåæåäâóõíåçàâèñèìûõïåðåìåííûõòàêîåïðå-
îáðàçîâàíèåíåçàâèñèìûõïåðåìåííûõñóùåñòâóåòïðèâåñüìà
îáùèõïðåäïîëîæåíèÿõîêîýôôèöèåíòàõóðàâíåíèÿ,êàêáó-
äåòïîêàçàíîâñëåäóþùåìðàçäåëå.
1.2.5.Ïðèâåäåíèåêêàíîíè÷åñêîìóâèäóëèíåéíîãî
óðàâíåíèÿâòîðîãîïîðÿäêàñäâóìÿíåçàâèñèìûìè
ïåðåìåííûìè
Ðàññìîòðèìëèíåéíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìè
âòîðîãîïîðÿäêà(1.41)âñëó÷àå
n
=2
:
a
@
2
u
@x
2
+2
b
@
2
u
@[email protected]
+
c
@
2
u
@y
2
+
d
@u
@x
+
e
@u
@y
+
gu
=
f;
(1.61)
ãäåìûïðèíÿëèîáîçíà÷åíèÿ
a
=
a
(
x;y
)=
a
11
(
x;y
)
;b
=
b
(
x;y
)=
a
12
(
x;y
)=
a
21
(
x;y
)
;
34
c
=
c
(
x;y
)=
a
22
(
x;y
)
;d
=
d
(
x;y
)=
b
1
(
x;y
)
;
e
=
e
(
x;y
)=
b
2
(
x;y
)
;f
=
f
(
x;y
)
:
Îïðåäåëåíèå112.
Êðèâàÿ
'
(
x;y
)=const
,ãäå
'
ðå-
øåíèåóðàâíåíèÿ
a

@'
@x

2
+2
b

@'
@x

@'
@y

+
c

@'
@y

2
=0
(1.62)
íàçûâàåòñÿõàðàêòåðèñòè÷åñêîéêðèâîéóðàâíåíèÿ(1.61)(õà-
ðàêòåðèñòèêà).
Îïðåäåëåíèå113.
Íàïðàâëåíèå
(
dy;dx
)
,îïðåäåëåííîå
èçðàâåíñòâà
a
(
dy
)
2

2
bdxdy
+
c
(
dy
)
2
=0
íàçûâàåòñÿõàðàêòåðèñòè÷åñêèìíàïðàâëåíèåì(êàñàòåëüíûé
âåêòîðêõàðàêòåðèñòè÷åñêîéêðèâîé).
Óðàâíåíèþ(1.61)ñîîòâåòñòâóåòêâàäðàòè÷íàÿôîðìà
a
2
1
+2
b
1

2
+
c
2
2
:
(1.63)
Òàêèìîáðàçîìäèôôåðåíöèàëüíîåóðàâíåíèå(1.61)ïðèíàäëå-
æèò:
1)ãèïåðáîëè÷åñêîìóòèïó,åñëè
b
2

ac�
0
(êâàäðàòè÷íàÿ
ôîðìà(1.63)çíàêîïåðåìåííàÿ);âêàæäîéòî÷êåãèïåðáîëè÷íî-
ñòèñóùåñòâóåòäâàðàçëè÷íûõõàðàêòåðèñòè÷åñêèõíàïðàâëå-
íèÿ;
2)ïàðàáîëè÷åñêîìóòèïó,åñëè
b
2

ac
=0
(êâàäðàòè÷íàÿ
ôîðìà(1.63)çíàêîïîñòîÿííàÿ);âîáëàñòèïàðàáîëè÷íîñòèîäíî
äåéñòâèòåëüíîåõàðàêòåðèñòè÷åñêîåíàïðàâëåíèå;
3)ýëëèïòè÷åñêîìóòèïó,åñëè
b
2

ac
0
(êâàäðàòè÷-
íàÿôîðìà(1.63)çíàêîîïðåäåëåííàÿ);âîáëàñòèýëëèïòè÷íîñòè
äåéñòâèòåëüíûõõàðàêòåðèñòè÷åñêèõíàïðàâëåíèéíåò.
Ïðèìåð110.
ÄëÿäâóìåðíîãîóðàâíåíèÿËàïëàñà
u
xx
+
u
yy
=0
35
óðàâíåíèåäëÿîïðåäåëåíèÿõàðàêòåðèñòè÷åñêîéêðèâîéèìååò
âèä
(
'
x
)
2
+(
'
y
)
2
=0
:
Äåéñòâèòåëüíûõôóíêöèé,óäîâëåòâîðÿþùèõäàííîìóóðàâíå-
íèþíåñóùåñòâóåò.
Ïðèìåð111.
Äëÿîäíîðîäíîãîîäíîìåðíîãî(îäíàïðî-
ñòðàíñòâåííàÿïåðåìåííàÿ)âîëíîâîãîóðàâíåíèÿ
u
tt

u
xx
=0
óðàâíåíèå(1.62)èìååòâèä
(
'
t
)
2

(
'
x
)
2
=0
èëè,÷òîòîæåñàìîå,
'
t

'
x
=0
;'
t
+
'
x
=0
:
Ðåøåíèÿìèýòèõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèïåðâîãî
ïîðÿäêàî÷åâèäíîÿâëÿþòñÿ
'
1
(
x;t
)=
f
(
x
+
t
)
;'
2
(
x;t
)=
g
(
x

t
)
;
ãäå
f
è
g
ïðîèçâîëüíûåãëàäêèåäåéñòâèòåëüíûåôóíêöèè.
Òàêèìîáðàçîìäëÿîäíîìåðíîãîâîëíîâîãîóðàâíåíèÿêðèâûå
f
(
x
+
t
)=const
;g
(
x

t
)=const
ÿâëÿþòñÿõàðàêòåðèñòèêàìè.
Ïðèìåð112.
Äëÿîäíîìåðíîãîîäíîðîäíîãîóðàâíåíèÿ
òåïëîïðîâîäíîñòè
u
t

u
xx
=0
õàðàêòåðèñòè÷åñêîåóðàâíåíèåèìååòâèä

(
'
x
)
2
=0
:
Î÷åâèäíî,÷òîåãîðåøåíèå
'
=
f
(
t
)
;
36
ãäå
f
ïðîèçâîëüíàÿãëàäêàÿôóíêöèÿ.Ñëåäîâàòåëüíîèìååì
îäíóäåéñòâèòåëüíóþõàðàêòåðèñòèêó
f
(
t
)=const
:
Òåîðåìà16.
Ïóñòüêîýôôèöèåíòû
a
,
b
,
c
óðàâíåíèÿ
(1.61)äîñòàòî÷íîãëàäêèåâîáëàñòè
D

R
2
äåéñòâèòåëüíûé
ôóíêöèè.Òîãäàìîæíîíàéòèíåâûðîæäåííîåïðåîáðàçîâàíèå

=

(
x;y
)
,

=

(
x;y
)
,
;
2
C
2
(
D
)
ïðèïîìîùèêîòîðîãî
óðàâíåíèå(1.61)ïðèâîäèòñÿêîäíîìóèçñëåäóþùèõêàíîíè-
÷åñêèõâèäîâ:
@
2
v
@
2
+
@
2
v
@
2
+
A
@v
@
+
B
@v
@
+
Cv
=
H
(1.64)
âýëëèïòè÷åñêîìñëó÷àå,
@
2
v
@@
+
A
@v
@
+
B
@v
@
+
Cv
=
H
(1.65)
èëè
@
2
v
@
2

@
2
v
@
2
+
A
1
@v
@
+
B
1
@v
@
+
C
1
v
=
H
1
(1.66)
âãèïåðáîëè÷åñêîìñëó÷àåè
@
2
v
@
2
+
A
@v
@
+
B
@v
@
+
Cv
=
H
(1.67)
âïàðàáîëè÷åñêîìñëó÷àå,ãäå
v
(
;
)=
u
(
x
(
;
)
;y
(
;
))
.
Äîêàçàòåëüñòâî
.Ââåäåìâìåñòî
(
x;y
)
íîâûåíåçàâèñè-
ìûåïåðåìåííûå
(
;
)
.Ïóñòü

=

(
x;y
)
;
=

(
x;y
)
(1.68)
äâàæäûíåïðåðûâíîäèôôåðåíöèðóåìûåôóíêöèè,ïðè÷åì
ÿêîáèàí
D
(
;
)
D
(
x;y
)
=








@
@x
@
@y
@
@x
@
@y








6
=0
:
(1.69)
37
Âðåçóëüòàòåòàêîéçàìåíû,÷àñòíûåïðîèçâîäíûåïåðâîãîè
âòîðîãîïîðÿäêîâïî
x
è
y
ïðåîáðàçóþòñÿñëåäóþùèìîáðàçîì:
@
@x
=

x
@
@
+

x
@
@
;
@
@y
=

y
@
@
+

y
@
@
;
@
2
@x
2
=

2
x
@
2
@
2
+2

x

x
@
2
@@
+

2
x
@
2
@
2
+

xx
@
@
+

xx
@
@
;
@
2
@[email protected]
=

x

y
@
2
@
2
+(

x

y
+

y

x
)
@
2
@@
+

x

y
@
2
@
2
+

xy
@
@
+

xy
@
@
;
@
2
@y
2
=

2
y
@
2
@
2
+2

y

y
@
2
@@
+

2
y
@
2
@
2
+

yy
@
@
+

yy
@
@
;
ãäå

x
=
@
@x
,

xx
=
@
2

@x
2
èò.ä.
Âíîâûõíåçàâèñèìûõïåðåìåííûõ

è

óðàâíåíèå(1.61)
çàïèøåòñÿòàê:
a
1
@
2
v
@
2
+2
b
1
@
2
v
@@
+
c
1
@
2
v
@
2
+
d
1
@v
@
+
e
1
@v
@
+
g
1
v
=
f
1
;
(1.70)
ãäå
8











































:
a
1
(
;
)=
a

@
@x

2
+2
b
@
@x
@
@y
+
c

@
@y

2
;
b
1
(
;
)=
a
@
@x
@
@x
+
b

@
@x
@
@y
+
@
@y
@
@x

+
c
@
@y
@
@y
;
c
1
(
;
)=
a

@
@x

2
+2
b
@
@x
@
@y
+
c

@
@y

2
;
d
1
(
;
)=
a
@
2

@x
2
+2
b
@
2

@[email protected]
+
c
@
2

@y
2
+
d
@
@x
+
e
@
@y
;
e
1
(
;
)=
a
@
2

@x
2
+2
b
@
2

@[email protected]
+
c
@
2

@y
2
+
d
@
@x
+
e
@
@y
;
g
1
(
;
)=
g;
f
1
(
;
)=
f;
(1.71)
38
x
=
x
(
;
)
,
y
=
y
(
;
)
ïðåîáðàçîâàíèåîáðàòíîåê

=

(
x;y
)
,

=

(
x;y
)
.
Íåïîñðåäñòâåííîéïîäñòàíîâêîéíåòðóäíîïðîâåðèòü,÷òî
b
2
1

a
1
c
1
=(
b
2

ac
)

@
@x
@
@y

@
@y
@
@x

2
:
(1.72)
Îòñþäàëåãêîâèäåòü,÷òîïðåîáðàçîâàíèåíåçàâèñèìûõïåðå-
ìåííûõíåìåíÿåòòèïàóðàâíåíèÿ.
Âïðåîáðàçîâàíèè(1.68)âíàøåìðàñïîðÿæåíèèäâåôóíê-
öèè

(
x;y
)
è

(
x;y
)
.Ïîêàæåì,÷òîèõìîæíîâûáðàòüòàê,÷òî
ïðåîáðàçîâàííîåóðàâíåíèå(1.70)ïðèìåòîäèíèçêàíîíè÷åñêèõ
âèäîâ(1.64)(1.67).
1)
Ãèïåðáîëè÷åñêèéòèï
.Óðàâíåíèå(1.61)ãèïåðáîëè÷åñêî-
ãîòèïà,åñëè
b
2

ac�
0
.Óðàâíåíèÿòàêîãîòèïàèìåþòäâåõà-
ðàêòåðèñòèêè.Ïóñòü

(
x;y
)
;
(
x;y
)
2
C
2
ÿâëÿþòñÿðåøåíèÿìè
óðàâíåíèÿ
a'
2
x
+2
b'
x
'
y
+
c'
2
y
=0
,ïðè÷åìÿêîáèàí
D
(
;
)
D
(
x;y
)
6
=0
.
Òîãäà
a
x
+(
b
+
p
b
2

ac
)

y
=0
;
a
x
+(
b

p
b
2

ac
)

y
=0
:
(1.73)
Âñèëóâûáîðà

(
x;y
)
,

(
x;y
)
èç(1.71)ñëåäóåò,÷òî
a
1
=
c
1
=0
,
b
1
6
=0
.Ðàçäåëèâ(1.70)íà
2
b
1
,ïîëó÷àåìóðàâíåíèå(1.65),â
êîòîðîì
A
=
d
1
2
b
1
;B
=
e
1
2
b
1
;C
=
g
1
2
b
1
;H
=
f
1
2
b
1
:
Âðåçóëüòàòåçàìåíû

=

+

,

=



íåòðóäíîïîëó÷èòü
(1.66).
2)
Ïàðàáîëè÷åñêèéòèï.
Óðàâíåíèå(1.61)ïàðàáîëè÷åñêîãî
òèïà,åñëè
b
2

ac
=0
.Ôóíêöèè

(
x;y
)
,

(
x;y
)
,ïðèâîäÿùèå
óðàâíåíèå(1.61)êêàíîíè÷åñêîìóâèäó,âûáåðåìòàê,÷òîáû

(
x;y
)
ÿâëÿëàñüðåøåíèåìóðàâíåíèÿ
a'
2
x
+2
b'
x
'
y
+
c'
2
y
=0
;
39
ò.å.
a
x
+
b
y
=0
;
(1.74)
à

(
x;y
)
óäîâëåòâîðÿëàóñëîâèþ
a
2
x
+2
b
x
'
y
+
c
2
y
6
=0
:
Ïðèòàêîìâûáîðå

(
x;y
)
,

(
x;y
)
î÷åâèäíî,÷òî
a
1
=0
,
c
1
6
=0
.
Ïîêàæåì,÷òî
b
1
=0
.Èìååì
b
1
=
a
x

x
+
b
(

x

y
+

y

x
)+
c
y

y
=
=(
a
x
+
b
y
)

x
+(
b
x
+
c
y
)

y
=0
;
ò.ê.èìååòìåñòî(1.74)èðàâåíñòâî
b
x
+
c
y
,âûòåêàþùååèç
óðàâíåíèÿ
a
2
x
+2
b
x

y
+
c
2
y
=0
èóñëîâèÿ
b
2

ac
=0
.Ðàçäåëèâóðàâíåíèå(1.70)íà
c
1
6
=0
;
ïîëó÷àåìóðàâíåíèå(1.67),âêîòîðîì
A
=
d
1
c
1
;B
=
e
1
c
1
;C
=
g
1
c
1
;H
=
f
1
c
1
:
3)
Ýëëèïòè÷åñêèéòèï.
Óðàâíåíèå(1.61)ýëëèïòè÷åñêîãî
òèïà,åñëè
b
2

ac
0
.Áóäåìñ÷èòàòü,÷òîêîýôôèöèåíòû
a;b
è
c
ÿâëÿþòñÿàíàëèòè÷åñêèìèôóíêöèÿìèîò
x
è
y
4
.Ïóñòü
'
=

+
i
=const
ðåøåíèåóðàâíåíèÿ
a'
2
x
+2
b'
x
'
y
+
c'
2
y
=0
;
(1.75)
ò.å.
a
(

x
+
i
x
)
2
+2
b
(

x
+
i
x
)(

y
+
i
y
)+
c
(

y
+
i
y
)
2
=0
:
4
Ôóíêöèÿ
F
(
x;y
)
ïåðåìåííûõ
x
è
y
íàçûâàåòñÿàíàëèòè÷åñêîéâòî÷êå
(
x
0
;y
0
)
,åñëèîíàðàçëàãàåòñÿ
âñòåïåííîéðÿä
F
(
x;y
)=
1
X
p;q
=0
a
pq
(
x

x
0
)
p
(
y

y
0
)
q
;
ñõîäÿùèéñÿïðèäîñòàòî÷íîìàëûõ
(
x

x
0
)
;
(
y

y
0
)
.
40
Ïðèðàâíèâàÿäåéñòâèòåëüíóþèìíèìóþ÷àñòèïîñëåäíåãî
óðàâíåíèÿêíóëþ,ïîëó÷àåìñèñòåìó
(
a
2
x
+2
b
x

y
+
c
2
y

a
x

2
b
x

y

c
2
y
=0
;
a
x

x
+
b
(

x

y
+

y

x
)+
c
y

y
=0
è,ñëåäîâàòåëüíî,ðàçäåëèâóðàâíåíèå(1.70)íà
a
1
;
ïîëó÷àåì
óðàâíåíèå(1.64),âêîòîðîì
A
=
d
1
a
1
;B
=
e
1
a
1
;C
=
g
1
a
1
;H
=
f
1
a
1
:
Èñêîìóþçàìåíó

=

(
x;y
)
,

=

(
x;y
)
,ïðèâîäÿùååèñõîäíîå
óðàâíåíèåýëëèïòè÷åñêîãîòèïàêêàíîíè÷åñêîìóâèäóìîæíî
îïðåäåëèòüèçóðàâíåíèÿ(1.75).Îáîçíà÷èì

1
=

b
+
i
p
ac

b
2
a
;
2
=

b

i
p
ac

b
2
a
èïåðåïèøåìóðàâíåíèå(1.75)âñëåäóþùåìâèäå:
a
(
'
x


1
'
y
)(
'
x


2
'
y
)=0
:
Ò.ê.

1
=


2
,òî
a
(
'
x


1
'
y
)
2
=0
:
Ïîñêîëüêó
'
=

+
i
,òî

x
+
i
x



b
+
i
p
ac

b
2
a
!
(

y
+
i
y
)=0
:
Ïðèðàâíèâàÿêíóëþäåéñòâèòåëüíóþèìíèìóþ÷àñòü,ïîëó-
÷àåìñèñòåìóóðàâíåíèéäëÿíàõîæäåíèÿôóíêöèé

=

(
x;y
)
,

=

(
x;y
)
(
a
x
+
b
y
+
p
ac

b
2

y
=0
;
a
x
+
b
y

p
ac

b
2

y
=0
:
(1.76)
Ìûðàññìîòðåëèóðàâíåíèÿòðåõòèïîâèïîêàçàëè,÷òîâ
ðåçóëüòàòåíåâûðîæäåííîéçàìåíûïåðåìåííûõ(1.73),(1.74),
41
(1.76),ïðåäñòàâëÿþùèåñîáîéëèíåéíûåäèôôåðåíöèàëüíûå
óðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà,èñõîä-
íîåóðàâíåíèå(1.61)ìîæåòáûòüïðèâåäåíîêîäíîìóèçóðàâ-
íåíèé(1.65),(1.67),(1.64)ñîîòâåòñòâåííî.
Ïîñêîëüêóâîïðîñîñóùåñòâîâàíèèðåøåíèéëèíåéíûõ
äèôôåðåíöèàëüíûõóðàâíåíèéïåðâîãîïîðÿäêàñâÿçàíñòåî-
ðèåéîáûêíîâåííûõäèôôåðåíöèàëüíûõóðàâíåíèéïåðâîãîïî-
ðÿäêàèèçâåñòíî,÷òîïðèäîñòàòî÷íîéãëàäêîñòèêîýôôèöè-
åíòîâ
a
,
b
è
c
,ñèñòåìàóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìè
(1.76),ëèíåéíûåóðàâíåíèÿ(1.73),(1.74)âîêðåñòíîñòèòî÷êè
(
x;y
)
2
D
èìåþòðåøåíèÿíóæíîãîíàìâèäà,òîâîçìîæíîñòü
ïðèâåäåíèÿóðàâíåíèÿ(1.61)âîêðåñòíîñòèòî÷êèêêàíîíè÷å-
ñêèìâèäàì(1.64),(1.65),(1.67)äîêàçàíà.
Ïðèìåð113.
Ðàññìîòðèìóðàâíåíèå
@
2
u
@x
2

2sin
x
@
2
u
@[email protected]

cos
2
x
@
2
u
@y
2

cos
x
@u
@y
=0
:
(1.77)
Ýòîóðàâíåíèåãèïåðáîëè÷åñêîãîòèïà,ò.ê.
b
2

ac
=sin
2
x
+cos
2
x
=1
:
Ñîãëàñíîîáùåéòåîðèè,íàõîäèì

(
x;y
)
,

(
x;y
)
èçóðàâíåíèé

x
+(1

sin
x
)

y
=0
;
x

(1+sin
x
)

y
=0
:
Íåòðóäíîïðîâåðèòü,÷òîðåøåíèÿìèýòèõóðàâíåíèéÿâëÿþòñÿ
ôóíêöèè

(
x;y
)=
x

y
+cos
x;
(
x;y
)=
x
+
y

cos
x:
Òàêèìîáðàçîìââîäèìíîâûåïåðåìåííûå
(
;
)
ïîôîðìóëàì

=
x

y
+cos
x;
=
x
+
y

cos
x:
Òîãäàóðàâíåíèå(1.77)âíîâûõíåçàâèñèìûõïåðåìåííûõïðè-
âîäèòñÿêâèäó
@
2
v
@@
=0
:
(1.78)
42
Ïîëîæèâ

=

+
;
=



,ïðèâåäåìóðàâíåíèå(1.78)ê
êàíîíè÷åñêîìóâèäó
@
2
w
@
2

@
2
w
@
2
=0
:
1.2.6.Çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòîðîãî
ïîðÿäêàãèïåðáîëè÷åñêîãîòèïà
Çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿñ÷àñòíûìèïðîèç-
âîäíûìèâòîðîãîïîðÿäêàñäâóìÿíåçàâèñèìûìèïåðåìåííûìè
x
è
y
a
(
x;y
)
@
2
u
@x
2
+2
b
(
x;y
)
@
2
u
@[email protected]
+
c
(
x;y
)
@
2
u
@y
2
+
+
d
(
x;y
)
@u
@x
+
e
(
x;y
)
@u
@y
+
g
(
x;y
)
u
=
f
(
x;y
)
;
(1.79)
ãäå
a;b;c;d;e;g;f
èçâåñòíûåâíåêîòîðîéîáëàñòè
D

R
2
ãëàäêèåäåéñòâèòåëüíûåôóíêöèè,ïðè÷åìôóíêöèè
a;b;c
îäíî-
âðåìåííîâíóëüíåîáðàùàþòñÿèâ
D
âûïîëíÿåòñÿ
b
2

ac�
0
,
ñóñëîâèÿìè
u
j

=
'
(
x;y
)
;
@u
@
l





=

(
x;y
)
;
(1.80)
ñîñòîèòâñëåäóþùåì.Ïóñòüâîáëàñòè
D
çàäàíîóðàâíåíèå
(1.79)ãèïåðáîëè÷åñêîãîòèïàèíàêðèâîé

,êîòîðàÿïðèíàä-
ëåæèòîáëàñòè
D
èëèÿâëÿåòñÿ÷àñòüþãðàíèöûîáëàñòè
D
,çà-
äàíûôóíêöèè
'
(
x;y
)
,

(
x;y
)
èíàïðàâëåíèå
l
(
x;y
)
.Òðåáóåòñÿ
íàéòèôóíêöèþ
u
(
x;y
)
,êîòîðàÿâîáëàñòè
D
ÿâëÿåòñÿðåøå-
íèåìóðàâíåíèÿ(1.79)èíàêðèâîé

óäîâëåòâîðÿåòóñëîâèÿì
(1.80).
Åñëèâêàæäîéòî÷êåêðèâîé

íàïðàâëåíèå
l
íåÿâëÿåòñÿ
êàñàòåëüíûìêêðèâîé

èêàñàòåëüíîåíàïðàâëåíèåêêðèâîé

íåÿâëÿåòñÿõàðàêòåðèñòè÷åñêèì,òîâîáëàñòè
D
,îãðàíè÷åí-
íîéõàðàêòåðèñòèêàìè,ïðîõîäÿùèìè÷åðåçêîíöûêðèâîé

,
43
ïðèäîñòàòî÷íîéãëàäêîñòèêîýôôèöèåíòîâóðàâíåíèÿ(1.79)è
äàííûõóñëîâèé(1.80)ñóùåñòâóåòåäèíñòâåííîåðåøåíèåçàäà-
÷èÊîøè(1.79),(1.80).
Ðàññìîòðèìäâàìåòîäàðåøåíèÿçàäà÷èÊîøè(1.79),(1.80).
Ìåòîäõàðàêòåðèñòèê
Àëãîðèòìðåøåíèÿçàäà÷èÊîøèìåòîäîìõàðàêòåðèñòèê
ñîñòîèòâòîì,÷òîóðàâíåíèå(1.79)ïðèâîäèòñÿñíà÷àëà,åñëè
ýòîâîçìîæíî,êñëåäóþùåìóêàíîíè÷åñêîìóâèäó:
@
2
v
@@
=
:
(1.81)
Çàòåìïîñëåäîâàòåëüíûìèíòåãðèðîâàíèåìïî

è

,ìîæíîíàé-
òèîáùååðåøåíèåóðàâíåíèÿ(1.81)
v
(
;
)=M(
;
)+
f
(

)+
g
(

)
;
M

=
;
(1.82)
àòåìñàìûìèíàéòèîáùååðåøåíèåóðàâíåíèÿ(1.79)
u
(
x;y
)=
v
(

(
x;y
)
;
(
x;y
))
;
(1.83)
çàâèñÿùååîòäâóõïðîèçâîëüíûõãëàäêèõôóíêöèé
f
è
g
.Äàí-
íûåôóíêöèèîïðåäåëÿþòñÿîäíîçíà÷íî,åñëèïîäñòàâèòüîáùåå
ðåøåíèå(1.83)âóñëîâèÿ(1.80).
Åñëèóðàâíåíèå(1.79)ñïîñòîÿííûìèêîýôôèöèåíòàìèè
èìååòêàíîíè÷åñêèéâèä
@
2
v
@@
+
A
@v
@
+
B
@v
@
+
Cv
=
H;
(1.84)
ãäå
A;B;C
=const
òàêèå,÷òî
A
2
+
B
2
+
C
2
6
=0
è
AB

C
=0
,
òîåãîìîæíîóïðîñòèòü,ïîëîæèâ
v
(
;
)=
e

+

w
(
;
)
.Òàê
êàê
@v
@
=
e

+

w
(
;
)+
e

+

@w
@
(
;
)
;
44
@v
@
=
e

+

w
(
;
)+
e

+

@w
@
(
;
)
;
@
2
v
@@
(
;
)=
e

+

w
(
;
)+
e

+

@w
@
(
;
)+
+
e

+

@w
@
(
;
)+
e

+

@
2
w
@@
(
;
)
;
òîóðàâíåíèå(1.84),âðåçóëüòàòåäàííîéçàìåíû,ïðèìåòâèä
@
2
w
@@
+
(
A
+

)
@w
@
+
(
B
+

)
@w
@
+
+(

+
A
+
B
+
C
)
w
=
H
1
;H
1
=
He




:
(1.85)
Âûáèðàÿ

è

òàê,÷òîáû
A
+

=0
,
B
+

=0
,ò.å.

=

A
,

=

B
èó÷èòûâàÿðàâåíñòâî
C
=
AB
,óðàâíåíèå(1.85)ìîæ-
íîïåðåïèñàòüââèäå(1.81).
Ïðèìåð114.
Ðàññìîòðèìçàäà÷óÊîøè
u
xx

4
u
xy
+3
u
yy

2
u
x
+6
u
y
=0
;u
j
y
=0
=0
;u
y
j
y
=0
=1
:
Äèñêðèìèíàíò
b
2

ac
õàðàêòåðèñòè÷åñêîéôîðìûðàâåí
1

0
,ò.å.äàííîåóðàâíåíèåïðèíàäëåæèòãèïåðáîëè÷åñêî-
ìóòèïó.Ïðèâåäåìèñõîäíîåóðàâíåíèåêêàíîíè÷åñêîìóâèäó.
Ñîñòàâèìóðàâíåíèåõàðàêòåðèñòèê
'
2
x

4
'
x
'
y
+3
'
2
y
=0
:
Ïîëó÷àåìîòñþäàóðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèïåðâîãî
ïîðÿäêà
'
x

'
y
=0
;'
x

3
'
y
=0
:
Íåòðóäíîïðîâåðèòü,÷òî÷àñòíûìèðåøåíèÿìèýòèõóðàâíåíèé
ñîîòâåòñòâåííîÿâëÿþòñÿôóíêöèè
'
1
(
x;y
)=
x
+
y;'
2
(
x;y
)=3
x
+
y:
45
Ââåäåìíîâûåïåðåìåííûå


=3
x
+
y;

=
x
+
y:
Îáîçíà÷èì
v
(
;
)=
u
(
x
(
;
)
;y
(
;
))
.Âíîâûõïåðåìåííûõ
÷àñòíûåïðîèçâîäíûåïðåîáðàçóþòñÿñëåäóþùèìîáðàçîì:
u
x
=3
v

+
v

;u
y
=
v

+
v

u
xx
=9
v

+6
v

+
v

;u
yy
=
v

+2
v

+
v

;
u
xy
=3
v

+
v

+4
v

:
Ïîñëåïîäñòàíîâêèèõâèñõîäíîåóðàâíåíèå,ïîëó÷èìêàíîíè-
÷åñêèéâèä
v


v

=0
:
(1.86)
×òîáûóïðîñòèòüóðàâíåíèå(1.86),ââåäåìçàìåíó
v
(
;
)=
e

+

!
(
;
)
:
Ñó÷åòîìïåðåñ÷åòàïðîèçâîäíûõ,ïîëó÷èì
!

+
!

+(


1)
!

+(



)
!
=0
:
Ïîëàãàÿ

=1
;
=0
,èìååì
!

=0
èëè
@
@

@!
@

=0
:
Èíòåãðèðóÿïî

,ïîëó÷àåì÷òî
!

=
~
f
1
(

)
:
Èíòåãðèðóÿòåïåðüïîñëåäíååðàâåíñòâîïîïåðåìåííîé

,ïîëó-
÷èì
!
(
;
)=
F
(

)+
g
(

)
;
46
ãäå
F
(

)
òàêàÿ,÷òî
F
0
(

)=
~
f
1
(

)
.Òîãäàîáùååðåøåíèåóðàâíå-
íèÿ(1.86)èìååòâèä
v
(
;
)=
e

(
F
(

)+
g
(

))
:
Ïåðåõîäÿêïåðåìåííûì
x;y
,ïîëó÷èìîáùååðåøåíèåçàäàííîãî
óðàâíåíèÿ
u
(
x;y
)=
e
3
x
+
y
(
F
(3
x
+
y
)+
g
(
x
+
y
))
èëè
u
(
x;y
)=
f
(3
x
+
y
)+
e
3
x
+
y
g
(
x
+
y
)
;
ãäå
f
(3
x
+
y
)=
e
3
x
+
y
F
(3
x
+
y
)
,
f;g
ïðîèçâîëüíûåãëàäêèåäåé-
ñòâèòåëüíûåôóíêöèè.Ïîä÷èíèìîáùååðåøåíèåíà÷àëüíûì
óñëîâèÿì
u
(
x;
0)=
f
(3
x
)+
e
3
x
g
(
x
)=0
;
u
y
(
x;
0)=
f
0
(3
x
)+
e
3
x
g
(
x
)+
e
3
x
g
0
(
x
)=1
:
Ôóíêöèè
f
è
g
îïðåäåëÿþòñÿèçñëåäóþùåéñèñòåìû
(
f
(3
x
)+
e
3
x
g
(
x
)=0
;
f
0
(3
x
)+
e
3
x
g
(
x
)+
e
3
x
g
0
(
x
)=1
:
Äèôôåðåíöèðóÿïåðâîåóðàâíåíèåýòîéñèñòåìûïîïåðåìåííîé
x
,ïîëó÷èì
3
f
0
(3
x
)+3
e
3
x
g
(
x
)+
e
3
x
g
0
(
x
)=0
:
Âòîðîåóðàâíåíèåñèñòåìûóìíîæèìíà3èâû÷òåìèçïîñëåä-
íåãî,âðåçóëüòàòåáóäåìèìåòü
g
0
(
x
)=
3
2
e

3
x
:
Èíòåãðèðóÿ,íàõîäèì
g
(
x
)=

1
2
e

3
x
+
C;C
2
R
:
47
Èçïåðâîãîóðàâíåíèÿñèñòåìûíàéäåìòåïåðü
f
(3
x
)=



1
2
e

3
x
+
C

e
3
x
=
1
2

Ce
3
x
èëè
f
(
s
)=
1
2

Ce
s
;s
=3
x:
Èòàê,ðåøåíèåçàäà÷èÊîøèèìååòâèä
u
(
x;y
)=
1
2

Ce
y
+3
x
+
e
y
+3
x


1
2
e

3(
x
+
y
)
+
C

èëè
u
(
x;y
)=
1

e

2
y
2
:
Ìåòîäôàêòîðèçàöèè
Äàííûéìåòîäïîçâîëÿåòâíåêîòîðûõñëó÷àÿõñâåñòèëè-
íåéíîåóðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêà
ãèïåðáîëè÷åñêîãîèëèïàðàáîëè÷åñêîãîòèïîâêñèñòåìåäâóõ
ëèíåéíûõóðàâíåíèéïåðâîãîïîðÿäêà.Ðàññìîòðèìóðàâíåíèå
(1.79)ñ÷èòàÿ,÷òîêîýôôèöèåíòûýòîãîóðàâíåíèÿïîñòîÿííûè
a
=1
;b
2

c

0
.Ïðåäñòàâèìëåâóþ÷àñòüóðàâíåíèÿ(1.79)â
âèäåêîìïîçèöèèäâóõëèíåéíûõäèôôåðåíöèàëüíûõîïåðàòî-
ðîâïåðâîãîïîðÿäêàäåéñòâóþùèõíàôóíêöèþ
u
(
x;y
)

@
@x


1
@
@y
+

1

@
@x


2
@
@y
+

2

u
(
x;y
)=
f
(
x;y
)
:
(1.87)
Ïðèýòîì,ñðàâíèâàÿóðàâíåíèÿ(1.79)è(1.87)ìûâèäèì,÷òî
48
êîýôôèöèåíòû

1
;
2
;
1
;
2
äîëæíûóäîâëåòâîðÿòüóñëîâèÿì
8



















:

1
+

2
=

2
b;

1

2
=
c;

1
+

2
=
d;

1

2
=
g;

1

2
+

2

1
=

e:
(1.88)
Ïðèîïðåäåëåííûõäîïîëíèòåëüíûõóñëîâèÿõíàêîýôôèöèåí-
òûóðàâíåíèÿ(1.79)ñèñòåìàóðàâíåíèé(1.88)èìååòðåøåíèå,à
èìåííî
d
2

4
g

0
;
(
b

p
b
2

c
)(
d

p
d
2

4
g
)+
+(
b

p
b
2

c
)(
d

p
d
2

4
g
)=

2
e:
(1.89)
Îáîçíà÷èì

@
@x


2
@
@y
+

2

u
(
x;y
)=
v
(
x;y
)
.Òîãäàóðàâíå-
íèå(1.87)(èòåìñàìûìóðàâíåíèå(1.79))ñâåäåíîêñëåäóþùåé
ñèñòåìåëèíåéíûõóðàâíåíèéâ÷àñòíûõïðîèçâîäíûõïåðâîãî
ïîðÿäêà:
8





:
@u
@x


2
@u
@y
+

2
u
=
v;
@v
@x


1
@v
@y
+

1
v
=
f:
(1.90)
Êóðàâíåíèþ(1.79)äîáàâèìñëåäóþùèåíà÷àëüíûåóñëîâèÿ:
u
j
y
=
y
0
(
x
)
=
'
(
x
)
;u
y
j
y
=
y
0
(
x
)
=

(
x
)
;x
2
R
:
(1.91)
×òîáûâûäåëèòüåäèíñòâåííîåðåøåíèåâòîðîãîóðàâíåíèÿñè-
ñòåìû(1.91),íåîáõîäèìîïîñòàâèòüíà÷àëüíîåóñëîâèåäëÿ
ôóíêöèè
v
(
x;y
)
.Çàìåòèì,÷òîèçïåðâîãîóðàâíåíèÿñèñòåìû
49
(1.89)ñëåäóåò,÷òî
v
j
y
=
y
0
=
@u
@x
j
y
=
y
0


2
@u
@y
j
y
=
y
0
+

2
u
j
y
=
y
0
è,âñèëó(1.91),èìååì
v
j
y
=
y
0
=
@u
@x
j
y
=
y
0


2

(
x
)+

2
'
(
x
)
:
(1.92)
Äëÿòîãî,÷òîáûíàéòèçíà÷åíèå
@u
@x
j
y
=
y
0
,ïðîäèôôåðåíöèðóåì
ïåðâîåðàâåíñòâî(1.91)ïîïåðåìåííîé
x
.Ïîëó÷èì,÷òî
@u
@x
j
y
=
y
0
+
@u
@y
j
y
=
y
0
dy
0
dx
=
d'
dx
èëè,ñó÷åòîìâòîðîãîðàâåíñòâà(1.91),áóäåìèìåòü
@u
@x
j
y
=
y
0
=
d'
dx


(
x
)
dy
0
dx
:
Ïîäñòàâèâýòîçíà÷åíèåâ(1.92),âèòîãåïîëó÷àåì
v
j
y
=
y
0
=
d'
dx


(
x
)
dy
0
dx


2

(
x
)+

2
'
(
x
)
:
(1.93)
Òàêèìîáðàçîì,çàäà÷à(1.79),(1.91)ñâåäåíàêäâóìçàäà÷àì
Êîøèäëÿóðàâíåíèéâ÷àñòíûõïðîèçâîäíûõïåðâîãîïîðÿäêà
8





:
@v
@x


1
@v
@y
+

1
v
=
f;
v
j
y
=
y
0
=
d'
dx


(
x
)
dy
0
dx


2

(
x
)+

2
'
(
x
)
;
(1.94)
è
8



:
@u
@x


2
@u
@y
+

2
u
=
v;
u
j
y
=
y
0
(
x
)
=
'
(
x
)
:
(1.95)
Ðåøàÿïîñëåäîâàòåëüíîçàäà÷è(1.94),(1.95)íàéäåìåäèíñòâåí-
íîåðåøåíèåçàäà÷èÊîøè(1.79),(1.91).
50
Ïðèìåð115.
Ðàññìîòðèìçàäà÷óÊîøè
u
xx

4
u
xy
+3
u
yy

2
u
x
+6
u
y
=0
;u
j
y
=0
=0
;u
y
j
y
=0
=1
:
Ïðåäñòàâèìçàäàííîåóðàâíåíèåââèäå(1.87).Ñèñòåìà
(1.88)äëÿîïðåäåëåíèÿêîýôôèöèåíòîâ

1
;
2
;
1
;
2
âäàííîì
ñëó÷àåïðèìåòâèä
8



















:

1
+

2
=4
;

1

2
=3
;

1
+

2
=

2
;

1

2
=0
;

1

2
+

2

1
=

6
:
(1.96)
Èçïåðâîãîèâòîðîãîóðàâíåíèÿñèñòåìû(1.96)íàõîäèì

1
=3
;
2
=1
èëè

1
=1
;
2
=3
.Èçòðåòüåãîè÷åòâåðòî-
ãîóðàâíåíèÿñèñòåìû(1.96)ñëåäóåò,÷òî

1
=0
;
2
=

2
èëè

1
=

2
;
2
=0
.Ïîëîæèì

1
=1
;
2
=3
,

1
=

2
;
2
=0
.
Ïðèýòîìïîñëåäíååðàâåíñòâî(1.96)âûïîëíåíî.Òîãäà,âñîîò-
âåòñòâèèñ(1.90),äàííîìóóðàâíåíèþñîïîñòàâèìñèñòåìóëè-
íåéíûõóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìèïåðâîãîïîðÿäêà
8





:
@u
@x

3
@u
@y
=
v;
@v
@x

@v
@y

2
v
=0
:
(1.97)
Ïîëüçóÿñüíà÷àëüíûìèäàííûìèèôîðìóëîé(1.93),èìååìñëå-
äóþùóþçàäà÷óÊîøèäëÿîïðåäåëåíèÿôóíêöèè
v
:
@v
@x

@v
@y
=2
v;v
j
y
=0
=

3
:
Èçõàðàêòåðèñòè÷åñêîéñèñòåìû
dx
1
=
dy

1
=
dv
2
v
51
ñëåäóåò,÷òî
'
1
=
x
+
y;'
2
=
e
2
y
v:
Ïîýòîìó
v
(
x;y
)=
e

2
y
f
(
x
+
y
)
;
ãäå
f
ïðîèçâîëüíàÿãëàäêàÿäåéñòâèòåëüíàÿôóíêöèÿ.Èñ-
ïîëüçóÿíà÷àëüíûåóñëîâèÿäëÿôóíêöèè
v
,ïîëó÷èì
v
j
y
=0
=
f
(
x
)=

3
;
ñëåäîâàòåëüíî,
v
(
x;y
)=

3
e

2
y
:
Òåïåðüïåðåõîäèìêïîèñêóôóíêöèè
u
.Ñîîòâåòñòâóþùàÿçà-
äà÷àÊîøèèìååòâèä
8



:
@u
@x

3
@u
@y
=

3
e

2
y
;
u
j
y
=0
=0
:
Ñîñòàâèìõàðàêòåðèñòè÷åñêóþñèñòåìó
dx
1
=
dy

3
=
dv

3
e

2
y
èíàéäåìïåðâûåèíòåãðàëû
'
3
=3
x
+
y;'
4
=
v
+
1
2
e

2
y
:
Òîãäà
u
(
x;y
)=
f
1
(3
x
+
y
)

1
2
e

2
y
;
ãäå
f
1
ïðîèçâîëüíàÿãëàäêàÿäåéñòâèòåëüíàÿôóíêöèÿ.Ïîä-
ñòàâëÿÿíàéäåííîå
u
(
x;y
)
âñîîòâåòñòâóþùååíà÷àëüíîåóñëî-
âèå,íàõîäèì
f
1
=
1
2
è,îêîí÷àòåëüíîèìååì
u
(
x;y
)=
1

e

2
y
2
:
52
1.2.7.Çàäà÷àÊîøèäëÿëèíåéíîãîóðàâíåíèÿâòîðîãî
ïîðÿäêàñàíàëèòè÷åñêèìèäàííûìè.
ÔîðìóëèðîâêàòåîðåìûÊîøèÊîâàëåâñêîé
Âýòîìðàçäåëåìûâûäåëèìäîâîëüíîîáùèéêëàññçàäà÷
Êîøèäëÿóðàâíåíèÿñ÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿä-
êà,äëÿêîòîðûõðåøåíèåñóùåñòâóåòèåäèíñòâåííî.Ðàññìîò-
ðèìñëåäóþùååóðàâíåíèåñ÷àñòíûìèïðîèçâîäíûìèâòîðîãî
ïîðÿäêà:
@
2
u
@x
2
1
=
F

x
;u;
@u
@x
1
;:::;
@u
@x
n
;:::
@
2
u
@x
i
@x
j
;:::

;i;j
=1
;:::;n;
(1.98)
ãäå
i
=
j
6
=1
,
x
=(
x
1
;:::;x
n
)
2
D

R
n
,
n

2
.Óðàâíåíèÿ
âèäà(1.98)ïðèíÿòîíàçûâàòüíîðìàëüíûìèóðàâíåíèÿìèèëè
óðàâíåíèÿìèÊîøè-Êîâàëåâñêîé.
Íàïðèìåð,âîëíîâîåóðàâíåíèå,óðàâíåíèåËàïëàñàèóðàâ-
íåíèåòåïëîïðîâîäíîñòèíîðìàëüíûîòíîñèòåëüíîêàæäîéïå-
ðåìåííîé
x
i
,
i
=1
;:::;n
;âîëíîâîåóðàâíåíèå,êðîìåòîãî,íîð-
ìàëüíîîòíîñèòåëüíî
t
.
Åñëèôóíêöèÿ
F
ëèíåéíàîòíîñèòåëüíî
u
èâñåõïðîèçâîä-
íûõîò
u
âõîäÿùèõâ
F
,òîóðàâíåíèå(1.98)ìîæíîçàïèñàòüâ
âèäå
@
2
u
@x
2
1
=
n
X
i;j
=1
i
=
j
6
=1
a
ij
(
x
)
@
2
u
@x
i
@x
j
+
n
X
i
=1
b
i
(
x
)
@u
@x
i
+
c
(
x
)
u
+
f
(
x
)
:
(1.99)
Ïóñòüâíåêîòîðîéîáëàñòè
D
0
ïðîñòðàíñòâàïåðåìåííûõ
x
2
;:::;x
n
çàäàíûôóíêöèè
'
0
(
x
2
;:::;x
n
)
;'
1
(
x
2
;:::;x
n
)
:
Çàäà÷àîòûñêàíèÿâíåêîòîðîéîêðåñòíîñòèîáëàñòè
D
0
ðåøå-
íèÿ
u
óðàâíåíèÿ(1.98),óäîâëåòâîðÿþùåãîíà÷àëüíûìóñëîâè-
53
ÿì
@
i
u
@x
i
1



x
1
=
x
1
0
=
'
i
(
x
2
;:::;x
n
)
;i
=0
;
1
;
(1.100)
íàçûâàåòñÿçàäà÷åéÊîøè.
Îïðåäåëåíèå114.
Ôóíêöèÿ
f
(
z
1
;:::;z
m
)
íàçûâàåòñÿ
àíàëèòè÷åñêîéâîêðåñòíîñòèòî÷êè
(
z
1
0
;:::;z
m
0
)
,åñëèîíà
ðàçëàãàåòñÿâñòåïåííîéðÿä
f
(
z
1
;:::;z
m
)=
X
k
1

0
;:::;k
m

0
a
k
1
:::k
m
(
z
1

z
1
0
)
k
1
:::
(
z
m

z
m
0
)
k
m
;
ñõîäÿùèéñÿïðèäîñòàòî÷íîìàëûõ
j
z
i

z
i
0
j
,
i
=1
;:::;m
.Ïðè
ýòîìôóíêöèÿ
f
(
z
1
;:::;z
m
)
èìååòâòî÷êå
(
z
1
0
;:::;z
m
0
)
ïðîèç-
âîäíûåâñåõïîðÿäêîâè
a
k
1
:::k
m
=
1
k
1
!
:::k
m
!

@
k
1
+

+
k
m
f
@x
k
1
1
:::@x
k
m
m
!



z
=
z
0
;
ãäå
z
=(
z
1
;:::;z
m
)
,
z
0
=(
z
1
0
;:::;z
m
0
)
:
Èìååòìåñòîñëåäóþùååâàæíîåóòâåðæäåíèå,èçâåñòíîå
ïîäíàçâàíèåìòåîðåìûÊîøèÊîâàëåâñêîé.
Òåîðåìà17.
Ïóñòüôóíêöèè
'
i
,
i
=0
;
1
ÿâëÿþòñÿàíà-
ëèòè÷åñêèìèôóíêöèÿìèâíåêîòîðîéîêðåñòíîñòèòî÷êè
(
x
2
0
;:::;x
n
0
)
è
F
àíàëèòè÷íàâíåêîòîðîéîêðåñòíîñòèòî÷êè

x
1
0
;:::;x
n
0
;'
0
(
x
2
0
;:::;x
n
0
)
;'
1
(
x
2
0
;:::;x
n
0
)
;
@'
0
@x
2
(
x
2
0
;:::;x
n
0
)
;:::;
@'
0
@x
n
(
x
2
0
;:::;x
n
0
)
;
@'
1
@x
2
(
x
2
0
;:::;x
n
0
)
;:::;
@
2
'
0
@x
2
n
(
x
2
0
;:::;x
n
0
)

;
òîçà-
äà÷àÊîøè(1.98),(1.100)èìååòàíàëèòè÷åñêîåðåøåíèå
âíåêîòîðîéîêðåñòíîñòèòî÷êè
(
x
1
0
;:::;x
n
0
)
èïðèýòîì
åäèíñòâåííîåâêëàññåàíàëèòè÷åñêèõôóíêöèé.
Ïðèâåäåìèäåþäîêàçàòåëüñòâà.Ðåøåíèå
u
âîêðåñòíîñòè
54
òî÷êè
(
x
1
0
;:::;x
n
0
)
èùåòñÿââèäåñòåïåííîãîðÿäà
u
(
x
1
;:::;x
n
)=
X
k
1

0
;:::;k
n

0
1
k
1
!
:::k
n
!

@
k
1
+
:::
+
k
n
u
(
x
1
0
;:::;x
n
0
)
@x
k
1
1
:::@x
k
n
n


(
x
1

x
1
0
)
k
1

:::

(
x
n

x
n
0
)
k
n
;k
1
+
:::
+
k
n
6
2
;k
1
6
1
:
(1.101)
Èçíà÷àëüíûõóñëîâèé(1.100)èèçóðàâíåíèé(1.98)ïîñëå-
äîâàòåëüíîîïðåäåëÿþòñÿâñåïðîèçâîäíûå
@
k
1
+
:::
+
k
n
u
@x
k
1
1
:::@x
k
n
n
âòî÷-
êå
(
x
1
0
;:::;x
n
0
)
.Äîêàçûâàåòñÿðàâíîìåðíàÿñõîäèìîñòüðÿäîâ
(1.101)âíåêîòîðîéîêðåñòíîñòèòî÷êè
(
x
1
0
;:::;x
n
0
)
.Åäèíñòâåí-
íîñòüïîñòðîåííîãîðåøåíèÿâêëàññåàíàëèòè÷åñêèõôóíêöèé
ñëåäóåòèçòåîðåìûåäèíñòâåííîñòèäëÿàíàëèòè÷åñêèõôóíê-
öèé.
Çàìå÷àíèå14.
Èçòåîðåìû1.7ñëåäóåò,÷òîåñëèäàííûå
çàäà÷èÊîøè(1.99),(1.100)
a
ij
,
b
i
,
i;j
=1
;:::;n
,
c
,
f
àíàëèòè-
÷åñêèåôóíêöèèâíåêîòîðîéîêðåñòíîñòèòî÷êè
(
x
1
0
;:::;x
n
0
)
à,
'
0
,
'
1
àíàëèòè÷åñêèåôóíêöèèâíåêîòîðîéîêðåñòíîñòèòî÷-
êè
(
x
2
0
;:::;x
n
0
)
,òîâíåêîòîðîéîêðåñòíîñòèòî÷êè
(
x
1
0
;:::;x
n
0
)
çàäà÷àÊîøè(1.99),(1.100)èìååòåäèíñòâåííîåàíàëèòè÷åñêîå
ðåøåíèå.
Çàìå÷àíèå15.
ÒåîðåìàÊîøèÊîâàëåâñêîé,íåñìîòðÿ
íàååîáùèéõàðàêòåð,ïîëíîñòüþíåðåøàåòâîïðîñàêîððåêòíî-
ñòèïîñòàíîâêèçàäà÷èÊîøèäëÿíîðìàëüíîãîäèôôåðåíöèàëü-
íîãîóðàâíåíèÿ.Äåéñòâèòåëüíî,ýòàòåîðåìàãàðàíòèðóåòñóùå-
ñòâîâàíèåèåäèíñòâåííîñòüðåøåíèÿëèøüâäîñòàòî÷íîìàëîé
îêðåñòíîñòè,èëè,êàêãîâîðÿò,âìàëîì;îáû÷íîæåýòèôàêòû
òðåáóåòñÿóñòàíîâèòüâíàïåðåäçàäàííûõ(èîòíþäüíåìàëûõ)
îáëàñòÿõ,èëè,êàêãîâîðÿò,âöåëîì.Áîëååòîãî,íà÷àëüíûåäàí-
íûåèñâîáîäíûé÷ëåíóðàâíåíèÿ,êàêïðàâèëî,îêàçûâàþòñÿ
íåàíàëèòè÷åñêèìèôóíêöèÿìè.
55
1.2.8.Ïîíÿòèåêîððåêòíîñòèçàäà÷èìàòåìàòè÷åñêîé
ôèçèêè.ÏðèìåðÀäàìàðà
Òàêêàêçàäà÷èìàòåìàòè÷åñêîéôèçèêèïðåäñòàâëÿþòñî-
áîéìàòåìàòè÷åñêèåìîäåëèðåàëüíûõôèçè÷åñêèõïðîöåññîâ,
òîèõïîñòàíîâêèäîëæíûóäîâëåòâîðÿòüñëåäóþùèìåñòåñòâåí-
íûìòðåáîâàíèÿì.
à)Ðåøåíèåäîëæíîñóùåñòâîâàòüâêàêîì-ëèáîêëàññå
ôóíêöèé
M
1
.
á)Ðåøåíèåäîëæíîáûòüåäèíñòâåííûìâêàêîì-ëèáîêëàñ-
ñåôóíêöèé
M
2
.
â)Ðåøåíèåäîëæíîíåïðåðûâíîçàâèñåòüîòäàííûõçàäà÷è
(íà÷àëüíûõèãðàíè÷íûõäàííûõ,ñâîáîäíîãî÷ëåíà,êîýôôè-
öèåíòîâóðàâíåíèÿèò.ä.).
Íåïðåðûâíàÿçàâèñèìîñòüðåøåíèÿ
u
îòäàííûõçàäà÷è
Z
îçíà÷àåòñëåäóþùåå:ïóñòüïîñëåäîâàòåëüíîñòüäàííûõ
Z
k
,
k
=1
;
2
;:::;
âêàêîì-òîñìûñëåñòðåìèòñÿê
Z
ïðè
k
!1

u
k
;
k
=1
;
2
;:::;
ñîîòâåòñòâóþùèåðåøåíèÿçàäà÷è;òîãäàäîëæíî
áûòü
u
k
!
u
ïðè
k
!1
âñìûñëåíàäëåæàùèìîáðàçîìâû-
áðàííîéñõîäèìîñòè.Òðåáîâàíèåíåïðåðûâíîéçàâèñèìîñòèðå-
øåíèÿîáóñëîâëèâàåòñÿòåìîáñòîÿòåëüñòâîì,÷òîôèçè÷åñêèå
äàííûå,êàêïðàâèëî,îïðåäåëÿþòñÿèçýêñïåðèìåíòàïðèáëè-
æåíèþ,èïîýòîìóíóæíîáûòüóâåðåííûìâòîì,÷òîðåøåíèå
çàäà÷èâðàìêàõâûáðàííîéìàòåìàòè÷åñêîéìîäåëèíåáóäåò
ñóùåñòâåííîçàâèñåòüîòïîãðåøíîñòåéèçìåðåíèé.
Çàäà÷à,óäîâëåòâîðÿþùàÿïåðå÷èñëåííûìòðåáîâàíèÿì,íà-
çûâàåòñÿêîððåêòíîïîñòàâëåííîéçàäà÷åéìàòåìàòè÷åñêîéôè-
çèêè(ïîÀäàìàðó),àìíîæåñòâîôóíêöèé
M
1
T
M
2
íàçûâàåò-
ñÿêëàññîìêîððåêòíîñòè.Çàäà÷à,íåóäîâëåòâîðÿþùàÿõîòÿáû
îäíîìóèçóñëîâèéà)-â),íàçûâàåòñÿíåêîððåêòíîïîñòàâëåííîé.
Ïðèâåäåìïðèìåð,ïîêàçûâàþùèé,÷òîìîæåòâîâñåíåáûòü
íåïðåðûâíîéçàâèñèìîñòèðåøåíèÿîòíà÷àëüíûõäàííûõ.Ýòîò
56
ïðèìåðïîñòðîåíÀäàìàðîì.
Ïðèìåð116.
Ðåøåíèåçàäà÷èÊîøè
@
2
u
@t
2
+
@
2
u
@x
2
=0
;u
j
t
=0
=0
;
@u
@t
j
t
=0
=
1
k
sin
kx
åñòü
u
k
(
x;t
)=
sh
kt
k
2
sin
kx
.Åñëè
k
!1
,òî
1
k
sin
kx
!
0
;òåì
íåìåíååïðè
x
6
=
j
,
j
=0
;

1
;:::u
k
(
x;t
)
íåñòðåìèòñÿê
íóëþïðè
k
!1
.Òàêèìîáðàçîì,çàäà÷àÊîøèäëÿóðàâíåíèÿ
Ëàïëàñàïîñòàâëåíàíåêîððåêòíî.
1.3.Âûâîäîñíîâíûõóðàâíåíèéìàòåìàòè÷åñêîéôèçèêè
Ìàòåìàòè÷åñêîåîïèñàíèåìíîãèõôèçè÷åñêèõïðîöåññîâ
ïðèâîäèòêäèôôåðåíöèàëüíûìèèíòåãðàëüíûìóðàâíåíèÿì.
Âåñüìàøèðîêèéêëàññôèçè÷åñêèõïðîöåññîâîïèñûâàåòñÿëè-
íåéíûìèäèôôåðåíöèàëüíûìèóðàâíåíèÿìèâòîðîãîïîðÿäêà.
Âýòîìðàçäåëåìûðàññìîòðèìõàðàêòåðíûåôèçè÷åñêèåïðî-
öåññû,ñâîäÿùèåñÿêðàçëè÷íûìêðàåâûìçàäà÷àìäëÿäèôôå-
ðåíöèàëüíûõóðàâíåíèé.
1.3.1.Óðàâíåíèåêîëåáàíèéñòðóíû
Ìíîãèåçàäà÷èìåõàíèêè(êîëåáàíèÿñòðóí,ñòåðæíåé,ìåì-
áðàíèòðåõìåðíûõîáúåìîâ)èôèçèêè(ýëåêòðîìàãíèòíûåêî-
ëåáàíèÿ)îïèñûâàþòñÿóðàâíåíèåìêîëåáàíèéâèäà(1.9).Ïðî-
äåìîíñòðèðóåìâûâîäóðàâíåíèÿ(1.9)íàïðèìåðåìàëûõïîïå-
ðå÷íûõêîëåáàíèéñòðóíûèìåìáðàíû.
Ðàññìîòðèìíàòÿíóòóþñòðóíó,çàêðåïëåííóþíàêîíöàõ.
Ïîäñòðóíîéïîíèìàþòòîíêóþíèòü,êîòîðàÿìîæåòñâîáîä-
íîèçãèáàòüñÿ,ò.å.íåîêàçûâàåòñîïðîòèâëåíèÿèçìåíåíèþå¼
ôîðìû,íåñâÿçàííîìóñèçìåíåíèåìå¼äëèíû.
57
Ðèñ.1.
Ñèëàíàòÿæåíèÿ
T
0
;
äåéñòâóþùàÿíàñòðóíó,ïðåäïîëàãà-
åòñÿçíà÷èòåëüíîé,òàê÷òîìîæíîïðåíåáðå÷üäåéñòâèåìñèëû
òÿæåñòè.
Ïóñòüâïîëîæåíèèðàâíîâåñèÿñòðóíàíàïðàâëåíàïîîñè
Ox:
Áóäåìðàññìàòðèâàòüòîëüêîïîïåðå÷íûåêîëåáàíèÿñòðó-
íû,ïðåäïîëàãàÿ,÷òîäâèæåíèåïðîèñõîäèòâîäíîéïëîñêîñòè
è÷òîâñåòî÷êèñòðóíûäâèæóòñÿïåðïåíäèêóëÿðíîîñè
Ox:
Îáîçíà÷èì÷åðåç
u
(
x;t
)
ñìåùåíèåòî÷åêñòðóíûâìîìåíò
âðåìåíè
t
îòïîëîæåíèÿðàâíîâåñèÿ.Ïðèêàæäîìôèêñèðîâàí-
íîìçíà÷åíèè
t
ãðàôèêôóíêöèè
u
(
x;t
)
;
î÷åâèäíî,äàåòôîðìó
ñòðóíûâýòîòìîìåíòâðåìåíè(ñì.ðèñ.1).Ðàññìàòðèâàÿäàëåå
òîëüêîìàëûåêîëåáàíèÿñòðóíû,áóäåìñ÷èòàòü,÷òîñìåùåíèå
u
(
x;t
)
,àòàêæåïðîèçâîäíàÿ
@u
@x
ñòîëüìàëû,÷òîèõêâàäðàòà-
ìèèïðîèçâåäåíèÿìèìîæíîïðåíåáðå÷üïîñðàâíåíèþñýòèìè
âåëè÷èíàìè.
Âûäåëèìïðîèçâîëüíûéó÷àñòîê
(
x
1
;x
2
)
ñòðóíû(ñì.ðèñ.1),
êîòîðûéïðèêîëåáàíèèñòðóíûäåôîðìèðóåòñÿâó÷àñòîê
M
1
M
2
:
Äëèíàäóãèýòîãîó÷àñòêàâìîìåíòâðåìåíè
t
ðàâíà
S
0
=
x
2
Z
x
1
p
1+
u
2
x
dx

x
2

x
1
=
S;u
x
=
@u
@x
;
âñëåäñòâèå÷åãîìîæíîñ÷èòàòü,÷òîâïðîöåññåìàëûõêîëåáà-
íèéóäëèíåíèåó÷àñòêîâñòðóíûíåïðîèñõîäèò.Îòñþäàâñèëó
58
çàêîíàÃóêàñëåäóåò,÷òîâåëè÷èíàíàòÿæåíèÿ
T
âêàæäîéòî÷-
êåñòðóíûíåìåíÿåòñÿñîâðåìåíåì.Òàêèìîáðàçîì,ïðèíàøèõ
ïðåäïîëîæåíèÿõèçìåíåíèåìâåëè÷èíûíàòÿæåíèÿñòðóíû,âîç-
íèêàþùèìïðèå¼äâèæåíèè,ìîæíîïðåíåáðå÷üïîñðàâíåíèþ
ñíàòÿæåíèåì,êîòîðîìóîíàáûëàóæåïîäâåðãíóòàâïîëîæå-
íèèðàâíîâåñèÿ.Ïîêàæåì,÷òîâåëè÷èíóíàòÿæåíèÿ
T
ìîæíî
ñ÷èòàòüíåçàâèñÿùåéîò
x;
ò.å.
T

T
0
:
Äåéñòâèòåëüíî,íàó÷à-
ñòîê
M
1
M
2
ñòðóíûäåéñòâóþòñèëûíàòÿæåíèÿ,íàïðàâëåííûå
ïîêàñàòåëüíûìêñòðóíåâòî÷êàõ
M
1
è
M
2
;
âíåøíèåñèëûèñè-
ëûèíåðöèè.Ñóììàïðîåêöèéíàîñü
Ox
âñåõýòèõñèëäîëæíà
ðàâíÿòüñÿíóëþ.Òàêêàêìûðàññìàòðèâàåìòîëüêîïîïåðå÷-
íûåêîëåáàíèÿ,òîñèëûèíåðöèèèâíåøíèåñèëûíàïðàâëåíû
ïàðàëëåëüíîîñè
Ou;
òîãäà

T
(
x
1
)cos

(
x
1
)+
T
(
x
2
)cos

(
x
2
)=0
;
ãäå

(
x
)
óãîëìåæäóêàñàòåëüíîéâòî÷êåñàáñöèññîé
x
ê
ñòðóíåâìîìåíòâðåìåíè
t
ñïîëîæèòåëüíûìíàïðàâëåíèåìîñè
x:
Âñèëóìàëîñòèêîëåáàíèé
cos

(
x
)=
1
p
1+
tg
2

(
x
)
=
1
p
1+
u
2
x

1
;
è,ñëåäîâàòåëüíî,
T
(
x
1
)

T
(
x
2
)
:
Îòñþäàââèäóïðîèçâîëüíîñòè
x
1
è
x
2
ñëåäóåò,÷òîâåëè÷èíà
íàòÿæåíèÿ
T
íåçàâèñèòîò
x:
Òàêèìîáðàçîì,ìîæíîñ÷èòàòü,
÷òî
T

T
0
äëÿâñåõçíà÷åíèé
x
è
t:
Ïåðåéäåìêâûâîäóóðàâíåíèéêîëåáàíèÿñòðóíû.Äëÿýòîãî
âîñïîëüçóåìñÿïðèíöèïîìÄàëàìáåðà,íàîñíîâàíèèêîòîðîãî
âñåñèëû,äåéñòâóþùèåíàíåêîòîðûéâûäåëåííûéó÷àñòîêâ
ñòðóíå,âêëþ÷àÿñèëûèíåðöèè,äîëæíûóðàâíîâåøèâàòüñÿ.
Ðàññìîòðèìïðîèçâîëüíûéó÷àñòîê
M
1
M
2
ñòðóíûèñîñòà-
âèìóñëîâèåðàâåíñòâàíóëþñóììûïðîåêöèéíàîñü
Ou
âñåõ
59
ñèë,äåéñòâóþùèõíàíåãî:ñèëíàòÿæåíèÿ,ðàâíûõïîâåëè÷èíå
èíàïðàâëåííûõïîêàñàòåëüíûìêñòðóíåâòî÷êàõ
M
1
è
M
2
;
âíåøíåéñèëû,íàïðàâëåííîéïàðàëëåëüíîîñè
Ou;
èñèëûèíåð-
öèè.
Ñóììàïðîåêöèéíàîñü
Ou
ñèëíàòÿæåíèÿ,äåéñòâóþùèõâ
òî÷êàõ
M
1
è
M
2
;
ðàâíÿåòñÿ
Y
=
T
0
[sin

(
x
2
)

sin

(
x
1
)]
;
íîâñëåäñòâèåíàøèõïðåäïîëîæåíèé
sin

(
x
)=
tg
(
x
)
p
1+
tg
2

(
x
)
=
u
x
p
1+
u
2
x

@u
@x
;
è,ñëåäîâàòåëüíî,
Y
=
T
0
"

@u
@x





x
=
x
2


@u
@x





x
=
x
1
#
:
Çàìå÷àÿòåïåðü,÷òî

@u
@x





x
=
x
2


@u
@x





x
=
x
1
=
x
2
Z
x
1
@
2
u
@x
2
dx;
îêîí÷àòåëüíîïîëó÷èì
Y
=
T
0
x
2
Z
x
1
@
2
u
@x
2
dx:
(1.102)
Îáîçíà÷èì÷åðåç
p
(
x;t
)
âíåøíþþñèëó,äåéñòâóþùóþíà
ñòðóíóïàðàëëåëüíîîñè
Ou
èðàññ÷èòàííóþíàåäèíèöóäëè-
íû.Òîãäàïðîåêöèÿíàîñü
Ou
âíåøíåéñèëû,äåéñòâóþùèéíà
ó÷àñòîê
M
1
M
2
ñòðóíû,áóäåòðàâíà
x
2
Z
x
1
p
(
x;t
)
dx
(1.103)
60
Ïóñòü

(
x
)
ëèíåéíàÿïëîòíîñòüñòðóíû,òîãäàñèëàèíåð-
öèèó÷àñòêà
M
1
M
2
ñòðóíûáóäåòðàâíà

x
2
Z
x
1

(
x
)
@
2
u
@t
2
dx
(1.104)
Ñóììàïðîåêöèé(1.102)-(1.104)íàîñü
Ou
âñåõñèë,äåé-
ñòâóþùèõíàó÷àñòîê
M
1
M
2
ñòðóíûäîëæíàáûòüðàâíàíóëþ,
ò.å.
x
2
Z
x
1

T
0
@
2
u
@x
2


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

dx
=0
:
Îòñþäàââèäóïðîèçâîëüíîñòè
x
1
è
x
2
ñëåäóåòïðèïðåäïîëîæå-
íèèãëàäêîñòèïîäûíòåãðàëüíîéôóíêöèè,÷òîïîäûíòåãðàëü-
íàÿôóíêöèÿäîëæíàðàâíÿòüñÿíóëþäëÿêàæäîéòî÷êèñòðó-
íûâëþáîéìîìåíòâðåìåíè
t
,ò.å.

(
x
)
@
2
u
@t
2
=
T
0
@
2
u
@x
2
+
p
(
x;t
)
(1.105)
Ýòîåñòüèñêîìîåóðàâíåíèåêîëåáàíèéñòðóíû.
Åñëè

=
const;
ò.å.âñëó÷àåîäíîðîäíîéñòðóíû,óðàâíåíèå
(1.105)îáû÷íîçàïèñûâàåòñÿââèäå
@
2
u
@t
2
=
c
2
@
2
u
@x
2
+
f
(
x;t
)
;
(1.106)
ãäå
c
=
s
T
0

;f
(
x;t
)=
p
(
x;t
)

:
(1.107)
Åñëèâíåøíÿÿñèëàîòñóòñòâóåò,òî
p
(
x;t
)=0
èïîëó÷àåì
óðàâíåíèåñâîáîäíûõêîëåáàíèéñòðóíû
@
2
u
@t
2
=
c
2
@
2
u
@x
2
:
(1.108)
61
Óðàâíåíèå(1.105)èìååòáåñ÷èñëåííîåìíîæåñòâî÷àñòíûõ
ðåøåíèé.Ïîýòîìóîäíîãîóðàâíåíèÿ(1.105)íåäîñòàòî÷íîäëÿ
ïîëíîãîîïðåäåëåíèÿäâèæåíèÿñòðóíû.Íóæíûåù¼íåêîòîðûå
äîïîëíèòåëüíûåóñëîâèÿ,âûòåêàþùèåèçôèçè÷åñêîãîñìûñëà
çàäà÷è.Òàê,âíà÷àëüíûéìîìåíòâðåìåíè
t
=0
íóæíîçàäàòü
ïîëîæåíèåèñêîðîñòüâñåõòî÷åêñòðóíû
u
j
t
=0
=
'
0
(
x
)
;
@u
@t




t
=0
=
'
1
(
x
)
:
(1.109)
Óñëîâèÿ(1.109)íàçûâàþòñÿíà÷àëüíûìèóñëîâèÿìè.
Äàëåå,òàêêàêñòðóíàîãðàíè÷åíà,òîíóæíîóêàçàòü,÷òî
ïðîèñõîäèòíàå¼êîíöàõ.Äëÿçàêðåïëåííîéñòðóíûíàêîíöàõ
äîëæíîáûòü
u
j
x
=0
=0
;u
j
x
=
l
=0
(1.110)
ïðèâñÿêîì
t

0
:
Óñëîâèÿ(1.110)íàçûâàþòñÿêðàåâûìèèëè
ãðàíè÷íûìèóñëîâèÿìè.Âîçìîæíûèäðóãèåãðàíè÷íûåóñëî-
âèÿ.
Èòàê,ôèçè÷åñêàÿçàäà÷àîêîëåáàíèèñòðóíûñâåëàñüêìà-
òåìàòè÷åñêîéçàäà÷å:íàéòèðåøåíèåóðàâíåíèÿ(1.105),êîòîðîå
óäîâëåòâîðÿëîáûíà÷àëüíûìóñëîâèÿì(1.109)èãðàíè÷íûì
óñëîâèÿì(1.110).
Ìîæíîðàññìàòðèâàòüêîëåáàíèÿïîëóáåñêîíå÷íîéèëèáåñ-
êîíå÷íîéñòðóíû,êîãäàîäèíèëèîáàêîíöàíàõîäÿòñÿáåñêî-
íå÷íîäàëåêî.Îáàýòèñëó÷àÿÿâëÿþòñÿèäåàëèçàöèåéñëó÷àÿ
î÷åíüäëèííîéñòðóíû,ïðè÷åìïåðâûéèçíèõñîîòâåòñòâóåò
ðàññìîòðåíèþòî÷åê,ñðàâíèòåëüíîáëèçêèõîòîäíîãîèçêîí-
öîâñòðóíû,àâòîðîéðàññìîòðåíèþòî÷åê,ðàñïîëîæåííûõ
äàëåêîîòîáîèõêîíöîâ.Âïåðâîìèçýòèõñëó÷àåââêà÷åñòâå
ãðàíè÷íîãîóñëîâèÿîñòà¼òñÿòðåáîâàíèå
u
j
x
=0
=0
,àâîâòî-
ðîìñëó÷àåãðàíè÷íûåóñëîâèÿâîîáùåîòñóòñòâóþò.Íà÷àëüíûå
ôóíêöèè
'
0
(
x
)
è
'
1
(
x
)
äîëæíûáûòüâýòèõñëó÷àÿõçàäàíû
ñîîòâåòñòâåííîäëÿâñåõ
0
6
x
1
èëèäëÿâñåõ
�1
x
1
.
62
1.3.2.Óðàâíåíèåêîëåáàíèéìåìáðàíû
Ìåìáðàíîéíàçûâàþòñâîáîäíîèçãèáàþùóþñÿíàòÿíóòóþ
ïë¼íêó.
Ïóñòüâïîëîæåíèèðàâíîâåñèÿìåìáðàíàðàñïîëîæåíàâ
ïëîñêîñòè
xOy
èçàíèìàåòíåêîòîðóþîáëàñòü
D
,îãðàíè÷åí-
íóþçàìêíóòîéêðèâîé
L
.Äàëååïðåäïîëîæèì,÷òîìåìáðàíà
íàõîäèòñÿïîääåéñòâèåìðàâíîìåðíîãîíàòÿæåíèÿ
T
,ïðèëî-
æåííîãîêêðàÿììåìáðàíû.Ýòîîçíà÷àåò,÷òîåñëèïðîâåñòè
ëèíèþïîìåìáðàíåâëþáîìíàïðàâëåíèè,òîñèëàâçàèìîäåé-
ñòâèÿìåæäóäâóìÿ÷àñòÿìè,ðàçäåëåííûìèýëåìåíòàìèëèíèè,
ïðîïîðöèîíàëüíàäëèíåýëåìåíòàèïåðïåíäèêóëÿðíàåãîíà-
ïðàâëåíèþ.Âåëè÷èíàñèëû,äåéñòâóþùàÿíàýëåìåíò
dS
ëèíèè,
áóäåòðàâíà
TdS
.
Áóäåìðàññìàòðèâàòüòîëüêîïîïåðå÷íûåêîëåáàíèÿìåì-
áðàíû,ïðèêîòîðûõêàæäàÿå¼òî÷êàäâèæåòñÿïåðïåíäèêó-
ëÿðíîïëîñêîñòè
xOy
,ïàðàëëåëüíîîñè
Ou
.Òîãäàñìåùåíèå
u
òî÷êè
(
x;y
)
ìåìáðàíûáóäåòôóíêöèåéîò
x;y
è
t
.
Ðàññìàòðèâàÿäàëååòîëüêîìàëûåêîëåáàíèÿìåìáðàíû,áó-
äåìñ÷èòàòü,÷òîôóíêöèÿ
u
(
x;y;t
)
,àòàêæåå¼÷àñòíûåïðîèç-
âîäíûåïî
x
è
y
ìàëû,òàê÷òîêâàäðàòàìèèïðîèçâåäåíèÿìèèõ
ìîæíîïðèíåáðå÷üïîñðàâíåíèþññàìèìèýòèìèâåëè÷èíàìè.
Âûäåëèìïðîèçâîëüíûéó÷àñòîê

ìåìáðàíû,îãðàíè÷åí-
íûéâïîëîæåíèèðàâíîâåñèÿêðèâîé
l
.Êîãäàìåìáðàíàáóäåò
âûâåäåíàèçïîëîæåíèÿðàâíîâåñèÿ,ýòîòó÷àñòîêìåìáðàíûäå-
ôîðìèðóåòñÿâó÷àñòîê

0
ïîâåðõíîñòèìåìáðàíû,îãðàíè÷åí-
íûéïðîñòðàíñòâåííîéêðèâîé
l
0
.
Ïëîùàäüó÷àñòêà

0
âìîìåíòâðåìåíè
t
ðàâíà

0
=
ZZ

q
1+
u
2
x
+
u
2
y
dxdy

ZZ

dxdy
=
:
Òàêèìîáðàçîì,ïðèíàøèõïðåäïîëîæåíèÿõìîæíîïðåíå-
63
áðå÷üèçìåíåíèåìïëîùàäèïðîèçâîëüíîâçÿòîãîó÷àñòêàìåì-
áðàíûâïðîöåññåêîëåáàíèéèñ÷èòàòü,÷òîëþáîéó÷àñòîê

0
ìåìáðàíûáóäåòíàõîäèòñÿïîääåéñòâèåìïåðâîíà÷àëüíîãîíà-
òÿæåíèÿ
T
.
Ïåðåéä¼ìêâûáîðóóðàâíåíèÿïîïåðå÷íûõêîëåáàíèéìåì-
áðàíû.Ðàññìîòðèìïðîèçâîëüíûéó÷àñòîê

0
ìåìáðàíû.Ñî
ñòîðîíûîñòàëüíîé÷àñòèìåìáðàíûíàýòîòó÷àñòîêäåéñòâóåò
íàïðàâëåííîåïîíîðìàëèêêîíòóðó
l
ðàâíîìåðíîðàñïðåäåë¼í-
íîåíàòÿæåíèå
T
,ëåæàùååâêàñàòåëüíîéïëîñêîñòèêïîâåðõ-
íîñòèìåìáðàíû.Íàéäåìïðîåêöèþíàîñü
Ou
ñèëíàòÿæåíèÿ,
ïðèëîæåííûõêêðèâîé
l
0
,îãðàíè÷èâàþùåéó÷àñòîê

0
ìåìáðà-
íû.Îáîçíà÷èì÷åðåç
dS
0
ýëåìåíòäóãèêðèâîé
l
0
.Íàýòîòýëå-
ìåíòäåéñòâóåòíàòÿæåíèå,ðàâíîåïîâåëè÷èíå
TdS
0
.Êîñèíóñ
óãëà,îáðàçîâàííîãîâåêòîðîìíàòÿæåíèÿ
T
ñîñüþ
Ou
,î÷åâèä-
íî,ðàâåí,âñèëóíàøèõïðåäïîëîæåíèé
@u
@
n
,ãäå
n
íàïðàâëå-
íèåâíåøíåéíîðìàëèêêðèâîé
l
,îãðàíè÷èâàþùåéó÷àñòîê

ìåìáðàíûâïîëîæåíèèðàâíîâåñèÿ(ñì.ðèñ.2).
Ðèñ.2.
Îòñþäàñëåäóåò,÷òîïðîåêöèÿíàîñü
Ou
ñèëíàòÿæåíèÿ,
64
ïðèëîæåííûõêýëåìåíòó
dS
0
êîíòóðà
l
0
,ðàâíà
T
@u
@
n
dS
0
è,ñòàëîáûòü,ïðîåêöèÿíàîñü
Ou
íàòÿæåíèÿ,ïðèëîæåííûõ
êîâñåìóêîíòóðó
l
0
,ðàâíà
T
Z
l
0
@u
@
n
dS
0
:
(1.111)
Ò.ê.ïðèìàëûõêîëåáàíèÿõìåìáðàíûìîæíîñ÷èòàòü
dS

dS
0
,òîìûìîæåìâèíòåãðàëå(1.111)ïóòüèíòåãðèðîâà-
íèÿ
l
0
çàìåíèòüíà
l
.ÒîãäàïðèìåíÿÿôîðìóëóÃðèíà,ïîëó÷èì
T
Z
l
@u
@
n
dS
=
T
ZZ


@
2
u
@x
2
+
@
2
u
@y
2

dxdy:
(1.112)
Ïðåäïîëîæèìäàëåå,÷òîíàìåìáðàíóïàðàëëåëüíîîñè
Ou
äåéñòâóåòâíåøíÿÿñèëà
p
(
x;y;t
)
,ðàññ÷èòàííàÿíàåäèíèöó
ïëîùàäè.Òîãäàïðîåêöèÿíàîñü
Ou
âíåøíåéñèëû,äåéñòâó-
þùåéíàó÷àñòîê

0
ìåìáðàíû,áóäåòðàâíà
ZZ

p
(
x;y;t
)
dxdy:
(1.113)
Ñèëû(1.112)è(1.113)äîëæíûâëþáîéìîìåíòâðåìåíè
t
óðàâíîâåøèâàòüñÿñèëàìèèíåðöèèó÷àñòêà

0
ìåìáðàíû

ZZ


(
x;y
)
@
2
u
@t
2
dxdy;
ãäå

(
x;y
)
ïîâåðõíîñòíàÿïëîòíîñòüìåìáðàíû.
Òàêèìîáðàçîì,ìûïîëó÷àåìðàâåíñòâî
ZZ



(
x;y
)
@
2
u
@t
2

T

@
2
u
@x
2
+
@
2
u
@y
2

+
p
(
x;y;t
)

dxdy
=0
:
65
Îòñþäàâñèëóïðîèçâîëüíîñòèïëîùàäêè

ñëåäóåò,÷òî

(
x;y
)
@
2
u
@t
2
=
T

@
2
u
@x
2
+
@
2
u
@y
2

+
p
(
x;y;t
)
:
(1.114)
Ýòîåñòüäèôôåðåíöèàëüíîåóðàâíåíèåïîïåðå÷íûõêîëåáàíèé
ìåìáðàíû.Âñëó÷àåîäíîðîäíîéìåìáðàíû

=
const
óðàâíå-
íèÿìàëûõêîëåáàíèéìåìáðàíûìîæíîçàïèñàòüââèäå
@
2
u
@t
2
=
c
2

@
2
u
@x
2
+
@
2
u
@y
2

+
f
(
x;y;t
)
;
(1.115)
ãäå
c
=
s
T
p
;f
(
x;y;t
)=
p
(
x;y;t
)

:
(1.116)
Åñëèâíåøíÿÿñèëàîòñóòñòâóåò,ò.å.
p
(
x;y;t
)=0
,òîèç
(1.115)ïîëó÷àåìóðàâíåíèåñâîáîäíûõêîëåáàíèéîäíîðîäíîé
ìåìáðàíû
@
2
u
@t
2
=
c
2

@
2
u
@x
2
+
@
2
u
@y
2

:
(1.117)
Êàêèïðèðàññìîòðåíèèêîëåáàíèéñòðóíû,îäíîãîóðàâíå-
íèÿ(1.114)íåäîñòàòî÷íîäëÿïîëíîãîîïðåäåëåíèÿäâèæåíèÿ
ìåìáðàíû.Íóæíîçàäàòüñìåùåíèåèñêîðîñòüå¼òî÷åêâíà-
÷àëüíûéìîìåíòâðåìåíè
u
j
t
=0
=
'
0
(
x;y
)
;
@u
@t




t
=0
=
'
1
(
x;y
)
:
(1.118)
Äàëåå,òàêêàêíàêîíòóðå
l
ìåìáðàíàçàêðåïëåíà,òîäîëæíî
áûòü
u
j
L
=0
(1.119)
ïðèëþáîì
t

0
.
Âçàêëþ÷åíèåîòìåòèì,÷òîàíàëîãè÷íîâûâîäóóðàâíåíèé
êîëåáàíèéñòðóíûèìåìáðàíûâûâîäèòñÿòðåõìåðíîåâîëíîâîå
66
óðàâíåíèå
@
2
u
@t
2
=
c
2

@
2
u
@x
2
+
@
2
u
@y
2
+
@
2
u
@z
2

+
f
(
x;y;z;t
)
;
(1.120)
êîòîðîåîïèñûâàåòïðîöåññûðàñïðîñòðàíåíèÿçâóêàâîäíîðîä-
íîéñðåäåèýëåêòðîìàãíèòíûõâîëíâîäíîðîäíîéíåïðîâîäÿ-
ùåéñðåäå.Ýòîìóóðàâíåíèþóäîâëåòâîðÿþòïëîòíîñòüãàçà,
åãîäàâëåíèåèïîòåíöèàëñêîðîñòåé,àòàêæåñîñòàâëÿþùèå
íàïðÿæåííîñòèýëåêòðè÷åñêîãîèìàãíèòíîãîïîëåéèñîîòâåò-
ñòâóþùèåïîòåíöèàëû.
1.3.3.Óðàâíåíèåðàñïðîñòðàíåíèÿòåïëàâèçîòðîïíîì
òâåðäîìòåëå
Ïðîöåññûðàñïðîñòðàíåíèÿòåïëàèëèäèôôóçèè÷àñòèö
âñðåäåîïèñûâàþòñÿîáùèìóðàâíåíèåìòåïëîïðîâîäíîñòè
(1.10).Âûâåäåìóðàâíåíèåðàñïðîñòðàíåíèÿòåïëà(1.10)âñëó-
÷àåòðåõïðîñòðàíñòâåííûõïåðåìåííûõ.
Ðàññìîòðèìòâ¼ðäîåòåëî,òåìïåðàòóðàêîòîðîãîâòî÷êå
(
x;y;z
)
âìîìåíòâðåìåíè
t
îïðåäåëÿåòñÿôóíêöèåé
u
(
x;y;z;t
)
.
Åñëèðàçëè÷íûå÷àñòèòåëàíàõîäÿòñÿïðèðàçëè÷íîéòåìïå-
ðàòóðå,òîâòåëåáóäåòïðîèñõîäèòüäâèæåíèåòåïëàîòáîëåå
íàãðåòûõ÷àñòåéêìåíååíàãðåòûì.Âîçüì¼ìêàêóþ-íèáóäüïî-
âåðõíîñòü
S
âíóòðèòåëàèíàíåéìàëûéýëåìåíò

S
.Âòåî-
ðèèòåïëîïðîâîäíîñòèïðèíèìàåòñÿ,÷òîêîëè÷åñòâîòåïëà

Q
,
ïðîõîäÿùåãî÷åðåçýëåìåíò

S
çàâðåìÿ

t
,ïðîïîðöèîíàëüíî

t

S
èíîðìàëüíîéïðîèçâîäíîé
@u
@
n
,ò.å.

Q
=

k
@u
@
n

S

t
=

k

S

t
r
n
u;
(1.121)
ãäå
k�
0
êîýôôèöèåíòâíóòðåííåéòåïëîïðîâîäíîñòè,à
n
íîðìàëüêýëåìåíòóïîâåðõíîñòè

S
âíàïðàâëåíèèäâè-
æåíèÿòåïëà.Áóäåìñ÷èòàòü,÷òîòåëîèçîòðîïíîâîòíîøåííèå
67
òåïëîïðîâîäíîñòè,ò.å.÷òîêîýôôèöèåíòâíóòðåííåéòåïëîïðî-
âîäíîñòè
k
çàâèñèòòîëüêîîòòî÷êè
(
x;y;z
)
òåëàèíåçàâèñèò
îòíàïðàâëåíèÿíîðìàëèïîâåðõíîñòè
S
âýòîéòî÷êå.
Îáîçíà÷èì,÷åðåç
q
òåïëîâîéïîòîê,ò.å.êîëè÷åñòâîòåïëà,
ïðîõîäÿùåãî÷åðåçåäèíèöóïëîùàäèïîâåðõíîñòèçàåäèíèöó
âðåìåíè.Òîãäà(1.121)ìîæíîçàïèñàòüââèäå
q
=

k
@u
@
n
:
(1.122)
Äëÿâûâîäàóðàâíåíèÿðàñïðîñòðàíåíèÿòåïëàâûäåëèì
âíóòðèòåëàïðîèçâîëüíûéîáú¼ì
V
,îãðàíè÷åííûéãëàäêîéçà-
ìêíóòîéïîâåðõíîñòüþ
S
,èðàññìîòðèìèçìåíåíèåêîëè÷åñòâà
òåïëàâýòîìîáú¼ìåçàïðîìåæóòîêâðåìåíè
(
t
1
;t
2
)
.Íåòðóäíî
âèäåòü,÷òî÷åðåçïîâåðõíîñòü
S
çàïðîìåæóòîêâðåìåíè
(
t
1
;t
2
)
,
ñîãëàñíîôîðìóëå(1.121),âõîäèòêîëè÷åñòâîòåïëà,ðàâíîå
Q
1
=

t
2
Z
t
1
dt
ZZ
S
k
(
x;y;z
)
@u
@
n
dS;
ãäå
n
âíóòðåííÿÿíîðìàëüêïîâåðõíîñòè
S
.Ðàññìîòðèìýëå-
ìåíòîáú¼ìà

V
.Íàèçìåíåíèåòåìïåðàòóðûýòîãîîáú¼ìàíà

u
çàïðîìåæóòîêâðåìåíè

t
íóæíîçàòðàòèòüêîëè÷åñòâî
òåïëà

Q
2
=[
u
(
x;y;z;t
+
t
)

u
(
x;y;z;t
)]

(
x;y;z
)
V;
ãäå

(
x;y;z
)
;
(
x;y;z
)
ïëîòíîñòüèòåïëî¼ìêîñòüâåùåñòâà.
Òàêèìîáðàçîì,êîëè÷åñòâîòåïëà,íåîáõîäèìîåäëÿèçìåíåíèÿ
òåìïåðàòóðûîáú¼ìà
V
íà

u
=
u
(
x;y;z;t
2
)

u
(
x;y;z;t
1
)
,ðàâ-
íî
Q
2
=
ZZZ
V
[
u
(
x;y;z;t
2
)

u
(
x;y;z;t
1
)]
dV
68
èëè
Q
2
=
t
2
Z
t
1
dt
ZZZ
V

@u
@t
dV;
ò.ê.
u
(
x;y;z;t
2
)

u
(
x;y;z;t
1
)=
t
2
Z
t
1
@u
@t
dt:
Ïðåäïîëîæèì,÷òîâíóòðèðàññìàòðèâàåìîãîòåëàèìåþòñÿ
èñòî÷íèêèòåïëà.Îáîçíà÷èì÷åðåç
F
(
x;y;z;t
)
ïëîòíîñòü(êî-
ëè÷åñòâîïîãëîùàåìîãîèëèâûäåëÿåìîãîòåïëàâåäèíèöóâðå-
ìåíèâåäèíèöåîáú¼ìàòåëà)òåïëîâûõèñòî÷íèêîâ.Òîãäàêîëè-
÷åñòâîòåïëà,âûäåëÿåìîãîâîáú¼ìå
V
çàïðîìåæóòîêâðåìåíè
(
t
1
;t
2
)
,áóäåòðàâíî
Q
3
=
t
2
Z
t
1
dt
ZZZ
V
F
(
x;y;z;t
)
dV:
Ñîñòàâèìòåïåðüóðàâíåíèÿáàëàíñàòåïëàäëÿâûäåëåííîãî
îáú¼ìà
V
.Î÷åâèäíî,÷òî
Q
2
=
Q
1
+
Q
3
,ò.å.
t
2
Z
t
1
dt
ZZZ
V

@u
@t
dV
=

t
2
Z
t
1
dt
ZZ
S
k
@u
@
n
dS
+
+
t
2
Z
t
1
dt
ZZZ
V
F
(
x;y;z;t
)
dV;
èëè,ïðèìåíèâôîðìóëóÎñòðîãðàäñêîãîêîâòîðîìóèíòåãðàëó,
ïîëó÷èì
t
2
Z
t
1
dt
ZZZ
V


@u
@t

div
(
k
r
u
)

F
(
x;y;z;t
)

dV
=0
:
69
Òàêêàêïîäûíòåãðàëüíàÿôóíêöèÿíåïðåðûâíà,àîáú¼ì
V
è
ïðîìåæóòîêâðåìåíè
(
t
1
;t
2
)
ïðîèçâîëüíû,òîäëÿëþáîéòî÷êè
(
x;y;z
)
ðàññìàòðèâàåìîãîòåëàèäëÿëþáîãîìîìåíòàâðåìåíè
t
äîëæíîáûòü

@u
@t
=
div
(
k
r
u
)+
F
(
x;y;z;t
)
(1.123)
èëè

@u
@t
=
div
(
k
r
u
)+
F
(
x;y;z;t
)

@u
@t
=
=
@
@x

k
@u
@x

+
@
@y

k
@u
@y

+
@
@z

k
@u
@z

+
F
(
x;y;z;t
)
:
(1.124)
Ýòîóðàâíåíèåíàçûâàåòñÿóðàâíåíèåìòåïëîïðîâîäíîñòèíåîä-
íîðîäíîãîèçîòðîïíîãîòåëà.
Åñëèòåëîîäíîðîäíî,òî

,

è
k
ïîñòîÿííûåèóðàâíåíèå
(1.124)ìîæíîïåðåïèñàòüââèäå
@u
@t
=
a
2

@
2
u
@x
2
+
@
2
u
@y
2
+
@
2
u
@z
2

+
f
(
x;y;z;t
)
;
(1.125)
ãäå
a
=
s
k

;f
(
x;y;z;t
)=
F
(
x;y;z;t
)

:
Åñëèâðàññìàòðèâàåìîìîäíîðîäíîìòåëåíåòèñòî÷íèêîâ
òåïëà,ò.å.
F
(
x;y;z;t
)=0
,òîïîëó÷èìîäíîðîäíîåóðàâíåíèå
òåïëîïðîâîäíîñòè
@u
@t
=
a
2

@
2
u
@x
2
+
@
2
u
@y
2
+
@
2
u
@z
2

:
(1.126)
Â÷àñòíîìñëó÷àå,êîãäàòåìïåðàòóðàçàâèñèòòîëüêîîòêî-
îðäèíàò
x;y
è
t
,÷òîíàïðèìåð,èìååòìåñòîïðèðàñïðîñòðà-
íåíèèòåïëàâî÷åíüòîíêîéîäíîðîäíîéïëàñòèíå,óðàâíåíèå
70
(1.126)ïåðåõîäèòâñëåäóþùåå:
@u
@t
=
a
2

@
2
u
@x
2
+
@
2
u
@y
2

:
(1.127)
Íàêîíåö,äëÿòåëàëèíåéíîãîðàçìåðà,íàïðèìåð,äëÿòîí-
êîãîîäíîðîäíîãîñòåðæíÿ,óðàâíåíèåòåïëîïðîâîäíîñòèïðè-
ìåòâèä
@u
@t
=
a
2
@
2
u
@x
2
:
(1.128)
Îòìåòèì,÷òîïðèòàêîéôîðìåóðàâíåíèé(1.127)è(1.128)
íåó÷èòûâàåòñÿ,êîíå÷íî,òåïëîâîéîáìåíìåæäóïîâåðõíîñòüþ
ïëàñòèíêèèëèñòåðæíÿñîêðóæàþùèìïðîñòðàíñòâîì.
×òîáûíàéòèòåìïåðàòóðóâíóòðèòåëàâëþáîéìîìåíòâðå-
ìåíè,íåäîñòàòî÷íîîäíîãîóðàâíåíèÿ(1.124).Íåîáõîäèìî,êàê
ýòîñëåäóåòèçôèçè÷åñêèõñîîáðàæåíèé,çíàòüåù¼ðàñïðåäå-
ëåíèåòåìïåðàòóðûâíóòðèòåëàâíà÷àëüíûéìîìåíòâðåìåíè
(íà÷àëüíîåóñëîâèå)èòåïëîâîéðåæèìíàãðàíèöå
S
òåëà(ãðà-
íè÷íîåóñëîâèå).
Ãðàíè÷íîåóñëîâèåìîæåòáûòüçàäàíîðàçëè÷íûìèñïîñî-
áàìè:
1)âêàæäîéòî÷êåïîâåðõíîñòè
S
çàäàåòñÿòåìïåðàòóðà
u
j
S
=

1
(
P
;t
)
;
(1.129)
ãäå

1
(
P
;t
)
èçâåñòíàÿôóíêöèÿòî÷êèïîâåðõíîñòè
S
èâðå-
ìåíè
t
;
2)íàïîâåðõíîñòè
S
çàäàåòñÿòåïëîâîéïîòîê
q
=

k
@u
@
n
;
îòêóäà
@u
@
n




S
=

2
(
P
;t
)
;
(1.130)
71
ãäå

2
(
P
;t
)
èçâåñòíàÿôóíêöèÿ,âûðàæàþùàÿñÿ÷åðåççàäàí-
íûéòåïëîâîéïîòîêïîôîðìóëå

2
(
P
;t
)=

q
(
P
;t
)
k
;
3)íàïîâåðõíîñòèòâåðäîãîòåëàïðîèñõîäèòòåïëîîáìåíñ
îêðóæàþùåéñðåäîé,òåìïåðàòóðàêîòîðîé
u
0
èçâåñòíà.Çàêîí
òåïëîîáìåíàî÷åíüñëîæåí,íîäëÿóïðîùåíèÿçàäà÷èîíìî-
æåòáûòüïðèíÿòââèäåçàêîíàÍüþòîíà.ÏîçàêîíóÍüþòîíà,
êîëè÷åñòâîòåïëà,ïåðåäàâàåìîåâåäèíèöóâðåìåíèñåäèíèöû
ïëîùàäèïîâåðõíîñòèòåëàâîêðóæàþùóþñðåäó,ïðîïîðöèî-
íàëüíîðàçíîñòèòåìïåðàòóðïîâåðõíîñòèòåëàèîêðóæàþùåé
ñðåäû:
q
=
H
(
u

u
0
)
;
ãäå
H
êîýôôèöèåíòòåïëîîáìåíà.Êîýôôèöèåíòòåïëîîáìåíà
çàâèñèòîòðàçíîñòèòåìïåðàòóð
u

u
0
,îòõàðàêòåðàïîâåðõíî-
ñòèèîêðóæàþùåéñðåäû(îíìîæåòèçìåíÿòüñÿâäîëüïîâåðõ-
íîñòèòåëà).Ìûáóäåìñ÷èòàòüêîýôôèöèåíòòåïëîîáìåíà
H
ïîñòîÿííûì,íåçàâèñÿùèìîòòåìïåðàòóðûèîäèíàêîâûìäëÿ
âñåéïîâåðõíîñòèòåëà.
Ïîçàêîíóñîõðàíåíèÿýíåðãèè,ýòîêîëè÷åñòâîòåïëàäîëæ-
íîáûòüðàâíîòîìóêîëè÷åñòâóòåïëà,êîòîðîåïåðåäàåòñÿ÷å-
ðåçåäèíèöóïëîùàäèïîâåðõíîñòèçàåäèíèöóâðåìåíèâñëåä-
ñòâèåâíóòðåííåéòåïëîïðîâîäíîñòè.Ýòîïðèâîäèòêñëåäóþ-
ùåìóãðàíè÷íîìóóñëîâèþ:
H
(
u

u
0
)=

k
@u
@
n
íàS,
ãäå
n
âíåøíÿÿíîðìàëüêïîâåðõíîñòè
S
,èëè,ïîëîæèâ
h
=
H
k
,

@u
@
n
+
h
(
u

u
0
)





S
=0
:
(1.131)
72
Òàêèìîáðàçîì,çàäà÷àîðàñïðîñòðàíåíèèòåïëàâèçîòðîï-
íîìòâåðäîìòåëåñòàâèòñÿòàê:íàéòèðåøåíèåóðàâíåíèÿòåï-
ëîïðîâîäíîñòè(1.124),óäîâëåòâîðÿþùååíà÷àëüíîìóóñëîâèþ
u
j
t
=0
=
'
(
x;y;z
)
(1.132)
èîäíîìóèçãðàíè÷íûõóñëîâèé(1.129),(1.130)èëè(1.131).
1.3.4.Óðàâíåíèÿ,îïèñûâàþùèåñòàöèîíàðíûåïðîöåññûðàñïðîñòðàíåíèÿ
òåïëà
Äëÿñòàöèîíàðíûõïðîöåññîâ(ïðîöåññîâíåçàâèñÿùèõîò
âðåìåíè)âóðàâíåíèÿõêîëåáàíèé(1.9)èòåïëîïðîâîäíîñòè
(1.10)ïðèíèìàåòñÿ,÷òî
f
=
f
(
x
)
è
u
=
u
(
x
)
èýòèóðàâíåíèÿ
ïðèíèìàþòâèä

u
=

h
(
x
)
;
=
n
X
i
=1
@
2
@x
2
i
:
(1.133)
Ïðè
h
(
x
)=0
óðàâíåíèå(1.133)íàçûâàåòñÿóðàâíåíèåìËàïëà-
ñà:

u
=0
:
(1.134)
Äëÿïîëíîãîîïèñàíèÿñòàöèîíàðíîãîïðîöåññàíåîáõîäèìî
åùåçàäàòüðåæèìíàãðàíèöåîäíîèçãðàíè÷íûõóñëîâèé
(1.129)(1.131).
1.4.Âîïðîñûèçàäà÷è
1.Ïðîâåäèòåêëàññèôèêàöèþóðàâíåíèé
à)
u
t
=
u
xx
+2
u
x
+
u
,
á)
u
t
=
u
xx
+
e

t
,
â)
u
xx
+3
u
xy
+
u
yy
=cos
x
,
ã)
u
tt
=
uu
xxx
+2
u
x
+sin
x
ïîñëåäóþùèìïðèçíàêàì:
1)ëèíåéíîñòü,êâàçèëèíåéíîñòü,íåëèíåéíîñòü;
2)ïîðÿäîê;
73
3)âèäêîýôôèöèåíòîâ(ïîñòîÿííûå,ïåðåìåííûå);
4)îäíîðîäíîñòü(äëÿëèíåéíûõóðàâíåíèé);
5)òèï(äëÿëèíåéíûõóðàâíåíèé).
2.Ñêîëüêîñóùåñòâóåòðåøåíèéóðàâíåíèÿ
u
t
=
u
xx
?Íàéäèòå
÷àñòíûåðåøåíèÿýòîãîóðàâíåíèÿâèäà
u
=
e
ax
+
bt
.
3.Íàéäèòåâñåôóíêöèè,êîòîðûåÿâëÿþòñÿðåøåíèÿìèóðàâ-
íåíèÿ
@u
(
x;y
)
@x
=0
.
4.Åñëèôóíêöèè
u
1
=
u
1
(
x;t
)
è
u
2
=
u
2
(
x;t
)
óäîâëåòâîðÿþò
óðàâíåíèþ(1.41),òîóäîâëåòâîðÿåòëèýòîìóóðàâíåíèþèõ
ñóììà?
5.Êàêîéôàêòëèíåéíîéàëãåáðûëåæèòâîñíîâåêëàññèôèêà-
öèèóðàâíåíèé?
6.Âûÿñíèòå,êàêîìóòèïóïðèíàäëåæàòñëåäóþùèåóðàâíåíèÿ:
à)
u
xx

u
xy
=0
,
á)
u
tt
=
u
xx
+
u
x
+5
u
,
â)
u
xx
+
u
yy
=
f
(
x;y
)
,
ã)
u
rr
+
1
r
u
r
+
1
r
2
u
''
=
f
(
r;'
)
.
7.Â÷åìîñîáåííîñòüêëàññèôèêàöèèíåëèíåéíûõóðàâíåíèé?
8.Íàéäèòåõàðàêòåðèñòèêèóðàâíåíèÿ
u
xx
+4
u
xy
=0
.
9.Ìîæíîëèâîáùåìñëó÷àåïðèâåñòèëèíåéíîåóðàâíåíèå
âòîðîãîïîðÿäêàêêàíîíè÷åñêîìóâèäóâíåêîòîðîéîáëàñòè?
74
10.Ïðèâåäèòåóðàâíåíèå
3
u
xx
+7
u
xy
+2
u
yy
=0
êêàíîíè÷åñêîìóâèäóâêàæäîéèçîáëàñòåé,ãäåñîõðàíÿåòñÿ
åãîòèï.
11.Îõàðàêòåðèçóéòåóðàâíåíèå
@
2
u
(
x;y
)
@[email protected]
=0
ïîïðèçíàêàì,
ñôîðìóëèðîâàííûìâçàäà÷å1.Íàéäèòåâñåðåøåíèÿýòîãî
óðàâíåíèÿ.
12.Ïîñòàâüòåíà÷àëüíûåóñëîâèÿäëÿóðàâíåíèÿ
u
tt
+
u
xt

2
u
xx
=0
òàêèìîáðàçîì,÷òîáûåäèíñòâåííûìðåøåíèåìïîëó÷åííîé
çàäà÷èÊîøèÿâëÿëàñüôóíêöèÿ
u
(
x;t
)=2
t

x
.
13.Âåðíîëè,÷òîåñëèíåõàðàêòåðèñòè÷åñêàÿçàäà÷àÊîøè
äëÿëèíåéíîãîóðàâíåíèÿâòîðîãîïîðÿäêàèìååòíåêîòîðîå
àíàëèòè÷åñêîåðåøåíèå,òîäðóãèõàíàëèòè÷åñêèõðåøåíèéó
ýòîéçàäà÷èíåò?
14.Êàêèåôèçè÷åñêèåçàêîíûëåæàòâîñíîâåìîäåëèðîâàíèÿ
ôèçè÷åñêèõïðîöåññîâðàñïðîñòðàíåíèÿâîëíèòåïëà?
15.Ïðåäïîëîæèì,÷òîòåïëîèçîëèðîâàííûéòîíêèéîäíîðîä-
íûéñòåðæåíüäëèíû
l
=1
,áîêîâàÿïîâåðõíîñòüêîòîðîãî
òåïëîèçîëèðîâàíà,èìååòíà÷àëüíóþòåìïåðàòóðó
sin(3
x
)
,
àñëåâàèñïðàâàíàêîíöàõïîääåðæèâàþòñÿñîîòâåòñòâåííî
ôèêñèðîâàííûåòåìïåðàòóðû
0
0
C
è
10
0
C
.Êàêñôîðìóëèðîâàòü
ñìåøàííóþçàäà÷óäëÿóðàâíåíèÿòåïëîïðîâîäíîñòèâýòîì
ñëó÷àå?
75
2.Óðàâíåíèÿãèïåðáîëè÷åñêîãîòèïà
2.1.Îäíîðîäíîåâîëíîâîåóðàâíåíèå
2.1.1.Çàäà÷àÊîøèäëÿîäíîìåðíîãîâîëíîâîãîóðàâíåíèÿ.
ÔîðìóëàÄàëàìáåðà
Èçó÷åíèåìåòîäîâïîñòðîåíèÿðåøåíèéçàäà÷äëÿóðàâíå-
íèéãèïåðáîëè÷åñêîãîòèïàìûíà÷èíàåìñçàäà÷èÊîøèäëÿ
óðàâíåíèÿñâîáîäíûõêîëåáàíèéñòðóíû:
@
2
u
@t
2
=
a
2
@
2
u
@x
2
;
(2.1)
u
(
x;
0)=
'
(
x
)
;
@u
(
x;
0)
@t
=

(
x
)
:
(2.2)
Ïðåîáðàçóåìóðàâíåíèå(2.1)êêàíîíè÷åñêîìóâèäó,ñîäåð-
æàùåìóñìåøàííóþïðîèçâîäíóþ.Óðàâíåíèåõàðàêòåðèñòèê
'
2
t

a
2
'
2
x
=0
ðàñïàäàåòñÿíàäâàóðàâíåíèÿ
'
t

a'
x
=0
;'
t
+
a'
x
=0
;
÷àñòíûìèðåøåíèÿìèêîòîðûõÿâëÿþòñÿñîîòâåòñòâåííîôóíê-
öèè
'
1
(
x;t
)=
x

at;'
2
(
x;t
)=
x
+
at:
Òåïåðüïîëàãàÿ

=
x
+
at;
=
x

at;
óðàâíåíèå(2.1)ïðåîáðàçóåòñÿêâèäó
@
2
u
@@
=0
:
(2.3)
Îáùååðåøåíèåóðàâíåíèÿ(2.3)äàåòñÿôîðìóëîé
u
=
f
1
(

)+
f
2
(

)
;
76
ãäå
f
1
(

)
è
f
2
(

)
ïðîèçâîëüíûåãëàäêèåôóíêöèè.Âîçâðàùà-
ÿñüêïåðåìåííûì
x
,
t
ïîëó÷àåì
u
=
f
1
(
x
+
at
)+
f
2
(
x

at
)
:
(2.4)
Ïîëó÷åííîåðåøåíèåçàâèñèòîòäâóõïðîèçâîëüíûõôóíêöèé
f
1
è
f
2
.ÎíîíàçûâàåòñÿðåøåíèåìÄàëàìáåðà.
Äàëåå,ïîäñòàâëÿÿ(2.4)â(2.2),áóäåìèìåòü
f
1
(
x
)+
f
2
(
x
)=
'
(
x
)
;
(2.5)
af
0
1
(
x
)

af
0
2
(
x
)=

(
x
)
;
(2.6)
îòêóäà,èíòåãðèðóÿðàâåíñòâî(2.6),ïîëó÷èì
f
1
(
x
)

f
2
(
x
)=
1
a
x
Z
x
0

(
y
)
dy
+
C;
(2.7)
ãäå
x
0
è
C
äåéñòâèòåëüíûåïîñòîÿííûå.Èçôîðìóë(2.5)è
(2.7)íàõîäèì
f
1
(
x
)=
1
2
2
6
4
'
(
x
)+
1
a
x
Z
x
0

(
y
)
dy
+
C
3
7
5
;
f
2
(
x
)=
1
2
2
6
4
'
(
x
)

1
a
x
Z
x
0

(
y
)
dy

C
3
7
5
:
Ïðèýòîì,ó÷èòûâàÿ(2.4),èìååì
u
(
x;t
)=
1
2
h
'
(
x
+
at
)+
1
a
x
+
at
Z
x
0

(
y
)
dy
+
C
+
+
'
(
x

at
)

1
a
x

at
Z
x
0

(
y
)
dy

C
i
;
77
èîêîí÷àòåëüíîïîëó÷àåìôîðìóëó
u
(
x;t
)=
'
(
x
+
at
)+
'
(
x

at
)
2
+
1
2
a
x
+
at
Z
x

at

(
y
)
dy:
(2.8)
Ôîðìóëà(2.8)íàçûâàåòñÿôîðìóëîéÄàëàìáåðà.
Íåòðóäíîïðîâåðèòü,÷òîôîðìóëà(2.8)óäîâëåòâîðÿåò
óðàâíåíèþ(2.1)èíà÷àëüíûìóñëîâèÿì(2.2)ïðèóñëîâèè,÷òî
'
(
x
)
2
C
2
(
R
)
,a

(
x
)
2
C
1
(
R
)
.Òàêèìîáðàçîì,èçëîæåííûéìå-
òîääîêàçûâàåòñóùåñòâîâàíèåðåøåíèÿïîñòàâëåííîéçàäà÷è.
Ïîêàæåì,÷òîðåøåíèåçàäà÷è(2.1)-(2.2)íåïðåðûâíîçàâè-
ñèòîòíà÷àëüíûõäàííûõ(óñòîé÷èâî).Àèìåííî:êàêîâáûíè
áûëïðîìåæóòîêâðåìåíè
[0
;t
0
]
èêàêîâàáûíèáûëàñòåïåíü
òî÷íîñòè
"
,íàéäåòñÿòàêîå

(
";t
0
)
,÷òîâñÿêèåäâàðåøåíèÿ
u
1
(
x;t
)
è
u
2
(
x;t
)
óðàâíåíèÿ(2.1)âòå÷åíèåïðîìåæóòêàâðå-
ìåíè
[0
;t
0
]
áóäóòðàçëè÷àòüñÿìåæäóñîáîéìåíüøå÷åìíà
"
:
j
u
1
(
x;t
)

u
2
(
x;t
)
j
";
0
6
t
6
t
0
;
åñëèòîëüêîíà÷àëüíûåçíà÷åíèÿ
8



:
u
1
(
x;
0)=
'
1
(
x
)
;
@u
1
(
x;t
)
@t



t
=0
=

1
(
x
)
è
8



:
u
2
(
x;
0)=
'
2
(
x
)
;
@u
2
(
x;t
)
@t



t
=0
=

2
(
x
)
îòëè÷àþòñÿäðóãîòäðóãàìåíüøå÷åìíà

:
j
'
1
(
x
)

'
2
(
x
)
j
;
j

1
(
x
)


2
(
x
)
j
:
(2.9)
Äåéñòâèòåëüíî,ôóíêöèè
u
1
(
x;t
)
è
u
2
(
x;t
)
ñâÿçàíûñîñâî-
èìèíà÷àëüíûìèäàííûìèôîðìóëîé(2.8),ïîýòîìóèìååì
j
u
1
(
x;t
)

u
2
(
x;t
)
j
6
1
2
j
'
1
(
x
+
at
)

'
2
(
x
+
at
)
j
+
+
1
2
j
'
1
(
x

at
)

'
2
(
x

at
)
j
+
1
2
a
x
+
at
Z
x

at
j

1
(
y
)


2
(
y
)
j
dy:
78
Îòêóäà,âñèëóíåðàâåíñòâ(2.9)ïîëó÷àåì:
j
u
1
(
x;t
)

u
2
(
x;t
)
j
6

2
+

2
+
1
2
a


2
at
6

(1+
t
0
)
;
÷òîèäîêàçûâàåòíàøåóòâåðæäåíèå,åñëèïîëîæèòü

"
1+
t
0
:
2.1.2.Çàäà÷àñíà÷àëüíûìèóñëîâèÿìèäëÿâîëíîâîãî
óðàâíåíèÿñòðåìÿïðîñòðàíñòâåííûìèïåðåìåííûìè.
ÔîðìóëàÊèðõãîôà
Ðàññìîòðèìâîëíîâîåóðàâíåíèåñòðåìÿïðîñòðàíñòâåííû-
ìèïåðåìåííûìèïðèîòñóòñòâèèâíåøíèõâîçìóùåíèé
@
2
u
@t
2
=
a
2

u;x
2
R
3
;t�
0
;
(2.10)
ãäå
=

@
2
@x
2
1
+
@
2
@x
2
2
+
@
2
@x
2
3

.
Çàìå÷àíèå21.
Âìåñòîóðàâíåíèÿ(2.10),íåîãðàíè÷èâàÿ
îáùíîñòè,äàëååáóäåìðàññìàòðèâàòüóðàâíåíèå
@
2
u
@t
2
=
u;x
2
R
3
;t�
0
;
(2.11)
ò.å.âîçüìåì
a
=1
,ò.ê.óðàâíåíèå(2.11)ñâîäèòñÿê(2.10)çà-
ìåíîé
at
íà
t
.
Õàðàêòåðèñòè÷åñêàÿêâàäðàòè÷íàÿôîðìàäëÿóðàâíåíèÿ
(2.11)èìååòêàíîíè÷åñêèéâèä
Q
(

1
;
2
;
3
;
4
)=

2
1
+

2
2
+

2
3


2
4
;
è,ñëåäîâàòåëüíî,ýòîóðàâíåíèåâîâñåìïðîñòðàíñòâå
R
4
ÿâëÿ-
åòñÿãèïåðáîëè÷åñêèì.
Ïîêàæåì,÷òîôóíêöèÿ
u
(
x
;t
)=
ZZ
S

(
y
1
;y
2
;y
3
)
j
y

x
j
dS
y
;
(2.12)
79
ãäå
j
y

x
j
ðàññòîÿíèåìåæäóòî÷êàìè
x
=(
x
1
;x
2
;x
3
)
è
y
=(
y
1
;y
2
;y
3
)
,
S
=
f
y
2
R
3
:
j
y

x
j
2
=
t
2
g
ñôåðàñöåíòðîì
âòî÷êå
x
ðàäèóñà
t


çàäàííàÿïðîèçâîëüíàÿäåéñòâèòåëü-
íàÿäâàæäûíåïðåðûâíîäèôôåðåíöèðóåìàÿôóíêöèÿ,ÿâëÿåò-
ñÿðåøåíèåìóðàâíåíèÿ(2.11).
Âñàìîìäåëå,âðåçóëüòàòåçàìåíûïåðåìåííûõ
y
i

x
i
=
t
i
;i
=1
;
2
;
3
,ôîðìóëà(2.12)ïðèíèìàåòâèä
u
(
x
;t
)=
t
ZZ


(
x
1
+
t
1
;x
2
+
t
2
;x
3
+
t
3
)
d

;
(2.13)
ãäå

=
f

2
R
3
:
j

j
=1
g
åäèíè÷íàÿñôåðà,à
d

=
dS
y
t
2
=
dS
y
j
y

x
j
2
ýëåìåíòååïëîùàäè.Èç(2.13)èìå-
åì
3
X
i
=1
@
2
u
@x
2
i
=
t
ZZ

3
X
i
=1
@
2

@y
2
i
d

:
(2.14)
Êðîìåòîãî,
@u
@t
=
ZZ


(
x
1
+
t
1
;x
2
+
t
2
;x
3
+
t
3
)
d

+
+
t
ZZ

3
X
i
=1
@
@y
i

i
d

=
u
t
+
1
t
I;
(2.15)
ãäå
I
=
ZZ
S

@
@y
1

1
+
@
@y
2

2
+
@
@y
3

3

dS
y
;
(2.16)
à

(
y
)=(

1
;
2
;
3
)
âíåøíÿÿíîðìàëüê
S
âòî÷êå
y
.Äèôôå-
ðåíöèðóÿðàâåíñòâî(2.15)ïî
t
,íàõîäèì
@
2
u
@t
2
=

u
t
2
+
1
t
@u
@t

1
t
2
I
+
1
t
@I
@t
=

u
t
2
+
1
t

u
t
+
I
t


I
t
2
+
80
+
1
t
@I
@t
=
1
t
@I
@t
:
(2.17)
Èçêóðñàìàòåìàòè÷åñêîãîàíàëèçàèçâåñòíî,÷òîäëÿäåé-
ñòâèòåëüíûõôóíêöèé
A
i
(
y
)
;i
=1
;:::;n
,íåïðåðûâíûõâìåñòå
ñîñâîèìèïðîèçâîäíûìèïåðâîãîïîðÿäêàâçàìêíóòîéîáëà-
ñòè
D
[
S
ñãëàäêîéãðàíèöåé
S
,èìååòìåñòîôîðìóëàÃàóññà-
Îñòðîãðàäñêîãî
ZZZ
D
n
X
i
=1
@A
i
@y
i
d
y
=
ZZ
S
n
X
i
=1
A
i
(
y
)

i
(
y
)
dS
y
;
(2.18)
ãäå
d
x
ýëåìåíòîáú¼ìà,à

=(

1
;:::;
n
)
âíåøíÿÿíîðìàëü
ê
S
âòî÷êå
y
2
S
.
Ïðàâàÿ÷àñòü(2.16)ïîôîðìóëå(2.18)ïðåîáðàçóåòñÿâèí-
òåãðàëïîøàðó
f
y
2
R
3
:
j
y

x
j
2
t
2
g
I
=
ZZZ
j
y

x
j
2
t
2
3
X
i
=1
@
2

@y
2
i
d
y
=
ZZZ
j
y

x
j
2
t
2

d
y
;
(2.19)
ãäå
d
y
ýëåìåíòîáúåìàïîïåðåìåííîìóèíòåãðèðîâàíèÿ
y
,
d
y
=
dy
1
dy
2
dy
3
.
Ïåðåõîäÿîòäåêàðòîâûõêîîðäèíàò
y
1
;y
2
;y
3
êñôåðè÷åñêèì
;;'
,âûðàæåíèå(2.19)äëÿ
I
çàïèøåìââèäå
I
=
ZZZ
j
y

x
j
2
t
2


(
y
)
d
y
=
ZZZ
j
y

x
j
2
t
2


(
y
)
dy
1
dy
2
dy
3
=
=
2
4
y
1
=
x
1
+
cos'sin
y
2
=
x
2
+
sin'sin
y
3
=
x
3
+
cos
3
5
=
=
t
Z
0

Z
0
2

Z
0


(
;;'
)

2
sind'dd;
81
ãäå

2
sind'dd
=
d
y
=
dy
1
dy
2
dy
3
.Îòñþäà,ò.ê.
sindd'
=
d

,íàõîäèì
@I
@t
=
t
2

Z
0
2

Z
0


(
t;;'
)
sind'd
=
t
2
ZZ


d

=
=
t
2
ZZ

3
X
i
=1
@
2

@y
2
i
d

:
Ñëåäîâàòåëüíî,âñèëó(2.17)ìîæíîíàïèñàòü
@
2
u
@t
2
=
t
ZZ

3
X
i
=1
@
2

@y
2
i
d

=
t
ZZ


d

(2.20)
Íàîñíîâàíèè(2.14)è(2.20)çàêëþ÷àåì,÷òîïðåäñòàâëåííàÿ
ôîðìóëîé(2.12)ôóíêöèÿ
u
(
x
;t
)
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ
(2.11).
Èòàê,ìûäîêàçàëè,÷òîïðèòðåáîâàíèèñóùåñòâîâà-
íèÿíåïðåðûâíûõïðîèçâîäíûõâòîðîãîïîðÿäêàóôóíêöèè

(
x
1
;x
2
;x
3
)
çàäàííîéâïðîñòðàíñòâå
R
3
ïåðåìåííûõ
x
1
;x
2
;x
3
,
ôóíêöèÿ
u
(
x
1
;x
2
;x
3
;t
)=
tM
(

)
;
ãäå
M
(

)=
ZZ


(
x
1
+
t
1
;x
2
+
t
2
;x
3
+
t
3
)
d

;
(2.21)
ïðåäñòàâëÿåòñîáîéðåãóëÿðíîåðåøåíèåâîëíîâîãîóðàâíåíèÿñ
òðåìÿïðîñòðàíñòâåííûìèïåðåìåííûìè.
Çàìåòèì,÷òîò.ê.
d

=
1
t
2
dS
y
,òîâûðàæåíèå
1
4

M
(

)=
1
4
t
2
ZZ
j
y

x
j
2
=
t
2

(
y
1
;y
2
;y
3
)
dS
y
(2.22)
82
ÿâëÿåòñÿèíòåãðàëüíûìñðåäíèìôóíêöèè

ïîñôåðå
S
.
Íàðÿäóñ
tM
(

)
ðåãóëÿðíûìðåøåíèåìóðàâíåíèÿ(2.11)ÿâ-
ëÿåòñÿèôóíêöèÿ
@
@t
[
tM
(

)]
,åñëèòîëüêî

èìååòíåïðåðûâíûå
ïðîèçâîäíûåòðåòüåãîïîðÿäêà.
Ëåãêîâèäåòü,÷òîôóíêöèÿ
u
(
x
1
;x
2
;x
3
;t
)=
1
4

tM
(

)+
1
4

@
@t
[
tM
(
'
)]
(2.23)
ÿâëÿåòñÿðåãóëÿðíûìðåøåíèåìçàäà÷èÊîøèäëÿâîëíîâîãî
óðàâíåíèÿ(2.11)ñíà÷àëüíûìèóñëîâèÿìè
u
(
x
1
;x
2
;x
3
;
0)=
'
(
x
1
;x
2
;x
3
)
;
(2.24)
@u
(
x
1
;x
2
;x
3
;t
)
@t



t
=0
=

(
x
1
;x
2
;x
3
)
;
(2.25)
ãäå
'
(
x
1
;x
2
;x
3
)
è

(
x
1
;x
2
;x
3
)
çàäàííûåâïðîñòðàíñòâå
R
3
ïå-
ðåìåííûõ
x
1
;x
2
;x
3
äåéñòâèòåëüíûåôóíêöèè,èìåþùèåíåïðå-
ðûâíûå÷àñòíûåïðîèçâîäíûåòðåòüåãîèâòîðîãîïîðÿäêàñî-
îòâåòñòâåííî.
Äåéñòâèòåëüíî,êàêóæåáûëîîòìå÷åíîâûøå,êàæäîåñëà-
ãàåìîåâïðàâîé÷àñòè(2.23)ÿâëÿåòñÿðåãóëÿðíûìðåøåíèåì
óðàâíåíèÿ(2.11).Ïðè
t
=0
íàîñíîâàíèè(2.21)èç(2.23)èìååì
u
(
x
1
;x
2
;x
3
;
0)=
1
4

M
(
'
)



t
=0
=
=
1
4

ZZ

'
(
x
1
;x
2
;x
3
)
d

=
'
(
x
1
;x
2
;x
3
)
:
Äàëåå,òàêêàê
@u
(
x
1
;x
2
;x
3
;t
)
@t
=
1
4

@
@t
[
tM
(

)]+
1
4

@
2
@t
2
[
tM
(
'
)]=
=
1
4

@
@t
[
tM
(

)]+
1
4

t

M
(
'
)
;
83
òî
@u
(
x
1
;x
2
;x
3
;t
)
@t



t
=0
=
1
4

ZZ


(
x
1
;x
2
;x
3
)
d

=

(
x
1
;x
2
;x
3
)
:
Ðàâåíñòâî(2.23),äàþùååðåøåíèåçàäà÷èÊîøè(2.24),
(2.25)äëÿâîëíîâîãîóðàâíåíèÿ(2.11)âñëó÷àåòðåõïðîñòðàí-
ñòâåííûõïåðåìåííûõ
x
1
;x
2
;x
3
íàçûâàåòñÿôîðìóëîéÊèðõãî-
ôà.
Ôèçè÷åñêîåÿâëåíèå,îïèñûâàåìîåðåøåíèåì
u
(
x
;t
)
âîëíî-
âîãîóðàâíåíèÿ,íàçûâàåòñÿðàñïðîñòðàíåíèåìâîëíû,àñàìî
ðåøåíèå
u
(
x
;t
)
âîëíîé.
Ïîñêîëüêó
@
@t
[
tM
(
'
)]=
M
(
'
)+
t
@
@t
M
(
'
)=
M
(
'
)+
t
ZZ

3
X
i
=1
@'
@y
i

i
d

=
=
M
(
'
)+
1
t
ZZ
S
r
'

y

x
t
dS
y
=
M
(
'
)+
1
t
ZZ
S
r
'


dS
y
=
=
M
(
'
)+
1
t
ZZ
S
@'
@

dS
y
;
ãäå

âíåøíÿÿíîðìàëüê
S
âòî÷êå
y
,èçôîðìóëûÊèðõ-
ãîôàñëåäóåò,÷òîñîîòâåòñòâóþùàÿçàäà÷åÊîøè(2.11),(2.24),
(2.25)âîëíàâñëó÷àåòðåõïðîñòðàíñòâåííûõïåðåìåííûõâòî÷-
êå
(
x
;t
)=(
x
1
;x
2
;x
3
;t
)
ïðîñòðàíñòâà
R
4
âïîëíåîïðåäåëÿåòñÿ
çíà÷åíèÿìè
'
,
@'
@

è

íàñôåðå
S
.Ýòîòôàêòâòåîðèèçâóêà
íàçûâàåòñÿïðèíöèïîìÃþéãåíñà.
Ïîêàæåì,÷òîðåøåíèåçàäà÷èÊîøè(2.11),(2.24),(2.25)
íåïðåðûâíîçàâèñèòîòíà÷àëüíûõäàííûõ.Äëÿýòîãîâìåñòî
ôóíêöèé
'
è

â(2.23)âîçüìåìäðóãèå
'
0
è

0
,òàêèå,÷òî
j
'

'
0
j
;




@'
@x
i

@'
0
@x
i




;
j



0
j
;i
=1
;
2
;
3
:
84
Òîãäà
8
t
2
[0
;t
0
]
j
u

u
0
j
6
t
4

ZZ

j



0
j
d

+
1
4

ZZ

j
'

'
0
j
d

+
+
t
4

ZZ

h



@'
@y
1

@'
0
@y
1



j

1
j
+



@'
@y
2

@'
0
@y
2



j

2
j
+
+



@'
@y
3

@'
0
@y
3



j

3
j
i
d

6
t
+

+
t

3

6

(1+4
t
0
)
";
åñëè

"
1+4
t
0
:
Èçïîñëåäíåéôîðìóëûñëåäóåò,÷òîðåøåíèåçàäà÷èÊîøè
(2.11),(2.24),(2.25)íåïðåðûâíûìîáðàçîìçàâèñèòîòíà÷àëü-
íûõäàííûõíàëþáîìêîíå÷íîìâðåìåííîìèíòåðâàëå.
2.1.3.Çàäà÷àÊîøèäëÿâîëíîâîãîóðàâíåíèÿñäâóìÿ
ïðîñòðàíñòâåííûìèïåðåìåííûìè.Ìåòîäñïóñêà.
ÔîðìóëàÏóàññîíà
Ðåøåíèå
u
(
x
1
;x
2
;t
)
çàäà÷èÊîøèäëÿâîëíîâîãîóðàâíåíèÿ
ñäâóìÿïðîñòðàíñòâåííûìèïåðåìåííûìè
@
2
u
@t
2
=
@
2
u
@x
2
1
+
@
2
u
@x
2
2
;
(2.26)
êîãäàíà÷àëüíûåäàííûå
u
(
x
1
;x
2
;
0)=
'
(
x
1
;x
2
)
;
(2.27)
@u
(
x
1
;x
2
;t
)
@t




t
=0
=

(
x
1
;x
2
)
(2.28)
èìåþòíåïðåðûâíûå÷àñòíûåïðîèçâîäíûåòðåòüåãîèâòîðî-
ãîïîðÿäêàñîîòâåòñòâåííî,ìîæåòáûòüïîëó÷åíîèçôîðìóëû
Êèðõãîôà(2.23)ìåòîäîìñïóñêà.
Ñóùíîñòüýòîãîìåòîäàçàêëþ÷àåòñÿâòîì,÷òîêîãäàâïðà-
âîé÷àñòèôîðìóëû(2.23)ôóíêöèè
'
è

çàâèñÿòòîëüêîîò
85
äâóõïåðåìåííûõ
x
1
;x
2
,òîýòàôîðìóëàäàåòôóíêöèþ
u
(
x
1
;x
2
;t
)=
1
4
t
ZZ
j
y
j
2
=
t
2

(
x
1
+
y
1
;x
2
+
y
2
)
dS
y
+
+
1
4

@
@t
2
6
4
1
t
ZZ
j
y
j
2
=
t
2
'
(
x
1
+
y
1
;x
2
+
y
2
)
dS
y
3
7
5
;
(2.29)
íåçàâèñÿùóþîò
x
3
,èóäîâëåòâîðÿþùóþêàêóðàâíåíèþ(2.26),
òàêèóñëîâèÿì(2.27)è(2.28).
Êàêèçâåñòíî,ïðîåêöèÿ
dy
1
dy
2
ýëåìåíòàïëîùàäè
dS
y
ñôå-
ðû
f
y
2
R
3
:
j
y
j
2
=
t
2
g
íàêðóã
f
y
2
R
2
:
j
y
j
2
t
2
g
âûðàæà-
åòñÿ÷åðåç
dS
y
ôîðìóëîé
dy
1
dy
2
=
dS
y
cos(
i
3
^
;

)=
y
3
t
dS
y
;
ãäå
i
3
îðòîñè
x
3


íîðìàëüñôåðû
f
y
2
R
3
:
j
y
j
2
=
t
2
g
âòî÷êå
y
:
Ïîýòîìó,ó÷èòûâàÿòîîáñòîÿòåëüñòâî,÷òîïðèâû÷èñëåíèè
èíòåãðàëîââïðàâîé÷àñòèôîðìóëû(2.23)ñëåäóåòñïðîåêòè-
ðîâàòüíàêðóã
f
y
2
R
2
:
j
y
j
2
t
2
g
êàêâåðõíþþ
y
3

0
;
òàêè
íèæíþþ
y
3

0
ïîëîâèíûñôåðû
f
y
2
R
3
:
j
y
j
2
=
t
2
g
ôîðìóëà
(2.29)çàïèøåòñÿââèäå
u
(
x
1
;x
2
;t
)=
1
2

ZZ
d

(
y
1
;y
2
)
dy
1
dy
2
q
t
2

(
y
1

x
1
)
2

(
y
2

x
2
)
2
+
+
1
2

@
@t
ZZ
d
'
(
y
1
;y
2
)
dy
1
dy
2
q
t
2

(
y
1

x
1
)
2

(
y
2

x
2
)
2
;
(2.30)
ãäå
d
=
f
y
2
R
2
:
j
y

x
j
2
t
2
g
.
Ðàâåíñòâî(2.30)íîñèòíàçâàíèåôîðìóëûÏóàññîíà.Èç
ýòîéôîðìóëûâèäíî,÷òîäëÿîïðåäåëåíèÿâîëíû
u
(
x
1
;x
2
;t
)
âòî÷êå
(
x
1
;x
2
;t
)
íåäîñòàòî÷íîçíàíèÿçíà÷åíèé
'
(
x
1
;x
2
)
è

(
x
1
;x
2
)
íàîêðóæíîñòè
f
y
2
R
2
:
j
y

x
j
2
=
t
2
g
.Âîïðå-
äåëåíèè
u
(
x
1
;x
2
;t
)
âòî÷êå
(
x
1
;x
2
;t
)
ó÷àñòâóþòçíà÷åíèÿíà-
86
÷àëüíûõäàííûõ
'
(
x
1
;x
2
)
è

(
x
1
;x
2
)
âîâñåõòî÷êàõêðóãà
d
.
Àýòîîçíà÷àåò,÷òîâñëó÷àåäâóõïðîñòðàíñòâåííûõïåðåìåí-
íûõ
x
1
;x
2
ââîëíîâûõïðîöåññàõïðèíöèïÃþéãåíñàíåèìååò
ìåñòà.
2.1.4.Àíàëèçðåøåíèÿ(ïîíÿòèåîáëàñòèçàâèñèìîñòè,
îáëàñòèâëèÿíèÿèîáëàñòèîïðåäåëåíèÿ)
Âðàññìîòðåííîéâïðåäûäóùèõïóíêòàõçàäà÷åÊîøèäëÿ
âîëíîâîãîóðàâíåíèÿíîñèòåëåìíà÷àëüíûõäàííûõÿâëÿåòñÿ
âñåïðîñòðàíñòâî
R
n
ïåðåìåííûõ
x
=(
x
1
;:::;x
n
)
.
Ìíîæåñòâîòî÷åêïðîñòðàíñòâà
R
n
,ïîçàäàííûìçíà÷åíè-
ÿìôóíêöèé
'
(
x
)
è

(
x
)
,íàêîòîðîìâïîëíåîïðåäåëÿåòñÿçíà-
÷åíèåðåøåíèÿ
u
(
x
;t
)
âîëíîâîãîóðàâíåíèÿâòî÷êå
(
x
;t
)
ïðî-
ñòðàíñòâà
R
n
+1
,íàçûâàåòñÿîáëàñòüþçàâèñèìîñòèäëÿòî÷êè
(
x
;t
)
.Êîáëàñòèçàâèñèìîñòè,ðàçóìååòñÿ,íåîòíîñÿòñÿòî÷êè,
âêîòîðûõçíà÷åíèÿ
'
(
x
)
è

(
x
)
íåó÷àñòâóþòâîïðåäåëåíèè
u
(
x
;t
)
âòî÷êå
(
x
;t
)
(ñì.Ðèñ.1).
Ðèñ.1.Îáëàñòüçàâèñèìîñòè
(
n
=1)
.
Âçàâèñèìîñòèîòòîãî
n
=2
èëè
n
=1
,îáëàñòüþçàâèñèìî-
ñòèäëÿòî÷êè
(
x
;t
)
ÿâëÿåòñÿêðóã
f
y
2
R
2
:
j
x

y
j
2
6
t
2
g
èëè
87
îòðåçîê
f
y
2
R
:
j
x

y
j
6
t
g
,àïðè
n
=3
îáëàñòüçàâèñèìîñòè
îïðåäåëÿåòñÿïîïðèíöèïóÃþéãåíñà.
Ïóñòüòåïåðüíîñèòåëåìíà÷àëüíûõäàííûõÿâëÿåòñÿíåâñå
ïðîñòðàíñòâî
R
n
,àíåêîòîðàÿåãîîáëàñòü
G
,ò.å.
u
(
x
;
0)=
'
(
x
)
;
@u
(
x
;t
)
@t




t
=0
=

(
x
)
;
x
2
G:
(2.31)
Êàêâèäíîèçôîðìóë(2.8),(2.23),(2.30),çíà÷åíèÿ
'
(
x
)
è

(
x
)
íà
G
âëèÿþòíàçíà÷åíèÿ
u
(
x
;t
)
âîâñåõòî÷êàõ
(
x
;t
)
ïðîñòðàíñòâà
R
n
+1
,êîòîðûåîáëàäàþòòåìñâîéñòâîì,÷òîïåðå-
ñå÷åíèåäâóõìíîæåñòâ
G
è

j
y

x
j
2
t
2

íåÿâëÿåòñÿïóñòûì.
Ìíîæåñòâîâñåõòàêèõòî÷åêïðèíÿòîíàçûâàòüîáëàñòüþâëè-
ÿíèÿ(ñì.Ðèñ.2).
Ðèñ.2.Îáëàñòüâëèÿíèÿ
(
n
=1)
.
Ìíîæåñòâîòî÷åê
(
x
;t
)
2
R
n
+1
,âêîòîðûõçíà÷åíèÿ
u
(
x
;t
)
âïîëíåîïðåäåëÿþòñÿïîçàäàííûìçíà÷åíèÿì
'
(
x
)
è

(
x
)
íà
G
,íàçûâàåòñÿîáëàñòüþîïðåäåëåíèÿèëèîáëàñòüþðàñïðîñòðà-
íåíèÿâîëíû
u
(
x
;t
)
ñíà÷àëüíûìèäàííûìèíà
G
(ñì.Ðèñ.3).
88
Ðèñ.3.Îáëàñòüîïðåäåëåíèÿ
(
n
=1)
.
Èçôîðìóë(2.8),(2.23),(2.30)ñëåäóåò,÷òîïðèíà÷àëüíûõ
äàííûõ(2.31)îáëàñòüîïðåäåëåíèÿâîëíû
u
(
x
;t
)
ñîñòàâëÿþò
èñêëþ÷èòåëüíîòåòî÷êè
(
x
;t
)
ïðîñòðàíñòâà
R
n
+1
,êîòîðûå
îáëàäàþòñâîéñòâîì:
1)ïðè
n
=3
ñôåðà
fj
y

x
j
2
=
t
2
g
,ÿâëÿþùàÿñÿïåðå-
ñå÷åíèåìõàðàêòåðèñòè÷åñêîãîêîíóñà
j
y

x
j
2
=(


t
)
2
ñ
âåðøèíîéâòî÷êå
(
x
;t
)
ñãèïåðïëîñêîñòüþ

=0
,ïðèíàäëåæèò
G
;
2)ïðè
n
=2
íåòîëüêîîêðóæíîñòü
fj
y

x
j
2
=
t
2
g
,
ÿâëÿþùàÿñÿïåðåñå÷åíèåìõàðàêòåðèñòè÷åñêîãîêîíóñà
j
y

x
j
2
=(


t
)
2
ñâåðøèíîéâòî÷êå
(
x
;t
)
ñïëîñêîñòüþ

=0
,íîâåñüêðóã
j
y

x
j
2
6
t
2
ïðèíàäëåæèò
G
;
3)ïðè
n
=1
íåòîëüêîòî÷êè
x

t
è
x
+
t
ïåðåñå÷åíèÿ
õàðàêòåðèñòè÷åñêèõïðÿìûõ
y

x
=


t
,
y

x
=
t


(âû-
ðîæäåííîãîõàðàêòåðèñòè÷åñêîãîêîíóñà
j
y

x
j
2
=(


t
)
2
),
ïðîõîäÿùèõ÷åðåçòî÷êó
(
x;t
)
,ñïðÿìîé

=0
,íîèâåñüïðÿ-
ìîëèíåéíûéîòðåçîêìåæäóýòèìèòî÷êàìèïðèíàäëåæèò
G
.
89
2.2.Íåîäíîðîäíîåâîëíîâîåóðàâíåíèå
2.2.1.Ñëó÷àéîäíîéïðîñòðàíñòâåííîéïåðåìåííîé
Ðàññìîòðèìòåïåðüçàäà÷óÊîøèäëÿíåîäíîðîäíîãîóðàâ-
íåíèÿêîëåáàíèé
@
2
u
@t
2
=
a
2
@
2
u
@x
2
+
f
(
x;t
)
;x
2
R
;t�
0
;
(2.32)
u
(
x;
0)=
'
(
x
)
;
@u
(
x;
0)
@t
=

(
x
)
;x
2
R
:
(2.33)
Ëåãêîïðîâåðèòü,÷òîðåøåíèåçàäà÷è(2.32)-(2.33)ïðåäñòà-
âèìîâôîðìå
u
=
v
+
w;
(2.34)
ãäå
v
ðåøåíèåçàäà÷èÊîøè(2.1)-(2.2),à
w
ðåøåíèåñëå-
äóþùåéçàäà÷è:
8





:
@
2
w
@t
2
=
a
2
@
2
w
@x
2
+
f
(
x;t
)
;x
2
R
;t�
0
w
(
x;
0)=0
;
@w
(
x;
0)
@t
=0
;x
2
R
:
(2.35)
Ïóñòü
W
(
x;t
;

)
ðåøåíèåâñïîìîãàòåëüíîéçàäà÷èÊîøè
8





:
@
2
W
@t
2
=
a
2
@
2
W
@x
2
;x
2
R
;t�;
W
(
x;t
;

)
j
t
=

=0
;
@W
(
x;t
;

)
@t
j
t
=

=
f
(
x;
)
:
(2.36)
Ïîêàæåì,÷òîðåøåíèå
w
(
x;t
)
çàäà÷è(2.35)îïðåäåëÿåòñÿôîð-
ìóëîé
w
(
x;t
)=
t
Z
0
W
(
x;t
;

)
d;
(2.37)
ãäå
W
(
x;t
;

)
ðåøåíèåçàäà÷è(2.36).Äåéñòâèòåëüíî
w
(
x;
0)=0
;
@w
(
x;t
)
@t
=
W
(
x;t
;

)+
t
Z
0
@W
(
x;t
;

)
@t
d
90
è,ñëåäîâàòåëüíî,
@w
(
x;
0)
@t
=0
âñèëóíà÷àëüíîãîóñëîâèÿâ
(2.36).È,íàêîíåö,
@
2
w
@t
2

a
2
@
2
w
@x
2
=
@W
(
x;t
;

)
@t



t
=

+
t
Z
0

@
2
W
(
x;t
;

)
@t
2


a
2
@
2
W
(
x;t
;

)
@x
2

d
=
f
(
x;t
)
:
Ðåøåíèåçàäà÷è(2.36)äàåòñÿôîðìóëîéÄàëàìáåðà
W
(
x;t
;

)=
1
2
a
x
+
a
(
t


)
Z
x

a
(
t


)
f
(
;
)
d:
(2.38)
Òåïåðü,èñïîëüçóÿôîðìóëû(2.8),(2.34),(2.37)è(2.38),íà-
õîäèì,÷òîðåøåíèåèñõîäíîéçàäà÷è(2.32)-(2.33)çàäàåòñÿôîð-
ìóëîé
u
(
x;t
)=
'
(
x
+
at
)+
'
(
x

at
)
2
+
1
2
a
x
+
at
Z
x

at

(
y
)
dy
+
+
1
2
a
x
+
a
(
t


)
Z
x

a
(
t


)
f
(
;
)
dd:
(2.39)
2.2.2.Ñëó÷àéòðåõïðîñòðàíñòâåííûõïåðåìåííûõ.
Çàïàçäûâàþùèéïîòåíöèàë
Çàíîñèòåëÿíà÷àëüíûõäàííûõâìåñòîïëîñêîñòè
t
=0
ïðè-
ìåìïëîñêîñòü
t
=

1
,ãäå

1
íåêîòîðûéïàðàìåòð,èîáîçíà-
÷èì÷åðåç
v
(
x
1
;x
2
;x
3
;t;
1
)
ðåøåíèåâîëíîâîãîóðàâíåíèÿ(2.11),
91
óäîâëåòâîðÿþùååíà÷àëüíûìóñëîâèÿì
v
(
x
1
;x
2
;x
3
;t;
1
)
j
t
=

1
=0
;
@
@t
v
(
x
1
;x
2
;x
3
;t;
1
)



t
=

1
=
g
(
x
1
;x
2
;x
3
;
1
)
;
(2.40)
ãäå
g
(
x
1
;x
2
;x
3
;
1
)
çàäàííàÿäåéñòâèòåëüíàÿôóíêöèÿ,èìåþ-
ùàÿíåïðåðûâíûå÷àñòíûåïðîèçâîäíûåâòîðîãîïîðÿäêà.
Çàìåíîé
t
÷åðåç
t


èçôîðìóëûÊèðõãîôà(2.23)äëÿ
v
ïîëó÷àåìâûðàæåíèå
v
(
x
1
;x
2
;x
3
;t;
1
)=
1
4

(
t


1
)
ZZ
j
y

x
j
2
=(
t


1
)
2
g
(
y
1
;y
2
;y
3
;
1
)
dS
y
;
'
=0
;
=
g
(
y
;
1
)
:
Ïîêàæåì,÷òîôóíêöèÿ
u
(
x
1
;x
2
;x
3
;t
)=
t
Z
0
v
(
x
1
;x
2
;x
3
;t;
1
)
d
1
(2.41)
ÿâëÿåòñÿðåøåíèåìçàäà÷èÊîøè
u
(
x
1
;x
2
;x
3
;
0)=0
;
@
@t
u
(
x
1
;x
2
;x
3
;t
)



t
=0
=0
(2.42)
äëÿíåîäíîðîäíîãîâîëíîâîãîóðàâíåíèÿ
@
2
u
@t
2
=
u
+
g
(
x
1
;x
2
;x
3
;t
)
;
=
@
2
@x
2
1
+
@
2
@x
2
2
+
@
2
@x
2
3
:
(2.43)
Âñàìîìäåëå,âñèëó(2.40)ñðàçóâèäíî,÷òîôóíêöèÿ
u
(
x
1
;x
2
;x
3
;t
)
óäîâëåòâîðÿåòíà÷àëüíûìóñëîâèÿì(2.42).Äàëåå,
íàîñíîâàíèè(2.40)èç(2.41)ïîëó÷àåì
@
2
u
@t
2
=
g
(
x
1
;x
2
;x
3
;t
)+
t
Z
0
@
2
v
(
x
1
;x
2
;x
3
;t;
1
)
@t
2
d
1
:
(2.44)
92
Èç(2.41)è(2.44)èìååì

@
2
u
@t
2
+
u
=

g
(
x
1
;x
2
;x
3
;t
)+
t
Z
0



@
2
@t
2

v
(
x
1
;x
2
;x
3
;t;
1
)
d
1
=

g
(
x
1
;x
2
;x
3
;t
)
;
÷òîèäîêàçûâàåòñïðàâåäëèâîñòüíàøåãîóòâåðæäåíèÿ.
Âðåçóëüòàòåçàìåíûïåðåìåííîãî
t


1
=

ôîðìóëà(2.41)
çàïèøåòñÿââèäå
u
(
x
;t
)=
t
Z
0
1
4

(
t


1
)
ZZ
j
y

x
j
2
=(
t


1
)
2
g
(
y
1
;
y
2
;y
3
;

1
)
dS
y
d
1
=
=
t
Z
0
1
4

ZZ
j
y

x
j
2
=

2
g
(
y
1
;
y
2
;y
3
;
t


)
dS
y
d
=
=
1
4

t
Z
0
ZZ
j
y

x
j
2
=

2
g
(
y
1
;
y
2
;y
3
;
t


)
dS
y

d
=
(2.45)
=
1
4

ZZZ
j
y

x
j
2

2
g
(
y
1
;
y
2
;y
3
;
t
�j
y

x
j
)
j
y

x
j
d
y
:
Îïðåäåë¼ííàÿïîôîðìóëå(2.45)ôóíêöèÿ
u
(
x
;t
)
,äàþùàÿ
ðåøåíèåçàäà÷è(2.42)-(2.43),ñîâïàäàåòcïîòåíöèàëîìîáúåì-
íûõìàññ,ðàñïðåäåë¼ííûõïîøàðó
f
y
2
R
3
:
j
y

x
j
2
t
2
g
ñ
ïëîòíîñòüþ
g
(
y
1
;y
2
;y
3
;
t
�j
y

x
j
)
.Ââèäóòîãî,÷òîôóíêöèÿ
g
ó÷àñòâóåòâôîðìóëå(2.45)äëÿçíà÷åíèÿâðåìåíè
t
�j
y

x
j
,
îòñòàþùåãîîòìîìåíòàíàáëþäåíèÿçàâîëíîé,ýòîòïîòåíöèàë
íàçûâàåòñÿçàïàçäûâàþùèì.
93
2.2.3.Ñëó÷àéäâóõïðîñòðàíñòâåííûõïåðåìåííûõ
Ïðèâåä¼ííàÿâûøåïðîöåäóðàïîñòðîåíèÿðåøåíèÿçàäà÷è
Êîøèäëÿóðàâíåíèÿ(2.43)ïðèìåíèìàèâñëó÷àåäâóõïðî-
ñòðàíñòâåííûõïåðåìåííûõ.Òàêêàêôóíêöèÿ
v
(
x
1
;x
2
;;t
)=
1
2

ZZ
d
g
(
y
1
;y
2
;
)
dy
1
dy
2
p
(
t


)
2

(
y
1

x
1
)
2

(
y
2

x
2
)
2
âñèëó(2.30)ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ(2.26),óäîâëåòâî-
ðÿþùèìóñëîâèÿì
v
(
x
1
;x
2
;t;
)=0
;
@
@t
v
(
x
1
;x
2
;t;
)



t
=

=
g
(
x
1
;x
2
;
)
;
òîâûðàæåíèå
u
(
x
1
;x
2
;t
)=
1
2

t
Z
0
ZZ
d
g
(
y
1
;y
2
;
)
dy
1
dy
2
p
(
t


)
2

(
y
1

x
1
)
2

(
y
2

x
2
)
2
d;
(2.46)
ãäå
d
=
f
y
2
R
2
:
j
y

x
j

(
t


)
2
g
,ïðåäñòàâëÿåòñîáîé
ðåøåíèåçàäà÷èÊîøè
u
(
x
1
;x
2
;
0)=0
;
@
@t
u
(
x
1
;x
2
;t
)



t
=0
=0
äëÿíåîäíîðîäíîãîóðàâíåíèÿ
@
2
u
@t
2
=
@
2
u
@x
2
1
+
@
2
u
@x
2
2
+
g
(
x
1
;
x
2
;t
)
:
(2.47)
Âóðàâíåíèè(2.47)è,ñëåäîâàòåëüíî,âôîðìóëå(2.46)ïðåäïî-
ëàãàåòñÿ,÷òîôóíêöèÿ
g
(
x
1
;x
2
;t
)
èìååòíåïðåðûâíûå÷àñòíûå
ïðîèçâîäíûåâòîðîãîïîðÿäêà.
2.3.Êîððåêòíîïîñòàâëåííûåçàäà÷èäëÿãèïåðáîëè÷åñêèõ
óðàâíåíèé
2.3.1.Åäèíñòâåííîñòüðåøåíèÿçàäà÷èÊîøè
Äîêàæåì,÷òîçàäà÷àÊîøèâïðèâåä¼ííîéâûøåïîñòàíîâ-
êåäëÿâîëíîâîãîóðàâíåíèÿ(êàêîäíîðîäíîãî,òàêèíåîäíî-
94
ðîäíîãî)íåìîæåòèìåòüáîëååîäíîãîðåøåíèÿ.Äëÿïðîñòîòû
îãðàíè÷èìñÿñëó÷àåìîäíîãîïðîñòðàíñòâåííîãîèçìåðåíèÿ
x
.
Åñëè
u
1
(
x;t
)
è
u
2
(
x;t
)
ÿâëÿþòñÿðåøåíèÿìèçàäà÷èÊîøèäëÿ
óðàâíåíèÿ
@
2
u
@t
2
=
@
2
u
@x
2
+
g
(
x;t
)
;
(2.48)
òîèõðàçíîñòü
u
(
x;t
)=
u
1
(
x;t
)

u
2
(
x;t
)
áóäåòðåøåíèåìóðàâ-
íåíèÿêîëåáàíèéñòðóíû
@
2
u
@t
2
=
@
2
u
@x
2
(2.49)
óäîâëåòâîðÿþùèìíà÷àëüíûìóñëîâèÿì
u
(
x;
0)=0
;
@
@t
u
(
x;t
)



t
=0
=0
:
(2.50)
Èòàê,íàìñëåäóåòïîêàçàòü,÷òîîäíîðîäíîåóðàâíåíèå
(2.49)íåìîæåòèìåòüîòëè÷íîãîîòíóëÿðåøåíèÿ,óäîâëåòâî-
ðÿþùåãîîäíîðîäíûìíà÷àëüíûìóñëîâèÿì(2.50).Èíòåãðèðóÿ
òîæäåñòâî

2
@u
@t

@
2
u
@x
2

@
2
u
@t
2

=

2
@
@x

@u
@x
@u
@t

+
+
@
@t

@u
@x

2
+
@
@t

@u
@t

2
=0
ïîòðåóãîëüíîéîáëàñòè

ñâåðøèíàìèâòî÷êàõ
A
(
x

t;
0)
,
B
(
x
+
t;
0)
è
C
(
x;t
)
èèñïîëüçóÿôîðìóëóÃàóññà-
Îñòðîãðàäñêîãî,ïîëó÷àåì
Z

h

2
@
@

@u
@
@u
@

+
@
@

@u
@

2
+
@
@

@u
@

2
i
dd
=
=
Z
AB
+
BC
+
CA

2
@u
@
@u
@
d

h
@u
@

2
+

@u
@

2
i
d
=0
:
(2.51)
95
Âäîëü
AB
âñèëó(2.50)èìåþòìåñòîðàâåíñòâà
@u
@
=0
è
@u
@
=0
.Êðîìåòîãî,ò.ê.óðàâíåíèÿïðÿìîëèíåéíûõîòðåçêîâ
BC
,
CA
èìåþòâèä

=


+
x
+
t
,

=

+
x

t
,òîâäîëüýòèõ
îòðåçêîâèìååìñîîòâåòñòâåííî
d
=

d
,
d
=
d
.Ïîýòîìó
ðàâåíñòâî(2.43)ìîæíîïåðåïèñàòüââèäå
Z
BC

@u
@

@u
@

2
d

Z
CA

@u
@
+
@u
@

2
d
=0
èëè
t
Z
0

@u
@

@u
@

2
d
+
t
Z
0

@u
@
+
@u
@

2
d
=0
;
îòêóäàñëåäóåò,÷òî
@u
@

@u
@
=0
íà
BC
è
@u
@
+
@u
@
=0
íà
AC
.Ñëåäîâàòåëüíîââåðøèíå
C
(
x;t
)
òðåóãîëüíèêà

èìå-
þòìåñòîðàâåíñòâà
@u
@x

@u
@t
=0
,
@u
@x
+
@u
@t
=0
,ò.å.
@u
@x
=0
,
@u
@t
=0
.Ò.ê.òî÷êà
C
(
x;t
)
âûáðàíàïðîèçâîëüíî,òîðàâåíñòâà
@u
@x
=0
,
@u
@t
=0
èìåþòìåñòîâñþäóíàïëîñêîñòèïåðåìåííûõ
x;t
.Ýòîîçíà÷àåò,÷òî
u
(
x;t
)=const
.Íîâñèëó(2.50)ôóíêöèÿ
u
(
x;
0)=0
,îòêóäàñëåäóåò,÷òî
u
(
x;t
)=0
âñþäó.
2.3.2.Îáùàÿïîñòàíîâêàçàäà÷èÊîøè
Äîñèõïîðìûñ÷èòàåì,÷òîíîñèòåëåìíà÷àëüíûõäàí-
íûõÿâëÿåòñÿïëîñêîñòü
t
=0
ïðîñòðàíñòâà
R
n
+1
ïåðåìåííûõ
x
1
;::;x
n
;t
.Ñåé÷àñ,íàïðèìåðåóðàâíåíèÿ
@
2
u
@t
2
=
@
2
u
@x
2
(2.52)
ïîêàæåì,êàêèìóñëîâèÿìäîëæåíóäîâëåòâîðÿòüíîñèòåëü
L
íà÷àëüíûõäàííûõ,îòëè÷íûéîò
t
=0
,èêàêîéâèääîëæíû
96
èìåòüñàìèíà÷àëüíûåäàííûåäëÿòîãî,÷òîáûïîëó÷åííàÿâ
èòîãåçàäà÷àáûëàïîñòàâëåíàêîððåêòíî.
Îáîçíà÷èì÷åðåç
D
îáëàñòüïëîñêîñòèïåðåìåííûõ
x
,
t
ñ
êóñî÷íî-ãëàäêîéãðàíèöåé
S
.Ïóñòü
u
(
x;t
)
ðåãóëÿðíîåâîá-
ëàñòè
D
ðåøåíèåóðàâíåíèÿ(2.52),èìåþùååíåïðåðûâíûå÷àñò-
íûåïðîèçâîäíûåâ
D
[
S
.
Èíòåãðèðóÿòîæäåñòâî
@
@x

@u
@x


@
@t

@u
@t

=0
(2.53)
ïîîáëàñòè
D
èèñïîëüçóÿôîðìóëóÃàóññà-Îñòðîãðàäñêîãî,ïî-
ëó÷àåì
Z
D

@
@x

@u
@x


@
@t

@u
@t

dxdt
=
Z
S
@u
@x
dt
+
@u
@t
dx
=0
:
(2.54)
Ïóñòü
L
ðàçîìêíóòàÿêðèâàÿñíåïðåðûâíîéêðèâèçíîé,
óäîâëåòâîðÿþùàÿäâóìòðåáîâàíèÿì:à)êàæäàÿïðÿìàÿèçäâóõ
ñåìåéñòâ
x
+
t
=const
,
x

t
=const
õàðàêòåðèñòèêóðàâíåíèÿ
(2.52)ïåðåñåêàåòñÿñêðèâîé
L
íåáîëåå÷åìâîäíîéå¼òî÷êåè
á)íàïðàâëåíèåêàñàòåëüíîéêêðèâîéíèâîäíîéòî÷êåíåñîâ-
ïàäàåòñõàðàêòåðèñòè÷åñêèìíàïðàâëåíèåì,ñîîòâåòñòâóþùèì
óðàâíåíèþ(2.52).
97
Ïðåäïîëîæèì,÷òîõàðàêòåðèñòèêè
x

x
1
=
t

t
1
,
x

x
1
=
t
1

t
,âûõîäÿùèåèçòî÷êè
C
(
x
1
;t
1
)
ïåðåñåêàþòñÿñ
êðèâîé
L
âòî÷êàõ
A
è
B
.Ïðèìåíÿÿôîðìóëó(2.54)âîáëàñòè,
îãðàíè÷åííîéäóãîé
AB
èêðèâîé
L
èîòðåçêàìèõàðàêòåðèñòèê
CA
è
CB
,ïîëó÷àåì
Z
AB
+
BC
+
CA
@u
@x
dt
+
@u
@t
dx
=0
:
(2.55)
Òàêêàêâäîëü
CA
è
BC
èìååì
dx
=
dt
,
dx
=

dt
ñîîòâåò-
ñòâåííî,òî(2.55)ìîæíîçàïèñàòüââèäå
Z
AB
@u
@x
dt
+
@u
@t
dx

2
u
(
C
)+
u
(
A
)+
u
(
B
)=0
;
îòêóäàíàõîäèì
u
(
C
)=
1
2
u
(
A
)+
1
2
u
(
B
)+
1
2
Z
AB
@u
@x
dt
+
@u
@t
dx:
(2.56)
Åñëèðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(2.52)óäîâëåòâîðÿåòóñëî-
âèÿì
u
j
L
=
';
@u
@
l




L
=
;
(2.57)
ãäå
'
è

çàäàííûåäåéñòâèòåëüíûåñîîòâåòñòâåííîäâàæäû
èîäèíðàçíåïðåðûâíîäèôôåðåíöèðóåìûåôóíêöèè,à
l
çà-
äàííûéíà
L
äîñòàòî÷íîãëàäêèéâåêòîð,íèãäåíåñîâïàäàþùèé
ñêàñàòåëüíîéêêðèâîé
L
,òî,îïðåäåëÿÿ
@u
@x
,
@u
@t
èçðàâåíñòâ
@u
@x
@x
@s
+
@u
@t
@t
@s
=
d'
ds
,
@u
@x
@x
@l
+
@u
@t
@t
@s
=

,ãäå
S
äëèíàäóãè
L
,èïîäñòàâëÿÿèçâåñòíûåçíà÷åíèÿ
u
,
@u
@x
,
@u
@t
âïðàâóþ÷àñòü
(2.50),ïîëó÷èìðåãóëÿðíîåðåøåíèåóðàâíåíèÿ(2.52),óäîâëå-
òâîðÿþùååóñëîâèÿì(2.57).
98
Çàäà÷àîòûñêàíèÿðåãóëÿðíîãîðåøåíèÿóðàâíåíèÿ(2.52),
óäîâëåòâîðÿþùåãîóñëîâèÿì(2.57),òàêæåíàçûâàåòñÿçàäà÷åé
Êîøè.Èçïðèâåä¼ííîãîâûøåðàññóæäåíèÿñëåäóåò,÷òîçàäà÷à
Êîøèâòîëüêî÷òîóêàçàííîéïîñòàíîâêåèìååòåäèíñòâåííîå
óñòîé÷èâîåðåøåíèå.
2.3.3.Çàäà÷àÃóðñà(õàðàêòåðèñòè÷åñêàÿçàäà÷à)
Ïóñòüòåïåðü
L
ïðåäñòàâëÿåòñîáîéñîâîêóïíîñòüîòðåçêîâ
OA
è
OB
õàðàêòåðèñòèê
x

t
=0
,
x
+
t
=0
ñîîòâåòñòâåí-
íî.Õàðàêòåðèñòèêè
x

x
1
=
t

t
1

x

x
1
=
t
1

t
,âûõî-
äÿùèåèçòî÷êè
C
(
x
1
;t
1
)
ïåðåñåêàþòñÿñ
OA
è
OB
âòî÷êàõ
A
1

x
1
+
t
1
2
;
x
1
+
t
1
2

è
B
1

x
1

t
1
2
;

x
1

t
1
2

ñîîòâåòñòâåííî.
Ðèñ.5.
Ïðèìåíÿÿôîðìóëó(2.54),âñëó÷àåõàðàêòåðèñòè÷åñêîãî
ïðÿìîóãîëüíèêà
OA
,
CB
1
,ïîëó÷àåì
Z
OA
1
+
A
1
C
+
CB
1
+
B
1
O
@u
@x
dt
+
@u
@t
dx
=0
èëè
Z
OA
1
@u
@x
dx
+
@u
@t
dt

Z
A
1
C
@u
@x
dx
+
@u
@t
dt
+
Z
CB
1
@u
@x
dx
+
@u
@t
dt

99

Z
B
1
O
@u
@x
dx
+
@u
@t
dt
=2
u
(
A
1
)

2
u
(
O
)

2
u
(
C
)+2
u
(
B
1
)=0
;
îòêóäàèìååì
u
(
C
)=
u
(
A
1
)+
u
(
B
1
)

u
(
O
)
:
(2.58)
Åñëèèçâåñòíî,÷òî
u
j
OA
=
'
(
x
1
)
;u
j
OB
=

(
x
1
)
;'
(0)=

(0)
;
(2.59)
òîèç(2.58)ïîëó÷àåì
u
(
x;t
)=
'

x
+
t
2

+


x

t
2


'
(0)
:
(2.60)
Èçýòîéôîðìóëûñëåäóåò,÷òîçíà÷åíèÿ
u
è
@u
@
l
âäîëüõà-
ðàêòåðèñòèêíåçàâèñèìîäðóãîòäðóãàçàäàâàòüíåëüçÿ.
Çàäà÷àîïðåäåëåíèÿðåãóëÿðíîãîðåøåíèÿóðàâíåíèÿ(2.52)
ïîóñëîâèÿì(2.59)íàçûâàåòñÿçàäà÷åéÃóðñà.Åäèíñòâåííîå
óñòîé÷èâîåðåøåíèåýòîéçàäà÷èäà¼òñÿôîðìóëîé(2.60).Ïî-
ñêîëüêóâçàäà÷åÃóðñàíîñèòåëÿìèäàííûõÿâëÿþòñÿõàðàêòå-
ðèñòèêèóðàâíåíèÿ(2.52),ýòàçàäà÷àíàçûâàåòñÿåù¼õàðàêòå-
ðèñòè÷åñêîéçàäà÷åé.
2.4.Âîïðîñûèçàäà÷è
1.Íàéäèòåðåøåíèåçàäà÷èÊîøèäëÿîäíîðîäíîãîóðàâíåíèÿ
êîëåáàíèéñòðóíû
8



:
u
tt
=
u
xx
;
u
(
x;
0)=0
;
u
t
(
x;
0)=
xe

x
2
:
2.ÏîëüçóÿñüôîðìóëîéÄàëàìáåðàäëÿðåøåíèÿ
u
(
x;t
)
çàäà÷è
Êîøè
8

:
u
tt
=
a
2
u
xx
;
u
(
x;
0)=
'
(
x
)
;u
t
(
x;
0)=

(
x
)
;
100
ïðîâåðüòå,÷òîâñëó÷àåíå÷åòíîñòèîáåèõôóíêöèé
'
(
x
)
è

(
x
)
u
(
x;t
)
j
x
=0
=0
,àâñëó÷àåèõ÷åòíîñòè
u
x
(
x;t
)
j
x
=0
=0
:
3.Íàéäèòåðåøåíèåçàäà÷èÊîøèäëÿíåîäíîðîäíîãîóðàâíåíèÿ
êîëåáàíèéñòðóíû
u
tt
=
u
xx
+
x
(
x

1)
ïðèíóëåâûõíà÷àëüíûõóñëîâèÿõ.
4.Ìîæåòëèîïèñûâàòüôóíêöèÿ
u
(
x
1
;x
2
;x
3
;t
)=
x
2
1
+
x
2
2
+
x
2
3

x
1
t
2
ïðîöåññðàñïðîñòðàíåíèÿâîëíû.
5.Ñôîðìóëèðóéòåîñíîâíóþèäåþìåòîäàñïóñêà.
6.×åìîáúÿñíÿåòñÿòîòôàêò,÷òîïðèíöèïÃþéãåíñàíåèìååò
ìåñòàâïëîñêîìñëó÷àå?
7.×òîòàêîåîáëàñòüâëèÿíèÿ,çàâèñèìîñòè,îïðåäåëåíèÿðåøå-
íèÿçàäà÷èÊîøèäëÿóðàâíåíèÿêîëåáàíèéñòðóíû?
8.Óêàæèòåîáëàñòüçàâèñèìîñòèòî÷êè
A
(2;3)
äëÿðåøåíèÿçà-
äà÷èÊîøè
(
u
tt
=
u
xx
;
u
j
t
=0
=

1
;u
t
j
t
=0
=3
:
9.×òîòàêîåçàïàçäûâàþùèéïîòåíöèàë?
101
10.Êîððåêòíîëèïîñòàâëåíàçàäà÷àîáîòûñêàíèèâïåðâîé÷åò-
âåðòèïëîñêîñòè
x;t
ðåøåíèÿ
u
(
x;t
)
óðàâíåíèÿ
u
tt
=
u
xx
;
åñëè
u
(
x;
0)=
'
(
x
)
;
0
6
x
1
;
u
(0
;t
)=

(
t
)
;
0
6
t
1
;
'
(0)=

(0)
;'
00
(0)=

00
(0)?
102
3.Óðàâíåíèÿïàðàáîëè÷åñêîãîòèïà
3.1.Ïðèíöèïìàêñèìóìà
Ïðîñòåéøèìïðèìåðîìóðàâíåíèéïàðàáîëè÷åñêîãîòèïà
ÿâëÿåòñÿóðàâíåíèåòåïëîïðîâîäíîñòè
@u
@t
=
@
2
u
@x
2
:
(3.1)
Ïîñêîëüêóäèôôåðåíöèàëüíîåóðàâíåíèåõàðàêòåðèñòèê,
ñîîòâåòñòâóþùèõóðàâíåíèþ(3.1),èìååòâèä
dt
2
=0
,òî
ýòîóðàâíåíèåèìååòåäèíñòâåííîåñåìåéñòâîõàðàêòåðèñòèê
t
=const
,ïðåäñòàâëÿþùèõñîáîéïðÿìûå,ïàðàëëåëüíûåîñè
x
.
Ðàññìîòðèìîáëàñòü
D
ïëîñêîñòèïåðåìåííûõ
x;t
,îãðà-
íè÷åííóþîòðåçêàìè
OA
è
BN
ïðÿìûõ
t
=0
;t
=
T
,
ãäå
T
ïîëîæèòåëüíîå÷èñëî,èêðèâûìè
OB
è
AN
,êàæ-
äàÿèçêîòîðûõïåðåñåêàåòñÿñïðÿìûìè
t
=const
âîäíîé
òî÷êå,ïðè÷åì,åñëèóðàâíåíèÿýòèõêðèâûõçàäàíûñîîòâåò-
ñòâåííîââèäå
x
=

(
t
)
;x
=

(
t
)
,òîïðåäïîëàãàåòñÿ,÷òî

(
t
)

(
t
)
;
0
6
t
6
T:
Ðèñ.6.
Îáîçíà÷èì÷åðåç
S
÷àñòüãðàíèöûîáëàñòè
D
,ñîñòîÿùóþ
103
èç
OA;OB
è
AN
,ïðè÷åìñ÷èòàåòñÿ,÷òî
B
2
S;N
2
S
(ñì.
Ðèñ.6).
Ôóíêöèþ
u
(
x;t
)
,èìåþùóþíàìíîæåñòâå
D
[
BN
íåïðå-
ðûâíûå÷àñòíûåïðîèçâîäíûå
@
2
u
@x
2
è
@u
@t
èóäîâëåòâîðÿþùóþ
óðàâíåíèþ(3.1)âîáëàñòè
D
,áóäåìíàçûâàòüðåãóëÿðíûìðå-
øåíèåìýòîãîóðàâíåíèÿ.
Òåîðåìà31.
Ðåãóëÿðíîåðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(3.1),
íåïðåðûâíîåâ
D
[
S
[
BN
,ñâîåãîýêñòðåìóìàäîñòèãàåòíà
S
.
Äîêàçàòåëüñòâî.
Îãðàíè÷èìñÿðàññìîòðåíèåìñëó÷àÿ
ìàêñèìóìà.Îáîçíà÷èì÷åðåç
M
ìàêñèìóì
u
(
x;t
)
íàçàìêíó-
òîììíîæåñòâå
D
[
S
[
BN
.Äîïóñòèì,÷òî
u
(
x;t
)
äîñòèãàåò
ìàêñèìóìà
M
íåíà
S
,àâíåêîòîðîéòî÷êå
(
x
0
;t
0
)
2
D
[
BN
.
Ïîêàæåì,÷òîýòîäîïóùåíèåïðèâîäèòêïðîòèâîðå÷èþ.
Âñàìîìäåëå,ââåäåìâðàññìîòðåíèåôóíêöèþ
v
(
x;t
)=
u
(
x;t
)+
a
(
T

t
)
;
(3.2)
ãäå
a
ïîëîæèòåëüíàÿïîñòîÿííàÿ.Ââèäóòîãî,÷òî
0
6
t
6
T
,
èç(3.2)èìååì
u
(
x;t
)
6
v
(
x;t
)
6
u
(
x;t
)+
aT
(3.3)
âñþäóâ
D
[
S
[
BN
.
Ïóñòü
M
S
u
;M
S
v
ìàêñèìóìûñîîòâåòñòâåííî
u
(
x;t
)
è
v
(
x;t
)
íà
S
.Ïîäîïóùåíèþ
M
S
u
M
.×èñëî
a
ïîäáåðåìòàê,÷òîáû
èìåëîìåñòîíåðàâåíñòâî
a
M

M
S
u
T
:
(3.4)
Íàîñíîâàíèè(3.3)è(3.4)ïîëó÷àåì
M
S
v
6
M
S
u
+
aTM
S
u
+
M

M
S
u
T
T
=
M
=
u
(
x
0
;t
0
)
6
v
(
x
0
;t
0
)
:
104
Îòñþäàñëåäóåò,÷òîôóíêöèÿ
v
(
x;t
)
íåìîæåòäîñòèãàòüìàê-
ñèìóìàíà
S
.Ñëåäîâàòåëüíî,ýòàôóíêöèÿñâîåãîìàêñèìóìà
íà
D
[
S
[
BN
äîñòèãàåòâíåêîòîðîéòî÷êå
(
x
1
;t
1
)
2
D
[
BN
.
Ñíà÷àëàïðåäïîëîæèì,÷òî
(
x
1
;t
1
)
2
D
.Òàêêàê
(
x
1
;t
1
)
ÿâëÿåòñÿòî÷êîéìàêñèìóìàôóíêöèè
v
(
x;t
)
íà
D
[
S
[
BN
,òî
âýòîéòî÷êå
@v
@t
=0
;
@
2
v
@x
2
6
0
,ò.å.
@v
@t

@
2
v
@x
2

0
:
(3.5)
Ïóñòüòåïåðü
(
x
1
;t
1
)
2
BN
.Ââèäóòîãî,÷òî
v
(
x;t
)
äîñòèãà-
åòâòî÷êå
(
x
1
;t
1
)
ñâîåãîìàêñèìóìàíàìíîæåñòâå
D
[
S
[
BN
,
âýòîéòî÷êå
@v
@t

0
.Ó÷èòûâàÿòîîáñòîÿòåëüñòâî,÷òî
(
x
1
;T
)
ÿâëÿåòñÿòî÷êîéìàêñèìóìà
v
(
x;T
)
êàêôóíêöèè
x
,ìûäîëæíû
èìåòü
@
2
v
(
x
1
;T
)
@x
2
1
6
0
.Ñëåäîâàòåëüíî,íåðàâåíñòâî(3.5)èìååò
ìåñòîâòî÷êå
(
x
1
;T
)
.
Ïîäñòàâëÿÿâëåâóþ÷àñòü(3.5)çíà÷åíèÿ
@v
@t
è
@
2
v
@x
2
,íàéäåí-
íûåèçðàâåíñòâà(3.2),ïîëó÷àåì
@u
@t
1

a

@
2
u
@x
2

0
;x
=
x
1
;t
=
t
1
;
èëè,ò.ê.
u
(
x;t
)
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ(3.1),

a

0
,a
ýòîíåâîçìîæíî,èáî
a�
0
.
Ïîëó÷åííîåïðîòèâîðå÷èåäîêàçûâàåòñïðàâåäëèâîñòüïåð-
âîé÷àñòèòåîðåìû.Àíàëîãè÷íîäîêàçûâàåòñÿèâòîðàÿåå
÷àñòü.
3.2.Ïåðâàÿêðàåâàÿçàäà÷àäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè
Äîêàçàííàÿòåîðåìàïîçâîëÿåòóñòàíîâèòüåäèíñòâåííîñòü
èóñòîé÷èâîñòüðåøåíèÿñëåäóþùåéòàêíàçûâàåìîéïåðâîé
105
êðàåâîéçàäà÷èäëÿóðàâíåíèÿòåïëîïðîâîäíîñòè:èùåòñÿðå-
ãóëÿðíîåâîáëàñòè
D
ðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(3.1),íåïðå-
ðûâíîåâ
D
[
S
[
BN
èóäîâëåòâîðÿþùååóñëîâèÿì
u
j
OB
=

1
(
t
)
;u
j
AN
=

2
(
t
)
;u
j
OA
=
'
(
x
)
;

1
(0)=
'
(0)
;
2
(
A
)=
'
(
A
)
;
(3.6)
ãäå

1
;
2
è
'
çàäàííûåäåéñòâèòåëüíûåíåïðåðûâíûåôóíê-
öèè.
Âñàìîìäåëå,åñëè
u
1
(
x;t
)
è
u
2
(
x;t
)
ðåãóëÿðíûåðåøåíèÿ
óðàâíåíèÿ(3.1),óäîâëåòâîðÿþùèåêðàåâûìóñëîâèÿì(3.6),òî
ôóíêöèÿ
u
(
x;t
)=
u
1
(
x;t
)

u
2
(
x;t
)
áóäåòðåãóëÿðíûìðåøåíèåì
óðàâíåíèÿ(3.1),îáðàùàþùèìñÿâíóëüíà
S
.Ñëåäîâàòåëüíî,â
ñèëóòåîðåìû3.1,
u
(
x;t
)=0
â
D
[
S
[
BN
,îòêóäàèñëåäóåò
åäèíñòâåííîñòüðåøåíèÿïåðâîéêðàåâîéçàäà÷è(3.1),(3.6).
Åñëèðàçíîñòüìåæäóêðàåâûìèçíà÷åíèÿìèíà
S
ðåãóëÿð-
íûõðåøåíèé
u
1
(
x;t
)
è
u
2
(
x;t
)
óðàâíåíèÿ(3.1)ïîìîäóëþìåíü-
øå
"�
0
,òîâñèëóòåîðåìû3.1,
j
u
1
(
x;t
)

u
2
(
x;t
)
j
"
âñþäó
â
D
[
S
[
BN
,èòåìñàìûìíåïðåðûâíàÿçàâèñèìîñòüðåøåíèÿ
ïåðâîéêðàåâîéçàäà÷èîòêðàåâûõäàííûõíà
S
(ò.å.óñòîé÷è-
âîñòüðåøåíèÿýòîéçàäà÷è)äîêàçàíà.
Òåïåðüìûäîêàæåìñóùåñòâîâàíèåðåøåíèÿïåðâîéêðà-
åâîéçàäà÷èäëÿóðàâíåíèÿ(3.1)âïðåäïîëîæåíèÿõ,÷òî
OB
è
AN
ïðÿìîëèíåéíûåîòðåçêè,ñîåäèíÿþùèåòî÷êè
O
(0
;
0)
;B
(0
;T
)
è
A
(
l;
0)
;N
(
l;T
)
ñîîòâåòñòâåííî,ïðè÷åì
u
(0
;t
)=0
;u
(
l;t
)=0
;
0
6
t
6
T;
(3.7)
u
(
x;
0)=
'
(
x
)
;
0
6
x
6
l;
(3.8)
ãäå
'
(
x
)
íåïðåðûâíîäèôôåðåíöèðóåìàÿíàîòðåçêå
0
6
x
6
l
ôóíêöèÿ,îáðàùàþùàÿñÿâíóëüïðè
x
=0
;x
=
l:
Êàêèçâåñòíîèçêóðñàìàòåìàòè÷åñêîãîàíàëèçà,ôóíêöèþ
'
(
x
)
íàîòðåçêå
0
6
x
6
l
ìîæíîðàçëîæèòüâàáñîëþòíîè
106
ðàâíîìåðíîñõîäÿùèéñÿðÿäÔóðüå
'
(
x
)=
1
X
k
=1
a
k
sin
k
l
x;
(3.9)
ãäå
a
k
=
2
l
l
Z
0
'
(
x
)sin
k
l
xdx;k
=1
;
2
;:::
Íåïîñðåäñòâåííûìâû÷èñëåíèåììîæíîóáåäèòüñÿ,÷òî
ôóíêöèÿ
u
(
x
)=
X
k

0
x
k
n
k
!

k

(
x
1
;:::;x
n

1
)
(3.10)
äëÿëþáîéçàäàííîéáåñêîíå÷íîäèôôåðåíöèðóåìîéäåéñòâè-
òåëüíîéôóíêöèè

ïåðåìåííûõ
x
1
;:::;x
n

1
ïðèðàâíîìåðíîé
ñõîäèìîñòèðÿäàâïðàâîé÷àñòè(3.10)èðÿäîâ,ïîëó÷åííûõèç
íåãîäèôôåðåíöèðîâàíèåìïî÷ëåííîîäèíðàçïî
x
n
èäâàæäû
ïî
x
i
;i
=1
;:::;n

1
,óäîâëåòâîðÿåòóðàâíåíèþ
@u
@x
n
=
n

1
X
i
=1
@
2
u
@x
2
i
:
Ïîëüçóÿñüôîðìóëîé(3.10)ïðè
n
=2
;x
1
=
x;x
2
=
t;

k
(
x
)=
sin
k
l
x
,ïîëó÷àåìðåãóëÿðíîåðåøåíèåóðàâíåíèÿ(3.1):
u
k
(
x;t
)=
e


2
k
2
t
l
2
sin
k
l
x;
(3.11)
óäîâëåòâîðÿþùååêðàåâûìóñëîâèÿì
u
k
(0
;t
)=
u
(
l;t
)=0
;
u
k
(
x;
0)=
sin
k
l
x
.
Î÷åâèäíî,÷òîôóíêöèÿ
u
(
x;t
)
,ïðåäñòàâëÿþùàÿñóììóðÿ-
äà
u
(
x;t
)=
1
X
k
=1
a
k
e


2
k
2
t
l
2
sin
k
l
x;
(3.12)
107
áóäåòèñêîìûìðåøåíèåìêðàåâîéçàäà÷è(3.1),(3.7),(3.8).
Ïðè
t�
0
àáñîëþòíàÿèðàâíîìåðíàÿñõîäèìîñòüðÿäà
(3.12)èðÿäîâ,ïîëó÷åííûõèçíåãîäèôôåðåíöèðîâàíèåìïî
x
è
t
ñêîëüóãîäíîðàçâîêðåñòíîñòèòî÷êè
(
x;t
)
,ñëåäóåòèç
òîãî,÷òî
lim
k
!1

k
l

m
e


2
k
2
l
2
t
=0
;m
=0
;
1
;:::
Êîãäàíîñèòåëåìäàííûõ(3.8)ÿâëÿåòñÿîòðåçîêïðÿìîé
t
=
t
0
,
t
0
6
t
6
T
âóñëîâèÿõ(3.7),òîðåøåíèåïåðâîéêðà-
åâîéçàäà÷èâïðÿìîóãîëüíèêå
0
xl;t
0
tT
äàåòñÿ
îïÿòüôîðìóëîé(3.12),âêîòîðîéâìåñòî
t
ñëåäóåòïèñàòü
t

t
0
.
Çàìåòèì,÷òîðÿäâïðàâîé÷àñòèýòîéôîðìóëûïðè
t
0
ìîæåòâîâñåíåèìåòüñìûñëà.Ïîýòîéïðè÷èíåíåðàññìàòðè-
âàåòñÿïåðâàÿêðàåâàÿçàäà÷àäëÿóðàâíåíèÿ(3.1),êîãäà
tt
0
,
ãäå
t
=
t
0
íîñèòåëüäàííûõâêðàåâûõóñëîâèÿõ.
Âñåñêàçàííîåâûøå,î÷åâèäíîîñòàåòñÿâñèëåèòîãäà,êîãäà
÷èñëîïðîñòðàíñòâåííûõïåðåìåííûõáîëüøååäèíèöû,ëèøüñ
òîéðàçíèöåé,÷òîâïîñëåäíåìñëó÷àåâìåñòîïðîñòûõðÿäîâ
(3.9),(3.12)ñëåäóåòáðàòüêðàòíûåðÿäû.
3.3.Ïîñòàíîâêàçàäà÷èÊîøèèäîêàçàòåëüñòâîñóùåñòâîâàíèÿ
ååðåøåíèÿ
Ïóñòü
D
ïðåäñòàâëÿåòñîáîéáåñêîíå÷íóþïîëîñó
�1
x
1
;
0
6
t
6
T
,ãäå
T
ôèêñèðîâàííîåïîëî-
æèòåëüíîå÷èñëî,ïðè÷åìñëó÷àé
T
=
1
íåèñêëþ÷àåòñÿ(ñì.
Ðèñ7.)
108
Ðèñ.7.
Îãðàíè÷åííóþ,íåïðåðûâíóþâîáëàñòè
D
ôóíêöèþ
u
(
x;t
)
ñíåïðåðûâíûìèâíóòðè
D
÷àñòíûìèïðîèçâîäíûìè
@
2
u
@x
2
;
@u
@t
;
óäîâëåòâîðÿþùóþóðàâíåíèþ(3.1),áóäåìíàçûâàòüðåãóëÿð-
íûìðåøåíèåìýòîãîóðàâíåíèÿ.
Ïîäçàäà÷åéÊîøèïîíèìàåòñÿñëåäóþùàÿçàäà÷à:èùåòñÿ
ðåãóëÿðíîåâïîëîñå
D
ðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(3.1),óäîâëå-
òâîðÿþùååóñëîâèþ
u
(
x;
0)=
'
(
x
)
;
�1
x
1
;
(3.13)
ãäå
'
(
x
)
;
�1
x
1
;
çàäàííàÿäåéñòâèòåëüíàÿíåïðå-
ðûâíàÿîãðàíè÷åííàÿôóíêöèÿ.
Íåïîñðåäñòâåííûìèâû÷èñëåíèåììîæíîóáåäèòüñÿ÷òîðå-
øåíèåìóðàâíåíèÿ
@u
@x
n
=
n

1
X
i
=1
@
2
u
@x
2
i
ÿâëÿåòñÿôóíêöèÿ
E
(
x
;

)=(
x
n


n
)
1

n
2
exp
"

1
4(
x
n


n
)
n

1
X
i
=1
(
x
i


i
)
2
#
;
(3.14)
ãäå

1
;:::;
n
äåéñòâèòåëüíûåïàðàìåòðû,ïðè÷åì
x
n
�
n
.
Ïðåäñòàâëåííàÿôîðìóëîé(3.14)ôóíêöèÿíàçûâàåòñÿýëåìåí-
109
òàðíûì(ôóíäàìåíòàëüíûì)ðåøåíèåìóðàâíåíèÿ
@u
@x
n
=
n

1
X
i
=1
@
2
u
@x
2
i
:
Òàêèìîáðàçîìôóíêöèÿ
E
(
x;;t;
0)=
1
p
t
e

(


x
)
2
4
t
;t�
0
;
(3.15)
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ(3.1)âîâñåõòî÷êàõ
(
x;t
)
ïîëó-
ïëîñêîñòè
t�
0
.
Äîêàæåì,÷òîôóíêöèÿ
u
(
x;t
)
,îïðåäåëåííàÿïîôîðìóëå
u
(
x;t
)=
1
2
p
t
1
Z
�1
'
(

)
e

(


x
)
2
4
t
d;
(3.16)
ÿâëÿåòñÿðåøåíèåìçàäà÷èÊîøè.
Íåòðóäíîïðîâåðèòü,÷òîèíòåãðàëâïðàâîé÷àñòè(3.16)
ñõîäèòüñÿðàâíîìåðíîâîêðåñòíîñòèëþáîéâíóòðåííåéòî÷êè
(
x;t
)
ïîëîñû
D
.
Âðåçóëüòàòåçàìåíûïåðåìåííîãîèíòåãðèðîâàíèÿ

=
x
+2

p
t
ôîðìóëà(3.16)ïðèíèìàåòâèä
u
(
x;t
)=
1
p

1
Z
�1
'

x
+2

p
t

e


2
d:
(3.17)
Òàêêàê
sup
�1
x
1
j
'
(
x
)
j
M
,ãäå
M
ïîëîæèòåëüíîå÷èñ-
ëî,èèíòåãðàëâïðàâîé÷àñòè(3.17)ñõîäèòñÿàáñîëþòíî,òî
j
u
(
x;t
)
j

M
p

1
Z
�1
e


2
d;
îòêóäàââèäóòîãî,÷òî
1
Z
�1
e


2
d
=
p
;
(3.18)
110
èìååì
j
u
(
x;t
)
j
6
M:
Ó÷èòûâàÿòîîáñòîÿòåëüñòâî,÷òîèíòåãðàëû,ïîëó÷åííûå
ïðèâíåñåíèèïîäçíàêèíòåãðàëàâïðàâîé÷àñòè(3.16)îïåðà-
öèèäèôôåðåíöèðîâàíèÿïî
x
è
t
(ëþáîå÷èñëîðàç),ñõîäÿò-
ñÿðàâíîìåðíîâáëèçèêàæäîéòî÷êå
(
x;t
)
;t�
0
,èôóíêöèÿ
E
(
x;;t;
0)
ïðè
t�
0
óäîâëåòâîðÿåòóðàâíåíèþ(3.1),çàêëþ÷à-
åì,÷òîîïðåäåëåííàÿôîðìóëîé(3.16)ôóíêöèÿ
u
(
x;t
)
âïîëîñå
D
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ(3.1).
Ïðåäåëüíûìïåðåõîäîìïðè
t
!
0
(ýòàîïåðàöèÿçàêîííà
èç-çàðàâíîìåðíîéñõîäèìîñòèèíòåãðàëàâáëèçèêàæäîéòî÷êè
(
x;
0)
ïðè
t

0
)èç(3.17)âñèëó(3.18)ïîëó÷àåì
lim
t
!
0
u
(
x;t
)=
'
(
x
)
:
Åäèíñòâåííîñòüèóñòîé÷èâîñòüðåøåíèÿçàäà÷èÊî-
øèíåïîñðåäñòâåííîïîëó÷àåòñÿèçñëåäóþùåãîóòâåðæäåíèÿ
(ïðèíöèïýêñòðåìóìàäëÿïîëîñû).
Òåîðåìà32.
Äëÿðåãóëÿðíîãîâïîëîñå
D
ðåøåíèÿ
u
(
x;t
)
óðàâíåíèÿ(3.1)èìåþòìåñòîîöåíêè
m
6
u
(
x;t
)
6
M;
m
=inf
u
(
x;
0)
;M
=sup
u
(
x;
0)
;
�1
x
1
:
(3.19)
Äëÿäîêàçàòåëüñòâàïåðâîãîíåðàâåíñòâàâ(3.19),ðàññìîò-
ðèìôóíêöèþ
v
(
x;t
)=
x
2
+2
t
,ÿâëÿþùóþñÿðåøåíèåìóðàâíå-
íèÿ(3.1).
Îáîçíà÷èì÷åðåç
n
íèæíþþãðàíü
u
(
x;t
)
ïðè
(
x;t
)
2
D
è
ââåäåìâðàññìîòðåíèåôóíêöèþ
w
(
x;t
)=
u
(
x;t
)

m
+
"
v
(
x;t
)
v
(
x
0
;t
0
)
;
(3.20)
111
ãäå
"
ïðîèçâîëüíîåïîëîæèòåëüíîå÷èñëî,
a
(
x
0
;t
0
)
ïðîèç-
âîëüíàÿôèêñèðîâàííàÿòî÷êàâíóòðèïîëîñû
D
.
Ïðåäñòàâëåííàÿôîðìóëîé(3.20)ôóíêöèÿ
w
(
x;t
)
ÿâëÿåòñÿ
ðåøåíèåìóðàâíåíèÿ(3.1),ïðè÷åìïðè
t
=0
w
(
x;
0)=
u
(
x;
0)

m
+
"
x
2
x
2
0
+2
t
2
0

0
(3.21)
èïðè
j
x
j
=
j
x
0
j
+
r
(
m

n
)
v
(
x
0
;t
0
)
"
w
(
x;t
)

u
(
x;t
)

n

0
:
(3.22)
Èçîöåíîê(3.21),(3.22)íàîñíîâàíèèäîêàçàííîãîâûøå
ïðèíöèïàýêñòðåìóìà,ïðèìåíåííîãîâïðÿìîóãîëüíèêå
0
6
t
6
T;
�j
x
0
j�
r
(
m

n
)
v
(
x
0
;t
0
)
"
6
x
6
6
j
x
0
j
+
r
(
m

n
)
v
(
x
0
;t
0
)
"
;
ñîäåðæàùåìòî÷êó
(
x
0
;t
0
)
,çàêëþ÷àåì,÷òî
w
(
x
0
;t
0
)=
u
(
x
0
;t
0
)

m
+
"

0
,ò.å.
u
(
x
0
;t
0
)

m

"
.
Îòñþäàâñâîþî÷åðåäüâñèëóïðîèçâîëüíîñòè
"
ñëåäóåò,÷òî
u
(
x
0
;t
0
)

m
.Òàêèìîáðàçîì
u
(
x;t
)

m
âñþäóâ
D
.
Çàìåíÿÿ
u
(
x;t
)
íà

u
(
x;t
)
èïîâòîðÿÿïðèâåäåííîåâûøå
ðàññóæäåíèå,óáåæäàåìñÿâñïðàâåäëèâîñòèèâòîðîãîíåðàâåí-
ñòâàâ(3.19).
3.4.Ãëàäêîñòüðåøåíèé
Âðàçäåëå3.2áûëîïîêàçàíî,÷òîðåøåíèå
u
(
x;t
)
ïåð-
âîéêðàåâîéçàäà÷è(3.7),(3.8)äëÿóðàâíåíèÿòåïëîïðî-
âîäíîñòè(3.1)ïðèòðåáîâàíèèíåïðåðûâíîñòèïåðâîéïðî-
èçâîäíîéôóíêöèè
u
(
x;t
)=
'
(
x
)
èìååòâîáëàñòè
D
:
0
xl;
0
tT
÷àñòíûåïðîèçâîäíûåâñåõïîðÿäêîâ
112
ïîïåðåìåííûì
x;t
.Òî÷íîòàêæåèçôîðìóëû(3.16)ðàçäå-
ëà3.3áûëñäåëàíâûâîä,÷òîîãðàíè÷åííîñòüèíåïðåðûâíîñòü
ôóíêöèè
'
(
x
)=
u
(
x;
0)
;
�1
x
1
ãàðàíòèðóåòñóùå-
ñòâîâàíèå÷àñòíûõïðîèçâîäíûõâñåõïîðÿäêîâðåøåíèÿ
u
(
x;t
)
çàäà÷èÊîøè(3.13)äëÿóðàâíåíèÿ(3.1).
3.5.Íåîäíîðîäíîåóðàâíåíèåòåïëîïðîâîäíîñòè
Ïóñòü
g
(
x;t
)
;
�1
x
1
;
0
6
t
6
1
çàäàííàÿäåé-
ñòâèòåëüíàÿîãðàíè÷åííàÿíåïðåðûâíàÿôóíêöèÿ.Çàíîñèòåëÿ
äàííûõâìåñòîïðÿìîé
t
=0
ïðèìåìïðÿìóþ
t
=

,ãäå


ôèêñèðîâàííîåïîëîæèòåëüíîå÷èñëî,èââåäåìôóíêöèþ
v
(
x;t;
)=
1
2
p

1
p
t


1
Z
�1
e

(


x
)
2
4(
t


)
g
(
;
)
d;t�:
Íåòðóäíîïðîâåðèòü,÷òî
@
2
v
@x
2

@v
@t
=0
;t�;v
(
x;;
)=
g
(
x;
)
:
Íàîñíîâàíèèýòèõðàâåíñòâçàêëþ÷àåì,÷òîôóíêöèÿ
u
(
x;t
)=
t
Z
0
v
(
x;t;
)
d
ÿâëÿåòñÿðåøåíèåìíåîäíîðîäíîãîóðàâíåíèÿ
@u
@t
=
@
2
u
@x
2
+
g
(
x;t
)
;
óäîâëåòâîðÿþùèìóñëîâèþ
u
(
x;
0)=0
.
3.6.Âîïðîñûèçàäà÷è
1.Ïîêàæèòå,÷òîâîáëàñòè
D
=
f
0
tT;
0
x
1
l
1
;
0
x
2
l
2
g
;
ôóíêöèÿ
u
(
x
1
;x
2
;t
)=exp



2

i
2
l
2
1
+
j
2
l
2
2

t

sin
ix
1

l
1
sin
jx
2

l
2
;
113
ãäå
i
è
j
íàòóðàëüíûå÷èñëà,ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ
@u
@t
=
@
2
u
@x
2
1
+
@
2
u
@x
2
2
èóäîâëåòâîðÿåòóñëîâèÿì
u
(
x
1
;x
2
;
0)=sin
ix
1

l
1
sin
jx
2

l
2
;u
j

=0
;
ãäå

áîêîâàÿïîâåðõíîñòüîáëàñòè
D
.
2.Â÷åìçàêëþ÷àåòñÿïðèíöèïìàêñèìóìàäëÿðåøåíèÿ
óðàâíåíèÿòåïëîïðîâîäíîñòè?
3.Äîêàæèòå,÷òîðåãóëÿðíîåâîáëàñòè
D
ðåøåíèåóðàâíå-
íèÿ(3.1),íåïðåðûâíîåâ
D
[
S
,ñâîåãîìèíèìóìàäîñòèãàåòíà
S
.
4.Ïîñòàâüòåïåðâóþêðàåâóþçàäà÷óäëÿóðàâíåíèÿòåïëî-
ïðîâîäíîñòè.
5.Íà÷åìîñíîâàíîäîêàçàòåëüñòâîåäèíñòâåííîñòèè
óñòîé÷èâîñòèðåøåíèÿïåðâîéêðàåâîéçàäà÷èäëÿóðàâíåíèÿ
òåïëîïðîâîäíîñòè?
6.Ñâåäèòåïåðâóþêðàåâóþçàäà÷ó
u
t
=
u
xx
+
f
(
x;t
)
;
u
(0
;t
)=

(
t
)
;u
(1
;t
)=

(
t
)
;
0
6
t
6
T
0
âïðÿìîóãîëüíèêå
f
(
x;t
)
2
R
2
:0
x
1
;
0
tT
0
g
;
ê
ïåðâîéêðàåâîéçàäà÷åñîäíîðîäíûìèêðàåâûìèóñëîâèÿìèíà
áîêîâûõñòîðîíàõ.
114
4.Óðàâíåíèÿýëëèïòè÷åñêîãîòèïà
4.1.Îñíîâíûåñâîéñòâàãàðìîíè÷åñêèõôóíêöèé
4.1.1.Èíòåãðàëüíîåïðåäñòàâëåíèåãàðìîíè÷åñêèõôóíêöèé
ÐàññìîòðèìóðàâíåíèåËàïëàñà

u
=0
;
(4.1)
ãäå

äèôôåðåíöèàëüíûéîïåðàòîðñ÷àñòíûìèïðîèçâîäíû-
ìèâòîðîãîïîðÿäêà
=
n
X
i
=1
@
2
@x
2
i
;
íàçûâàåìûéîïåðàòîðîìËàïëàñà.
Õàðàêòåðèñòè÷åñêàÿêâàäðàòè÷íàÿôîðìà,ñîîòâåòñòâóþ-
ùàÿóðàâíåíèþ(4.1),
Q
=
n
X
i
=1

2
i
ïîëîæèòåëüíîîïðåäåëåíàâîâñåõòî÷êàõïðîñòðàíñòâà
R
n
.Ñëå-
äîâàòåëüíî,ýòîóðàâíåíèåïðèíàäëåæèòýëëèïòè÷åñêîìóòèïó
âñþäóâ
R
n
.
Ôóíêöèÿ
u

x

;
x
=

x
1
;:::;x
n

,îáëàäàþùàÿíåïðåðûâíû-
ìè÷àñòíûìèïðîèçâîäíûìèâòîðîãîïîðÿäêàâîáëàñòè
D

R
n
èóäîâëåòâîðÿþùàÿóðàâíåíèþËàïëàñà,íàçûâàåòñÿãàðìîíè-
÷åñêîéôóíêöèåéâîáëàñòè
D
.
Êîãäàîáëàñòü
D
ñîäåðæèòáåñêîíå÷íîóäàëåííóþòî÷êó,
ïðèâåäåííîåâûøåîïðåäåëåíèåãàðìîíè÷åñêîéôóíêöèèíóæ-
äàåòñÿâóòî÷íåíèè,ò.ê.ïîíÿòèåïðîèçâîäíîéâáåñêîíå÷íîóäà-
ëåííîéòî÷êåíåèìååòñìûñëà.
Áóäåìãîâîðèòü,÷òîôóíêöèÿ
u
(
x
)
ãàðìîíè÷íàâîêðåñò-
íîñòèáåñêîíå÷íîóäàëåííîéòî÷êè(ò.å.âíåçàìêíóòîãîøàðà
f
x
2
D
:
j
x
j
6
R
g
äîñòàòî÷íîáîëüøîãîðàäèóñà),åñëèôóíê-
115
öèÿ
v
(
y
)=
j
y
j
2

n
u

y
j
y
j
2

;
äîîïðåäåëåííàÿâòî÷êå
y
=
0
êàê
lim
y
!
0
v
(
y
)
,ãàðìîíè÷íàâ
îêðåñòíîñòèòî÷êè
y
=
0
.
Âðåçóëüòàòåçàìåíû
y
=
x
j
x
j
2
èìååì
u
(
x
)=
j
x
j
2

n
v

x
j
x
j
2

:
Âñîîòâåòñòâèèñýòèìïîäðåãóëÿðíûìâîêðåñòíîñòèáåñ-
êîíå÷íîóäàëåííîéòî÷êèðåøåíèåìóðàâíåíèÿËàïëàñàïîíè-
ìàåòñÿãàðìîíè÷åñêàÿâýòîéîêðåñòíîñòèâñþäó,êðîìåáåñêî-
íå÷íîóäàëåííîéòî÷êè,ôóíêöèÿ
u
(
x
)
,êîòîðàÿïðè
j
x
j!1
îñòàåòñÿîãðàíè÷åííîéâñëó÷àå
n
=2
èñòðåìèòñÿêíóëþíå
ìåäëåííåå,÷åì
j
x
j
2

n
ïðè
n�
2
.
Ïóñòü
D
îáëàñòüïðîñòðàíñòâà
R
n
ñäîñòàòî÷íîãëàäêîé
ãðàíèöåé
S

u
(
x
)
è
v
(
x
)
çàäàííûåâíåéäåéñòâèòåëüíûå
ãàðìîíè÷åñêèåôóíêöèè,íåïðåðûâíûåâìåñòåñîñâîèìèïðîèç-
âîäíûìèïåðâîãîïîðÿäêàâ
D
[
S
.
Ñâîéñòâàãàðìîíè÷åñêèõôóíêöèé:
1.Íàðÿäóñãàðìîíè÷åñêîéôóíêöèåé
u
(
x
)
âîáëàñòè
D
ãàð-
ìîíè÷åñêîéÿâëÿåòñÿèôóíêöèÿ
u
(
C
x
+
h
)
,ãäå

äåéñòâè-
òåëüíàÿïîñòîÿííàÿ,
C
ïîñòîÿííàÿäåéñòâèòåëüíàÿîðòîãî-
íàëüíàÿìàòðèöàïîðÿäêà
n;
h
=(
h
1
;:::;h
n
)
ïîñòîÿííûé
äåéñòâèòåëüíûéâåêòîð,ïðè÷åìïðåäïîëàãàåòñÿ,÷òîòî÷êè
x
è
C
x
+
h
ïðèíàäëåæàòîáëàñòè
D
.
2.Êîíå÷íàÿëèíåéíàÿêîìáèíàöèÿãàðìîíè÷åñêèõôóíêöèé
åñòüãàðìîíè÷åñêàÿôóíêöèÿ.
3.Åñëè
u
(
x
)
ãàðìîíè÷íàÿâîáëàñòè
D
,òîèôóíêöèÿ
v
(
x
)=
j
x
j
2

n
u

x
j
x
j
2

116
ãàðìîíè÷åñêàÿâñþäó,ãäåîíàîïðåäåëåíà.
4.Åñëèãàðìîíè÷åñêàÿâîáëàñòè
D
ôóíêöèÿ
u
(
x
)
íåïðå-
ðûâíàâ
D
[
S
âìåñòåñîñâîèìèïðîèçâîäíûìèïåðâîãîïîðÿäêà
èðàâíàíóëþíàãðàíèöå
S
îáëàñòè
D
,òî
u
(
x
)=0
äëÿâñåõ
x
2
D
[
S
.
5.Åñëèíîðìàëüíàÿïðîèçâîäíàÿ
@u
(
x
)
@

x
ãàðìîíè÷åñêîéâ
îáëàñòè
D
ôóíêöèè
u
(
x
)
,íåïðåðûâíîéâìåñòåñîñâîèìèïðî-
èçâîäíûìèïåðâîãîïîðÿäêàâ
D
[
S
,ðàâíàíóëþíàãðàíèöå
S
îáëàñòè
D
,òî
u
(
x
)=const
äëÿâñåõ
x
2
D
.
6.Èíòåãðàë,âçÿòûéïîãðàíèöå
S
îòíîðìàëüíîéïðîèçâîä-
íîé
@u
(
x
)
@

x
ãàðìîíè÷åñêîéâîáëàñòè
D
ôóíêöèè
u
(
x
)
,íåïðå-
ðûâíîéâìåñòåñîñâîèìè÷àñòíûìèïðîèçâîäíûìèïåðâîãîïî-
ðÿäêàâ
D
[
S
,ðàâåííóëþ.
Äëÿãàðìîíè÷åñêîéâîáëàñòè
D
ôóíêöèè
u
(
x
)
,íåïðåðûâ-
íîéâìåñòåñîñâîèìèïðîèçâîäíûìèïåðâîãîïîðÿäêàâ
D
[
S
,
èìååòìåñòîèíòåãðàëüíîåïðåäñòàâëåíèå
u
(
x
)=
1
w
n
Z
S
E
(
x
;
y
)
@u
(
y
)
@

y
dS
y

1
w
n
Z
S
u
(
y
)
@E
(
x
;
y
)
@

y
dS
y
;
(4.2)
ãäå
E
(
x
;
y
)
ýëåìåíòàðíîåðåøåíèåóðàâíåíèÿËàïëàñà
E
(
x
;
y
)=
8

:
1
n

2
j
y

x
j
2

n
;n�
2
;

ln
j
y

x
j
;n
=2
;
(4.3)
w
n
=
2

n
2
�(
n
2
)
ïëîùàäüåäèíè÷íîéñôåðûâ
R
n


ãàììà-
ôóíêöèÿÝéëåðà.
Äëÿâûâîäàôîðìóëû(4.2)âûäåëèìòî÷êó
x
èçîáëàñòè
D
âìåñòèñçàìêíóòûìøàðîì
f
y
2
D
:
j
y

x
j
6
"
g
,ëåæàùèìâ
D
,èäëÿîñòàâøåéñÿ÷àñòè
D
"
îáëàñòè
D
,îãðàíè÷åííîéïîâåðõ-
117
íîñòüþ
S
èñôåðîé
f
y
2
D
:
j
y

x
j
=
"
g
,ïðèìåíèìôîðìóëó
Z
S

v
(
y
)
@u
(
y
)
@

y

u
(
y
)
@v
(
y
)
@

y

dS
y
=0
;
(4.4)
âêîòîðîé
v
(
y
)=
E
(
x
;
y
)
:
Z
S

E
(
x
;
y
)
@u
(
y
)
@

y

u
(
y
)
@E
(
x
;
y
)
@

y

dS
y
=
=
Z
j
y

x
j
=
"

E
(
x
;
y
)
@u
(
y
)
@

y

u
(
y
)
@E
(
x
;
y
)
@

y

dS
y
=
(4.5)
=
Z
j
y

x
j
=
"
E
(
x
;
y
)
@u
(
y
)
@

y
dS
y

Z
j
y

x
j
=
"
[
u
(
y
)

u
(
x
)]
@E
(
x
;
y
)
@

y
dS
y


u
(
x
)
Z
j
y

x
j
=
"
@E
(
x
;
y
)
@

y
dS
y
:
Ðèñ.1.
Ó÷èòûâàÿòîîáñòîÿòåëüñòâî,÷òîíàñôåðå
f
y
2
D
:
j
y

x
j
=
"
g
E
(
x
;
y
)=
8



:
1
(
n

2)
"
n

2
;n�
2
;

ln
";n
=2
;
118
@E
(
x
;
y
)
@

y
=

1
"
n

1
;n

2
;
lim
"
!
0
Z
j
y

x
j
=
"
[
u
(
y
)

u
(
x
)]
@E
(
x
;
y
)
@

y
dS
y
=0
;
Z
j
y

x
j
=
"
d
y
y
"
n

1
=
w
n
;
âñèëóðàâåíñòâà
Z
S
@u
(
y
)
@

y
dS
y
=0
(4.6)
èçôîðìóëû(4.5)âïðåäåëåïðè
"
!
0
ïîëó÷àåìèíòåãðàëüíîå
ïðåäñòàâëåíèå(4.2).
4.1.2.Òåîðåìàîñðåäíåì
Ñïðàâåäëèâîñëåäóþùååóòâåðæäåíèå.
Òåîðåìà41.
Åñëèøàð
f
y
2
D
:
j
y

x
j
6
R
g
ëåæèò
öåëèêîìâîáëàñòè
D
ãàðìîíè÷íîñòèôóíêöèè
u
(
x
)
,òîçíà-
÷åíèåýòîéôóíêöèèâöåíòðåøàðàðàâíîñðåäíåìóàðèôìåòè-
÷åñêîìóå¼çíà÷åíèéíàñôåðå
f
y
2
D
:
j
y

x
j
=
R
g
.
Äåéñòâèòåëüíî,ò.ê.íàñôåðå
f
y
2
D
:
j
y

x
j
6
R
g
èìåþò
ìåñòîðàâåíñòâà
E
(
x
;
y
)=
8



:
1
(
n

2)
R
n

2
;n�
2
;

ln
R;n
=2
;
@E
(
x
;
y
)
@

y
=

1
R
n

1
;n

2
;
òîâñèëó(4.6)èçôîðìóëû(4.2),íàïèñàííîéäëÿøàðà
f
y
2
D
:
j
y

x
j
R
g
,ïîëó÷àåì
u
(
x
)=
1
w
n
R
n

1
Z
j
y

x
j
=
R
u
(
y
)
dS
y
:
(4.7)
119
Çàïèñûâàÿôîðìóëó(4.7)äëÿñôåðû
f
y
2
D
:
j
y

x
j
=

6
R
g
ââèäå

n

1
u
(
x
)=
1
w
n
Z
j
y

x
j
=

u
(
y
)
dS
y
èèíòåãðèðóÿïî

âïðîìåæóòêå
0
6

6
R
,ïîëó÷àåì
u
(
x
)=
n
w
n
R
n
Z
j
y

x
j
R
u
(
y
)
d
y
;
(4.8)
ãäå

y
ýëåìåíòîáúåìàïîïåðåìåííîìó
y

w
n
R
n
n
îáúåì
øàðà
f
y
2
D
:
j
y

x
j
R
g
.
Ôîðìóëû(4.7)è(4.8)èçâåñòíûïîäíàçâàíèåìôîðìóëî
ñðåäíåìàðèôìåòè÷åñêîìäëÿãàðìîíè÷åñêèõôóíêöèéïîñôå-
ðåèïîøàðóñîîòâåòñòâåííî.
Ïðè
n
=2
è
n
=3
,ïîëüçóÿñüïîëÿðíûìèêîîðäèíàòàìè,
ôîðìóëó(4.7)ìîæíîçàïèñàòüåùåââèäå
u
(
x
1
;x
2
)=
1
2

2

Z
0
u
(
x
1
+
R
cos
;x
2
+
R
sin

)
d
(4.9)
è
u
(
x
1
;x
2
;x
3
)=
1
4


Z
0
d
2

Z
0
u
(
x
1
+
y
1
;x
2
+
y
2
;x
3
+
y
3
)sin
d ;
ãäå
y
1
=
R
sin

cos
;y
2
=
R
sin

sin
;y
3
=
R
cos
:
4.1.3.Ïðèíöèïýêñòðåìóìàèåãîñëåäñòâèÿ
Îáîçíà÷èì÷åðåç
M
è
m
âåðõíþþèíèæíþþãðàíèçíà÷å-
íèéâîáëàñòè
D
ãàðìîíè÷åñêîéôóíêöèè
u
(
x
)
.
Íàîñíîâàíèèôîðìóëû(4.8)ëåãêîóñòàíîâèòüñëåäóþùåå
ñâîéñòâîãàðìîíè÷åñêèõôóíêöèé,èçâåñòíîåïîäíàçâàíèåì
ïðèíöèïàýêñòðåìóìàäëÿãàðìîíè÷åñêèõôóíêöèé.
120
Òåîðåìà42.
Îòëè÷íàÿîòïîñòîÿííîéãàðìîíè÷åñêàÿâ
îáëàñòè
D
ôóíêöèÿ
u
(
x
)
íèâîäíîéòî÷êå
x
2
D
íåìîæåò
ïðèíèìàòüíèçíà÷åíèÿ
M
,íèçíà÷åíèÿ
m
.
Êîãäà
M
=+
1
èëè
m
=
�1
,ñïðàâåäëèâîñòüýòîãîóòâåð-
æäåíèÿî÷åâèäíà,èáîâêàæäîéòî÷êåîáëàñòè
D
ôóíêöèÿ
u
(
x
)
ïðèíèìàåòëèøüêîíå÷íîåçíà÷åíèå.Êîãäàæå
M
6
=+
1
,äîïó-
ñòèìîáðàòíîå,ò.å.
u
(
x
0
)=
M;
x
0
2
D
,èðàññìîòðèìøàð
f
x
2
D
:
j
x

x
0
j
"
g
,ëåæàùèéâ
D
.Âêàæäîéòî÷êåýòî-
ãîøàðà
u
(
x
)=
M
.Äåéñòâèòåëüíî,åñëèáûâòî÷êå
y
ïðè
f
y
2
D
:
j
y

x
0
j
"
g
èìåëîìåñòîíåðàâåíñòâî
u
(
y
)
M
(íåðàâåíñòâî
u
(
y
)
�M
èñêëþ÷åíî),òîâñèëóíåïðåðûâíî-
ñòè
u
(
x
)
ýòîíåðàâåíñòâîñîõðàíèëîñüáûâñþäóâíåêîòîðîé
îêðåñòíîñòè
j


x
j

òî÷êè
y
èíàîñíîâàíèèôîðìóëû(4.8),
ïðèìåíåííîéâñëó÷àåøàðà
f
x
2
D
:
j
x

x
0
j
"
g
,ïîëó÷è-
ëîñüáûáåññìûñëåííîåíåðàâåíñòâî
MM
.Ñëåäîâàòåëüíî,
u
(
x
)=
M
âñþäóâøàðå
f
x
2
D
:
j
x

x
0
j
"
g
.
Ïóñòüòåïåðü
x
ïðîèçâîëüíàÿôèêñèðîâàííàÿòî÷êàîá-
ëàñòè
D
è
l
íåïðåðûâíàÿêðèâàÿ,ñîåäèíÿþùàÿ
x
ñ
x
0
è
ëåæàùàÿâ
D
.Ïóñòü÷èñëî
"
ìåíüøå,÷åìðàññòîÿíèåìåæäó
ãðàíèöåé
S
îáëàñòè
D
èêðèâîé
l
.Ïåðåäâèãàÿöåíòð
y
øàðà
f

2
D
:
j


x
0
j
"
g
èçòî÷êè
x
0
âòî÷êó
x
ïîêðèâîé
l
è
ïîëüçóÿñüóæåäîêàçàííûìôàêòîì,÷òîïðèëþáîìïîëîæåíèè
y
âíóòðèýòîãîøàðà
u
=
M
,ïîëó÷àåì
u
(
x
)=
M
.Ñëåäîâàòåëü-
íî,
u
(
x
)=
M
âñþäóâîáëàñòè
D
.Ïîëó÷åííîåïðîòèâîðå÷èå
äîêàçûâàåòñïðàâåäëèâîñòüïåðâîé÷àñòèñôîðìóëèðîâàííîãî
óòâåðæäåíèÿ.Àíàëîãè÷íîäîêàçûâàåòñÿèâòîðàÿåãî÷àñòü.
Åñëèäîïîëíèòåëüíîèçâåñòíî,÷òîãàðìîíè÷åñêàÿâîáëà-
ñòè
D
ôóíêöèÿ
u
(
x
)
íåïðåðûâíàâ
D
[
S
,òîîíàîáÿçàòåëüíî
ïðèìåòñâîåìàêñèìàëüíîå(ìèíèìàëüíîå)çíà÷åíèåâíåêîòî-
ðîéòî÷êå
x
0
2
D
[
S
.Âñèëóòîëüêî÷òîäîêàçàííîãîñâîéñòâà
ãàðìîíè÷åñêèõôóíêöèéòî÷êàýêñòðåìóìà
x
0
íåìîæåòáûòü
121
âíóòðåííåéäëÿîáëàñòè
D
è,çíà÷èò,
x
0
2
S
.
Èçïðèíöèïàýêñòðåìóìàäëÿãàðìîíè÷åñêèõôóíêöèéñëå-
äóåò,÷òîçàäà÷àÄèðèõëå(èëèïåðâàÿêðàåâàÿçàäà÷à)

u
=0
;
u
j
S
=
';
(4.10)
ãäå
'
çàäàííàÿíà
S
äåéñòâèòåëüíàÿíåïðåðûâíàÿôóíêöèÿ,
íåìîæåòèìåòüáîëååîäíîãîðåãóëÿðíîãîâîáëàñòè
D
ðåøå-
íèÿ,íåïðåðûâíîãîâçàìêíóòîéîáëàñòè
D
[
S
.Âñàìîìäåëå,
åñëè
u
(
x
)
è
v
(
x
)
ÿâëÿþòñÿðåøåíèÿìèýòîéçàäà÷è,òîèõðàç-
íîñòü
w
(
x
)=
u
(
x
)

v
(
x
)
áóäåòðàâíàíóëþíàãðàíèöå
S
îá-
ëàñòè
D
èïîýòîìóâñèëóïðèíöèïàýêñòðåìóìà
w
(
x
)=0
,ò.å.
u
(
x
)=
v
(
x
)
âñþäóâ
D
[
S
.
4.2.ÔóíêöèÿÃðèíà.Ðåøåíèåçàäà÷èÄèðèõëåäëÿøàðàè
ïîëóïðîñòðàíñòâà
4.2.1.ÏîíÿòèåôóíêöèèÃðèíàçàäà÷èÄèðèõëå
äëÿóðàâíåíèÿËàïëàñà
ÔóíêöèåéÃðèíàçàäà÷èÄèðèõëåäëÿóðàâíåíèÿËàïëàñàâ
îáëàñòè
D
íàçûâàåòñÿôóíêöèÿ
G
(
x
;

)
äâóõòî÷åê
x
;

2
D
[
S
,
îáëàäàþùàÿñâîéñòâàìè:
1)Îíàèìååòâèä
G
(
x
;

)=
E
(
x
;

)+
g
(
x
;

)
;
(4.11)
ãäå
E
(
x
;

)
ýëåìåíòàðíîåðåøåíèåóðàâíåíèÿËàïëàñà(ñì.
ôîðìóëó(4.3)),
g
(
x
;

)
ãàðìîíè÷åñêàÿôóíêöèÿêàêïî
x
2
D
,òàêèïî

2
D
;
2)Êîãäàòî÷êà
x
èëè

ëåæèòíàãðàíèöå
S
îáëàñòè
D
,òî
G
(
x
;

)=0
:
(4.12)
Ëåãêîâèäåòü,÷òî
G
(
x
;

)

0
âñþäóâîáëàñòè
D
.
122
Âñàìîìäåëå,îáîçíà÷èì÷åðåç
D

÷àñòüîáëàñòè
D
,ëå-
æàùóþâíåøàðà
f

2
D
:
j
x


j
6

g
äîñòàòî÷íîìàëîãî
ðàäèóñà

.Ò.ê.
lim
x
!

G
(
x
;

)=+
1
,òîïðèäîñòàòî÷íîìàëîì

èìååì
G
(
x
;

)

0
,êîãäà
j
x


j

.Ñëåäîâàòåëüíî,íàãðàíè-
öåîáëàñòè
D

èìååì
G
(
x
;

)

0
,èïîýòîìó,âñèëóïðèíöèïà
ýêñòðåìóìà,
G
(
x
;

)

0
äëÿâñåõ
x
2
D

.Îòñþäàçàêëþ÷àåì,
÷òî
G
(
x
;

)

0
âñþäóâ
D
[
S
.
Çàìåòèì,÷òîôóíêöèÿÃðèíà
G
(
x
;

)
ñèììåòðè÷íàîòíîñè-
òåëüíîòî÷åê
x
è

.
Åñëèâðàâåíñòâå(4.2)ïðèìåì,÷òî
u
(
x
)
ðåøåíèåçàäà-
÷èÄèðèõëåäëÿóðàâíåíèÿËàïëàñà,èâìåñòî
E
(
x
;
y
)
âîçüìåì
G
(
x
;

)
,òîïîâòîðåíèåìïðèâåäåííîãîâûøåðàññóæäåíèÿïðè
âûâîäåôîðìóëû(4.2)ñó÷åòîì(4.11)è(4.12)ïîëó÷èì
u
(
x
)=

1
w
n
Z
S
@G
(
x
;

)
@


'
(

)
dS

;
(4.13)
ãäå
'
çàäàííàÿäåéñòâèòåëüíàÿíåïðåðûâíàÿôóíêöèÿ.
ÊîãäàôóíêöèÿÃðèíàèçâåñòíà,ôîðìóëà(4.13)äàåòðåøå-
íèåçàäà÷èÄèðèõëåâñëåäóþùåéïîñòàíîâêå:èùåòñÿãàðìî-
íè÷åñêàÿâîáëàñòè
D
ôóíêöèÿ
u
(
x
)
,íåïðåðûâíàÿâ
D
[
S
è
óäîâëåòâîðÿþùàÿêðàåâîìóóñëîâèþ
u
j
S
=
':
(4.14)
Ãàðìîíè÷íîñòüïðåäñòàâëåííîéôîðìóëîé(4.13)ôóíêöèè
u
(
x
)
ñëåäóåòèçãàðìîíè÷íîñòèôóíêöèèÃðèíà
G
(
x
;

)
ïî
x
ïðè
x
6
=

.Òî,÷òîýòàôóíêöèÿóäîâëåòâîðÿåòèêðàåâîìóóñëî-
âèþ(4.14),òðåáóåòîñîáîãîäîêàçàòåëüñòâà.
4.2.2.Ðåøåíèåçàäà÷èÄèðèõëåäëÿøàðà.ÔîðìóëàÏóàññîíà
Ìûñåé÷àñÿâíîïîñòðîèìôóíêöèþÃðèíà,êîãäà
D
ïðåä-
ñòàâëÿåòñîáîéøàð,èâýòîìñëó÷àåïîêàæåì,÷òîïðåäñòàâ-
123
ëåííàÿôîðìóëîé(4.13)ãàðìîíè÷åñêàÿôóíêöèÿäåéñòâèòåëüíî
óäîâëåòâîðÿåòêðàåâîìóóñëîâèþ(4.14).
Ïóñòüîáëàñòü
D
ïðåäñòàâëÿåòñîáîéøàð
f
x
2
R
n
:
j
x
j

1
g

x
è

âíóòðåííèåòî÷êèýòîãî
øàðà.Òî÷êà

0
=

j

j
2
ñèììåòðè÷íàòî÷êå

îòíîñèòåëüíîñôå-
ðû
S
=
f
x
2
R
n
:
j
x
j
=1
g
.Ïîêàæåì,÷òîôóíêöèÿÃðèíà
G
(
x
;

)
çàäà÷èÄèðèõëåäëÿøàðà
D
èìååòâèä
G
(
x
;

)=
E
(
x
;

)

E

j
x
j

;
x
j
x
j

:
(4.15)
Äåéñòâèòåëüíî,ò.ê.




j
x
j


x
j
x
j




=
h
j
x
j
2
j

j
2

2
x

+1
i
1
/
2
=




j

j
x


j

j




=
=
j

j




x


j

j
2




=
j
x
j






x
j
x
j
2




;
(4.16)
ôóíêöèÿ
g
(
x
;

)=

E

j
x
j

;
x
j
x
j

ïðè
j
x
j

1
;
j

j

1
ÿâëÿåò-
ñÿãàðìîíè÷åñêîéêàêïî
x
,òàêèïî

.Àïðè
j

j
=1
èìååì
j


x
j
=
h
j
x
j
2

2
x

+1
i
1
2
=




j

j
x


j

j




=




j
x
j


x
j
x
j




:
(4.17)
Ñëåäîâàòåëüíî,ïðåäñòàâëåííàÿôîðìóëîé(4.15)ôóíêöèÿ
G
(
x
;

)
óäîâëåòâîðÿåòâñåìòðåáîâàíèÿì,ïðåäúÿâëÿåìûìê
ôóíêöèèÃðèíà.
Òàêêàêïðè
j

j
=1
âñèëó(4.17)
@G
(
x
;

)
@


=

n
X
i
=1
8



:

i
(

i

x
i
)
j


x
j
n
�j
x
j

i

j
x
j

i

x
i
j
x
j




j
x
j


x
j
x
j



n
9

=

;
=
124
=

1
�j
x
j
2
j


x
j
n
;
òîôîðìóëà(4.13)âðàññìàòðèâàåìîìñëó÷àåçàïèøåòñÿââèäå
u
(
x
)=
1
w
n
Z
j

j
=1
1
�j
x
j
2
j


x
j
n
'
(

)
dS

:
(4.18)
ÝòàôîðìóëàíîñèòíàçâàíèåôîðìóëûÏóàññîíà.
Âñëó÷àå
n
=3
è
n
=2
ôîðìóëàÏóàññîíàâïîëÿðíûõ
êîîðäèíàòàõçàïèøåòñÿñîîòâåòñòâåííîââèäå
u
(
x
1
;x
2
;x
3
)=
1
4


Z
0
d
2

Z
0
1
�j
x
j
2

1

2
j
x
j
cos

+
j
x
j
2

3
/
2
'
sin
d ;
ãäå
'
=
'
(

1
;
2
;
3
)
,

1
=sin

cos

,

2
=sin

sin

,

3
=cos

,
j
x
j
cos

=
x

è
u
(
x
1
;x
2
)=
1
2

2

Z
0
1
�j
x
j
2
1

2
j
x
j
cos
(



)+
j
x
j
2
'
(cos
;
sin

)
d ;
(4.19)
ãäå
x
1
=
j
x
j
cos
;x
2
=
j
x
j
sin
;
1
=cos
;
2
=sin
:
Ôîðìóëà(4.18)áûëàâûâåäåíàäëÿåäèíè÷íîãîøàðàñ
öåíòðîìâòî÷êå
x
=0
.Åñëè
u
(
x
)
ãàðìîíè÷åñêàÿâøà-
ðå
f
x
2
R
n
:
j
x
j
R
g
ôóíêöèÿ,íåïðåðûâíàÿâøàðå
f
x
2
R
n
:
j
x
j
6
R
g
èóäîâëåòâîðÿþùàÿêðàåâîìóóñëîâèþ
lim
x
!
y
u
(
x
)=
'
(
y
)
;
j
x
j
R;
j
y
j
=
R;
òîôóíêöèÿ
v
(
z
)=
u
(
R
z
)
áóäåòãàðìîíè÷åñêîéâøàðå
f
z
2
R
n
:
j
z
j

1
g
,íåïðåðûâíîé
ïðè
j
z
j
6
1
èóäîâëåòâîðÿþùåéóñëîâèþ
lim
z
!
t
v
(
z
)=
'
(
R
t
)
;
j
z
j

1
;
j
t
j
=1
:
125
Ïîýòîìóâñèëóôîðìóëû(4.18)èìååì
v
(
z
)=
1
w
n
Z
j

j
=1
1
�j
z
j
2
j


z
j
n
'
(
R

)
dS

;
ò.å.
u
(
x
)=
v

x
R

=
1
w
n
R
Z
j

j
=1
R
2
�j
x
j
2
j
R


x
j
n
R
n

1
'
(
R

)
dS

;
èëè,ïîñëåçàìåíû
y
=
R

,
u
(
x
)=
1
w
n
R
Z
j
y
j
=
R
R
2
�j
x
j
2
j
y

x
j
n
'
(
y
)
dS
y
:
(4.20)
Ïóñòü
u
(
x
)
ãàðìîíè÷åñêàÿâøàðå
f
x
2
R
n
:
jj
x

x
0
j
R
g
ôóíêöèÿ,íåïðåðûâ-
íàÿâïëîòüäîãðàíèöûèóäîâëåòâîðÿþùàÿóñëîâèþ
lim
x
!
y
u
(
x
)=
'
(
y
)
;
j
x

x
0
j
R;
j
y

x
0
j
=
R:
Òàê
êàêôóíêöèÿ
w
(
z
)=
u
(
z
+
x
0
)
ãàðìîíè÷íàâøàðå
f
z
2
R
n
:
j
z
j
R
g
,íåïðåðûâíàïðè
j
z
j
6
R
èóäîâëå-
òâîðÿåòóñëîâèþ
lim
z
!
t
w
(
z
)=
'
(
t
+
x
0
)
;
j
z
j
R;
j
t
j
=
R;
òîâ
ñèëó(4.20)ìîæåìíàïèñàòü
w
(
z
)=
1
w
n
R
Z
j
t
j
=
R
R
2
�j
z
j
2
j
t

z
j
n
'
(
t
+
x
0
)
dS
t
;
îòêóäàñðàçóñëåäóåòôîðìóëàÏóàññîíàäëÿøàðà
f
x
2
R
n
:
jj
x

x
0
j
R
g
u
(
x
)=
w
(
x

x
0
)=
1
w
n
R
Z
j


x
0
j
=
R
R
2
�j
x

x
0
j
2
j


x
j
n
'
(

)
dS

:
(4.21)
Ïðè
x
=
x
0
èçôîðìóëû(4.21)ñíîâàïîëó÷àåìôîðìóëó
(4.7)îñðåäíåìàðèôìåòè÷åñêîì.
126
Òåïåðüïîêàæåì,÷òîîïðåäåëåííàÿïîôîðìóëåÏóàññîíà
ôóíêöèÿ
u
(
x
)
óäîâëåòâîðÿåòêðàåâîìóóñëîâèþ(4.14),èñòàëî
áûòü,ýòàôîðìóëàäàåòðåøåíèåçàäà÷èÄèðèõëåâïðèâåäåííîé
âûøåïîñòàíîâêå.
Äëÿïðîñòîòûîãðàíè÷èìñÿðàññìîòðåíèåìñëó÷àÿ
n
=2
.
Òàêêàê
u
(
x
)=1
ÿâëÿåòñÿãàðìîíè÷åñêîéôóíêöèåé,óäîâëå-
òâîðÿþùåéóñëîâèþ
lim
x
!
x
0
u
(
x
)=1
,
j
x
j

1
,ãäå
x
0
ïðîèçâîëü-
íàÿôèêñèðîâàííàÿòî÷êàíàîêðóæíîñòè
f
x
2
R
2
:
j
x
j
=1
g
,
òîäëÿâñåõòî÷åê
x
âêðóãå
f
x
2
R
2
:
j
x
j

1
g
èçôîðìóëû
(4.18)ïîëó÷àåì
1
2

2

Z
0
1
�j
x
j
2
j


x
j
2
d
=1
;
1
=cos
;
2
=sin
:
(4.22)
Íàîñíîâàíèèôîðìóë(4.18)è(4.22)èìååì
u
(
x
)

'
(
x
0
)=
1
2

2

Z
0
1
�j
x
j
2
j


x
j
2
[
'
(

)

'
(
x
0
)]
d ;
j
x
j

1
:
(4.23)
Âñèëóðàâíîìåðíîéíåïðåðûâíîñòèôóíêöèè
'
íàîêðóæíî-
ñòè
f
x
2
R
2
:
j
x
j
=1
g
äëÿëþáîãîíàïåðåäçàäàííîãî
"�
0
ñóùåñòâóåò

(
"
)

0
òàêîå,÷òîäëÿâñåõ

è

0
,

1
=cos

,

2
=sin

,
x
1
0
=cos

0
,
x
2
0
=sin

0
,óäîâëåòâîðÿþùèõóñëîâèþ
j



0
j

,èìååòìåñòîíåðàâåíñòâî
j
'
(

)

'
(
x
0
)
j
":
(4.24)
Ïåðåïèøåìâûðàæåíèå(4.23)ââèäå
u
(
x
)

'
(
x
0
)=
I
1
+
I
2
;
ãäå
I
1
=
1
2


0
+

Z

0


1
�j
x
j
2
j


x
j
2
[
'
(

)

'
(
x
0
)]
d ;
127
I
2
=
1
2

0
B
@

0


Z
0
+
2

Z

0
+

1
C
A
1
�j
x
j
2
j


x
j
2
[
'
(

)

'
(
x
0
)]
d :
Íàîñíîâàíèè(4.22)è(4.24)çàêëþ÷àåì,÷òî
j
I
1
j
"
.Ïîñëåâû-
áîðà

(
"
)
âîçüìåì
x
íàñòîëüêîáëèçêîê
x
0
,÷òîáûèìåëîìåñòî
íåðàâåíñòâî
0
B
@

0


Z
0
+
2

Z

0
+

1
C
A
1
�j
x
j
2
j


x
j
2
d
"
M
;M
=max
0


6
2

j
'
(

)
j
;
ò.å.
j
I
2
j
"
.Ñëåäîâàòåëüíî,
j
u
(
x
)

'
(
x
0
)
j

2
"
è,ïîýòîìó,
lim
x
!
x
0
u
(
x
)=
'
(
x
0
)
;
j
x
j

1
;
j
x
0
j
=1
:
4.2.3.Ðåøåíèåçàäà÷èÄèðèõëåäëÿïîëóïðîñòðàíñòâà
Ðàññìîòðèìòåïåðüñëó÷àé,êîãäàîáëàñòü
D
ñîâïàäàåòñïî-
ëóïðîñòðàíñòâîì
x
n

0
,àîòèñêîìîãîðåøåíèÿçàäà÷èÄèðè-
õëåïîòðåáóåì,÷òîáûîíîáûëîîãðàíè÷åíî.Ïóñòü
x
è

òî÷-
êèýòîãîïîëóïðîñòðàíñòâà,à

0
=(

1
;:::;
n

1
;


n
)
òî÷êà,
ñèììåòðè÷íàÿòî÷êå

îòíîñèòåëüíîïëîñêîñòè

n
=0
.Òàêêàê
ôóíêöèÿ
g
(
x
;

)=

E
(
x
;

0
)
ïðè
x
n

0
,

n

0
ÿâëÿåòñÿãàðìî-
íè÷åñêîéêàêïî
x
,òàêèïî

,è,êðîìåòîãî,
E
(
x
;

)=
E
(
x
;

0
)
ïðè

n
=0
,òî
G
(
x
;

)=
E
(
x
;

)

E
(
x
;

0
)
(4.25)
ÿâëÿåòñÿôóíêöèåéÃðèíàäëÿðàññìàòðèâàåìîãîïîëóïðî-
ñòðàíñòâà.
Áóäåìïðåäïîëàãàòü,÷òîäëÿèñêîìîãîðåøåíèÿ
u
(
x
)
çàäà-
÷èÄèðèõëåâðàññìàòðèâàåìîìñëó÷àåñïðàâåäëèâàôîðìóëà
(4.14).Ýòîçàâåäîìîòàê,åñëè
8
x
2
D
ïðè
j
x
j!1
j
u
(
x
)
j
A
j
x
j
h
;



@u
@x
i




A
j
x
j
h
+1
;i
=1
;:::;n;
128
ãäå
A
è
h
ïîëîæèòåëüíûåïîñòîÿííûå.Âñîîòâåòñòâèèñýòèì
ïðèáîëüøèõ

=

n

1
X
i
=1
y
2
i
!
1
2
èçàäàííàÿíàïëîñêîñòè
y
n
=0
ôóíêöèÿ
'
(
y
1
;:::;y
n

1
)
äîëæíàóäîâëåòâîðÿòüóñëîâèþ
j
'
j

A

h
:
Ïîäñòàâëÿÿâûðàæåíèå
G
(
x
;

)
èçôîðìóëû(4.25)âïðàâóþ
÷àñòüôîðìóëû(4.13)èó÷èòûâàÿ,÷òî
@G
(
x
;

)
@


=

@G
(
x
;

)
@
n
=

n

x
n
j


x
j
n


n
+
x
n
j

0

x
j
n
=
=

2
x
n

n

1
P
i
=1
(

i

x
i
)
2
+
x
2
n

n
2
ïðè

n
=0
,ïîëó÷àåìôîðìóëó
u
(
x
)=
�(
n
2
)

n
2
x
n
Z

n
=0
'
(

1
;:::;
n

1
)

n

1
P
i
=1
(
x
i


i
)
2
+
x
2
n

n
2
d
1
:::d
n

1
;
(4.26)
äàþùóþðåøåíèåçàäà÷èÄèðèõëåñóñëîâèåì
lim
x
!
y
u
(
x
)=
'
(
y
1
;:::;y
n

1
)
;x
n

0
;y
n
=0
;
(4.27)
äëÿïîëóïðîñòðàíñòâà
x
n

0
èíîñÿùóþòàêæåíàçâàíèåôîð-
ìóëûÏóàññîíà.
Òîòôàêò,÷òî,îïðåäåëåííàÿïîôîðìóëå(4.26)ôóíêöèÿ
óäîâëåòâîðÿåòêðàåâîìóóñëîâèþ(4.27),äîêàçûâàåòñÿòî÷íî
òàêæå,êàêýòîáûëîñäåëàíîâûøåâñëó÷àåçàäà÷èÄèðèõëå
äëÿêðóãà.
Ðåøåíèåçàäà÷èÄèðèõëåäëÿïîëóïðîñòðàíñòâà
n
P
k
=1
a
k
x
k

b�
0
ðåäóöèðóåòñÿêðàññìîòðåííîìóñëó÷àþ,
129
åñëèó÷åñòü,÷òîíàðÿäóñ
u
(
x
)
ãàðìîíè÷åñêîéÿâëÿåòñÿè
ôóíêöèÿ
u
(
C
x
+
h
)
,ãäå

ïîñòîÿííàÿ,
C
ïîñòîÿííàÿ
îðòîãîíàëüíàÿìàòðèöà,à
h
ïîñòîÿííûéâåêòîð.
4.2.4.Íåêîòîðûåñëåäñòâèÿ,âûòåêàþùèåèçôîðìóëû
Ïóàññîíà.ÒåîðåìûËèóâèëëÿèÃàðíàêà
Èçôîðìóëû(4.20)ñëåäóåòñïðàâåäëèâîñòüóòâåðæäåíèÿ.
Òåîðåìà43.
Åñëèãàðìîíè÷åñêàÿâïðîñòðàíñòâå
R
n
ôóíêöèÿâñþäóíåîòðèöàòåëüíà(íåïîëîæèòåëüíà),òîîíà
ïîñòîÿííà.
Äåéñòâèòåëüíî,ò.ê.ïðè
j
x
j
R
,
j
y
j
=
R
èìåþòìåñòîíåðà-
âåíñòâà
R
�j
x
j
6
j
y

x
j
6
R
+
j
x
j
èïîóñëîâèþ
u
(
x
)

0
,òî
èçôîðìóëû(4.20)âñèëó(4.7)ñëåäóåò,÷òî
R
n

2
R
�j
x
j
(
R
+
j
x
j
)
n

1
u
(0)
6
u
(
x
)
6
R
n

2
R
+
j
x
j
(
R
�j
x
j
)
n

1
u
(0)
(4.28)
äëÿëþáîãî
R�
0
.Îòñþäà,ïðîèçâîëüíîôèêñèðóÿòî÷êó
x
2
R
n
èçàòåìóñòðåìëÿÿ
R
!1
,ïîëó÷àåì,÷òîâêàæäîé
òî÷êå
x
ïðîñòðàíñòâà
R
n
ôóíêöèÿ
u
(
x
)=
u
(0)
.
Èçôîðìóëû(4.28)òàêæåíåïîñðåäñòâåííîâûòåêàåòñïðà-
âåäëèâîñòüñëåäóþùåéòåîðåìûËèóâèëëÿ.
Òåîðåìà44.
Åñëèãàðìîíè÷åñêàÿâ
R
n
ôóíêöèÿ
u
(
x
)
îãðàíè÷åíàñâåðõó(ñíèçó),òîîíàïîñòîÿííà.
Âñàìîìäåëå,ïóñòü
u
(
x
)
6
M
8
x
2
R
n
,ãäå
M
=
const
.
Òàêêàêôóíêöèÿ
M

u
(
x
)
ãàðìîíè÷íàâ
R
n
èíåîòðèöàòåëüíà,
òî,êàêòîëüêî÷òîáûëîäîêàçàíî,
M

u
(
x
)=
M

u
(0)
,ò.å.
u
(
x
)=
u
(0)
.
ÒåîðåìàËèóâèëëÿïîçâîëÿåòóòâåðæäàòü,÷òîðàññìîòðåí-
íàÿâïðåäûäóùåìðàçäåëåçàäà÷àÄèðèõëåäëÿïîëóïðîñòðàí-
ñòâà
x
n

0
âêëàññåîãðàíè÷åííûõôóíêöèéíåìîæåòèìåòü
áîëååîäíîãîðåøåíèÿ.
Äåéñòâèòåëüíî,ðàçíîñòü
v
(
x
)=
u
1
(
x
)

u
2
(
x
)
ëþáûõäâóõ
ðåøåíèé
u
1
(
x
)
è
u
2
(
x
)
ýòîéçàäà÷èóäîâëåòâîðÿåòêðàåâîìó
130
óñëîâèþ
v
(
x
)=0
ïðè
x
n
=0
.Ðàññìîòðèìôóíêöèþ
w
(
x
)=

v
(
x
1
;:::;x
n
)
;x
n

0
;

v
(
x
1
;:::;

x
n
)
;x
n
6
0
;
êîòîðàÿãàðìîíè÷íàêàêïðè
x
n

0
,òàêèïðè
x
n

0
.Áî-
ëååòîãî,ôóíêöèÿ
w
(
x
)
ãàðìîíè÷íàâîâñåìïðîñòðàíñòâå
R
n
,
èáîâøàðå
f
x
2
R
n
:
j
x
j
R
g8
R�
0
îíàñîâïàäàåòñãàðìî-
íè÷åñêîéôóíêöèåé
w

(
x
)
,óäîâëåòâîðÿþùåéêðàåâîìóóñëîâèþ
w

(
x
)=
w
(
x
)
ïðè
j
x
j
=
R
.Òàêêàêïîóñëîâèþ
w
(
x
)
îãðàíè÷åíà,
òîâñèëóòåîðåìûËèóâèëëÿîíàïîñòîÿííà.Íî
w
(
x
)=0
ïðè
x
n
=0
,ò.å.
w
(
x
)=0
âñþäóâ
R
n
è,ñòàëîáûòü,
u
1
(
x
)=
u
2
(
x
)
.
Ïîëüçóÿñüïðèíöèïîìýêñòðåìóìàäëÿãàðìîíè÷åñêèõ
ôóíêöèéèôîðìóëîéÏóàññîíà(4.21),ëåãêîìîæíîïîëó÷èòü
äîêàçàòåëüñòâîñëåäóþùåéòåîðåìûÃàðíàêà.
Òåîðåìà45.
Åñëèðÿä
1
P
k
=1
u
k
(
x
)
ãàðìîíè÷åñêèõâîáëàñòè
D
ôóíêöèé
u
k
(
x
)
,íåïðåðûâíûõâ
D
[
S
,ðàâíîìåðíîñõîäèòñÿ
íàãðàíèöå
S
îáëàñòè
D
,òîýòîòðÿäðàâíîìåðíîñõîäèòñÿâ
D
[
S
èåãîñóììà
u
(
x
)
ãàðìîíè÷íàâ
D
.
Äåéñòâèòåëüíî,èçðàâíîìåðíîéñõîäèìîñòèðÿäà
1
P
k
=1
u
k
(
y
)
,
y
2
S
,ñëåäóåò,÷òî
8
"�
0
9
N
(
"
)
:
8
p

1



p
X
i
=1
u
N
+
i
(
x
)



":
Îòñþäà,ââèäóòîãî,÷òîêîíå÷íàÿñóììà
p
P
i
=1
u
N
+
i
(
x
)
ãàðìîíè÷-
íàâ
D
èíåïðåðûâíàâ
D
[
S
,âñèëóïðèíöèïàýêñòðåìóìà
çàêëþ÷àåì,÷òî




p
P
i
=1
u
n
+
i
(
x
)




"
8
x
2
D
[
S
.Ïîñëåäíååíåðà-
âåíñòâî,êàêèçâåñòíîèçêóðñàìàòåìàòè÷åñêîãîàíàëèçà,ÿâëÿ-
åòñÿóñëîâèåì,íåîáõîäèìûìèäîñòàòî÷íûìäëÿðàâíîìåðíîé
ñõîäèìîñòèðÿäà
1
P
k
=1
u
k
(
x
)
â
D
[
S
.
131
Ïóñòü
x
0
ïðîèçâîëüíàÿòî÷êàîáëàñòè
D
èøàð
f
y
2
D
:
j
y

x
0
j
R
g
ëåæèòâíóòðè
D
.Êàæäóþãàðìîíè÷å-
ñêóþôóíêöèþ
u
k
(
x
)
âýòîìøàðåìîæíîïðåäñòàâèòüôîðìóëîé
Ïóàññîíà(4.21)
u
k
(
x
)=
1
w
n
R
Z
j
y

x
0
j
=
R
R
2
�j
x

x
0
j
2
j
y

x
j
n
u
k
(
y
)
dS
y
:
Ñëåäîâàòåëüíî,ò.ê.ðàâíîìåðíîñõîäÿùèéñÿðÿäìîæíîèíòå-
ãðèðîâàòüïî÷ëåííî,èìååì
u
(
x
)=
1
X
k
=1
1
w
n
R
Z
j
y

x
0
j
=
R
R
2

(
x

x
0
)
2
j
y

x
j
n
u
k
(
y
)
dS
y
=
=
1
w
n
R
Z
j
y

x
0
j
=
R
R
2
�j
x

x
0
j
2
j
y

x
j
n
u
(
y
)
dS
y
;
îòêóäàèñëåäóåòãàðìîíè÷íîñòü
u
(
x
)
âøàðå
f
x
2
D
:
j
x

x
0
j
R
g
.Òàêêàê
x
0
ïðîèçâîëüíàÿ
òî÷êàîáëàñòè
D
,òîòåìñàìûìãàðìîíè÷íîñòü
u
(
x
)
äîêàçàíà
âñþäóâ
D
.
4.3.Âîïðîñûèçàäà÷è
1.ÇàïèøèòåîïåðàòîðËàïëàñàâïîëÿðíîéñèñòåìåêîîð-
äèíàò.
2.Ñôîðìóëèðóéòåîïðåäåëåíèåãàðìîíè÷åñêîéôóíêöèè.
Äîêàæèòåîäíîèçñâîéñòâãàðìîíè÷åñêèõôóíêöèé.
3.Íàéäèòåóñëîâèå,ïðèñîáëþäåíèèêîòîðîãîâêðóãå
132
x
2
+
y
2
=
r
2
R
2
çàäà÷à
8



:

u
(
x;y
)=0
;
0
6
r
R
;
@u
(
x;y
)
@r
j
r
=1
=2
x
2
+
A;r
=
R
èìååòðåøåíèå.
4.Ïóñòüøàð
f
y
2
R
n
:
j
y

x
j
6
R
g
ëåæèòâîáëàñòè
ãàðìîíè÷íîñòèôóíêöèè
u
(
x
)
.Ïîêàæèòåñïðàâåäëèâîñòüôîð-
ìóëû,âûðàæàþùåéòåîðåìóîñðåäíåì
u
(
x
)=
1
w
n
R
n

1
Z
j
y

x
j
=
R
u
(
y
)=
dS
y
äëÿñôåðû
j
y

x
j
=
R:
5.Íàéäèòåóñëîâèå,ïðèñîáëþäåíèèêîòîðîãîäëÿðåøåíèÿ
u
(
x;y
)
çàäà÷èÄèðèõëå
8

:

u
=0
;x
2
+
y
2

1
;
u
j
x
2
+
y
2
=1
=
a
+
x
3
y
3
ñïðàâåäëèâàôîðìóëà
u
(0
;
0)=0
:
6.Äîêàæèòåñèììåòðè÷íîñòüôóíêöèèÃðèíà
G
(
x
;
y
)
,ò.å.
G
(
x
;
y
)=
G
(
y
;
x
)
:
133
7.Ïîêàæèòå,÷òîãàðìîíè÷åñêàÿôóíêöèÿâîáëàñòèãàð-
ìîíè÷íîñòèèìååòïðîèçâîäíûåâñåõïîðÿäêîâ.
8.Íàéäèòåãàðìîíè÷åñêóþâïîëóïëîñêîñòè
y�
0
ôóíêöèþ
u
(
x;y
)
,åñëèèçâåñòíî,÷òî
u
(
x;
0)=
x
x
2
+1
:
9.Ïîêàæèòåñïðàâåäëèâîñòüòîæäåñòâà
1
2

2

Z
0
1
�j
x
j
2
j
y

x
j
2
d
=1
;
ãäå
x
=(
x
1
;x
2
)
òî÷êàêðóãà
j
x
j

1

y
=(cos
;
sin

)

òî÷êàíàîêðóæíîñòè
j
y
j
=1
.
10.Ìîæåòëèñîõðàíÿòüçíàêãàðìîíè÷åñêàÿâ
R
n
ôóíêöèÿ,
îòëè÷íàÿîòïîñòîÿííîé?
134
5.Ìåòîäðàçäåëåíèÿïåðåìåííûõ(ìåòîäÔóðüå)
5.1.Ðåøåíèåñìåøàííûõçàäà÷äëÿóðàâíåíèé
ãèïåðáîëè÷åñêîãîòèïàìåòîäîìðàçäåëåíèÿïåðåìåííûõ
Ìåòîäðàçäåëåíèÿïåðåìåííûõ(ìåòîäÔóðüå)èñïîëüçóþò
ïðèïîñòðîåíèèðåøåíèéñìåøàííûõçàäà÷äëÿøèðîêîãîêëàñ-
ñàóðàâíåíèéñ÷àñòíûìèïðîèçâîäíûìè.
Èçëîæåíèåñõåìûìåòîäàâäàííîìðàçäåëåðàññìîòðèìíà
ïðèìåðåîäíîìåðíîãîâîëíîâîãîóðàâíåíèÿ,îïèñûâàþùåãîïðî-
öåññêîëåáàíèÿñòðóíûäëèíû
l
:
u
tt
=
u
xx
:
(5.1)
ÑóòüìåòîäàÔóðüåñîñòîèòâòîì,÷òîñíà÷àëàèùåìíåòðè-
âèàëüíîåðåøåíèåóðàâíåíèÿ(5.1),ââèäåïðîèçâåäåíèÿäâóõ
ôóíêöèé,îäíàèçêîòîðûõçàâèñèòîò
x
,äðóãàÿîò
t
,ò.å.â
âèäå
u
(
x;t
)=
X
(
x
)

T
(
t
)
:
(5.2)
Ïîäñòàâëÿÿâûðàæåíèå(5.2)äëÿ
u
(
x;t
)
âóðàâíåíèå(5.1),ïî-
ëó÷àåì
X
(
x
)
T
00
(
t
)=
X
00
(
x
)
T
(
t
)
èëè
T
00
(
t
)
T
(
t
)
=
X
00
(
x
)
X
(
x
)
:
(5.3)
Ò.ê.ëåâàÿ÷àñòü(5.3)íåçàâèñèòîò
t
,àïðàâàÿ÷àñòüîò
x
,òî
T
00
(
t
)
T
(
t
)
=
X
00
(
x
)
X
(
x
)
=
const:
(5.4)
Îáîçíà÷èâ÷åðåç


ïîñòîÿííóþâïðàâîé÷àñòè(5.4),ïåðåïè-
øåìýòèðàâåíñòâàââèäå
X
00
(
x
)+
X
(
x
)=0
(5.5)
135
è
T
00
(
t
)+
T
(
t
)=0
;
(5.6)
ïðåäñòàâëÿþùèåñîáîéîáûêíîâåííûåëèíåéíûåäèôôåðåíöè-
àëüíûåóðàâíåíèÿâòîðîãîïîðÿäêàñïîñòîÿííûìèêîýôôèöè-
åíòàìè.Èçêóðñàîáûêíîâåííûõäèôôåðåíöèàëüíûõóðàâíå-
íèéèçâåñòíî,÷òîîáùååðåøåíèå
X
(
x
)
óðàâíåíèÿ(5.5)èìååò
âèä
X
=
C
1
x
+
C
2
(5.7)
ïðè

=0
;
X
=
C
1
cos
p
x
+
C
2
sin
p
x
(5.8)
ïðè
�
0
è
X
=
C
1
e
p

x
+
C
2
e

p

x
(5.9)
ïðè

0
,ãäå
C
1
è
C
2
ïðîèçâîëüíûåäåéñòâèòåëüíûåïîñòî-
ÿííûå.Òî÷íîòàêæåâñîîòâåòñòâèèñòåì,÷òî

=0
,
�
0
èëè

0
,îáùååðåøåíèåóðàâíåíèÿ(5.6)çàïèøåòñÿââèäå
T
=
C
3
t
+
C
4
;
T
=
C
3
cos
p
t
+
C
4
sin
p
t;
(5.10)
T
=
C
3
e
p

t
+
C
4
e

p

t
;
ãäå
C
3
è
C
4
ïðîèçâîëüíûåäåéñòâèòåëüíûåïîñòîÿííûå.
Ïóñòüòðåáóåòñÿíàéòèíåòðèâèàëüíîå(íåðàâíîåòîæäå-
ñòâåííîíóëþ)ðåãóëÿðíîåâïîëóïîëîñå
0
xl
,
t�
0
ðå-
øåíèå
u
(
x;t
)
óðàâíåíèÿ(5.1),íåïðåðûâíîåïðè
0
6
x
6
l
,
t

0
èóäîâëåòâîðÿþùååêðàåâûìóñëîâèÿì
u
(0
;t
)=0
;u
(
l;t
)=0
;t

0
:
(5.11)
Áóäåìèñêàòüðåøåíèåçàäà÷è(5.1),(5.11)ââèäå(5.2).Òîãäà
ôóíêöèè
X
(
x
)
è
T
(
t
)
äîëæíûóäîâëåòâîðÿòüóðàâíåíèÿì(5.5)
è(5.6)ñîîòâåòñòâåííîèóñëîâèÿì
X
(0)
T
(
t
)=
X
(
l
)
T
(
t
)=0
èëè,
÷òîòîæåñàìîå,
X
(0)=0
;X
(
l
)=0
:
(5.12)
136
Çàäà÷àîòûñêàíèÿíåòðèâèàëüíîãîðåøåíèÿ
X
(
x
)
óðàâíå-
íèÿ(5.5),óäîâëåòâîðÿþùåãîóñëîâèÿì(5.12),ÿâëÿåòñÿ÷àñò-
íûìñëó÷àåìòàêíàçûâàåìîéîáùåéñïåêòðàëüíîéçàäà÷èèëè
çàäà÷èØòóðìàËèóâèëëÿ.Çíà÷åíèå

,äëÿêîòîðîãîóðàâíå-
íèå(5.5)èìååòíåòðèâèàëüíîåðåøåíèå
X
(
x
)
,óäîâëåòâîðÿþùåå
óñëîâèÿì(5.12),íàçûâàåòñÿñîáñòâåííûì÷èñëîì,àñàìîðåøå-
íèå
X
(
x
)
ñîîòâåòñòâóþùåé

ñîáñòâåííîéôóíêöèåé.
Íåòðèâèàëüíûåðåøåíèÿçàäà÷è(5.5),(5.12)âèäà(5.7)è
(5.9)íåñóùåñòâóþò,èáî,ïîäñòàâëÿÿâûðàæåíèÿ(5.7)è(5.9)
äëÿ
X
(
x
)
â(5.12),ïîëó÷àåì
C
2
=0
;lC
1
+
C
2
=0
âïåðâîìñëó-
÷àåè
C
1
+
C
2
=0
;e
p

l
C
1
+
e

p

l
C
2
=0
âîâòîðîìñëó÷àå,
ò.å.âîáîèõñëó÷àÿõ
C
1
=
C
2
=0
.Òåïåðü,ïîäñòàâëÿÿâûðàæå-
íèå(5.8)â(5.12),áóäåìèìåòü
C
1
=0
;C
2
sin
p
l
=0
.Îòñþäà
âèäíî,÷òîçàäà÷à(5.5),(5.12)áóäåòèìåòüíåòðèâèàëüíîåðå-
øåíèåâèäà(5.8)òîãäàèòîëüêîòîãäà,êîãäà
sin
p
l
=0
;
ò.å.êîãäà

=

n
l

2
,ãäå
n
=1
;
2
;:::
Òàêèìîáðàçîì,ìûïðèøëèêçàêëþ÷åíèþ,÷òî

=

n
l

2
,
ãäå
n
=1
;
2
;:::
,ÿâëÿþòñÿñîáñòâåííûìè÷èñëàìèçàäà÷è(5.5),
(5.12),àôóíêöèè
C
n
sin
n
l
x;n
=1
;
2
;:::
,ãäå
C
n
ïðîèçâîëü-
íûåäåéñòâèòåëüíûåïîñòîÿííûå,îòëè÷íûåîòíóëÿ,ñîîòâåò-
ñòâóþùèìèèìñîáñòâåííûìèôóíêöèÿìè.
Íèæåáóäåìïðåäïîëàãàòü,÷òîïîñòîÿííûå
C
n
=1
;n
=1
;
2
;:::
Âñîîòâåòñòâèèñýòèìñèñòåìàñîáñòâåí-
íûõôóíêöèéçàïèøåòñÿââèäå
X
n
(
x
)=sin
n
l
x;n
=1
;
2
;:::
Ñëåäîâàòåëüíî,îäíîðîäíàÿçàäà÷à(5.1),(5.11)èìååò
áåñêîíå÷íîåìíîæåñòâîëèíåéíîíåçàâèñèìûõðåøåíèé
137
u
n
(
x;t
)=sin
n
l
x

T
n
(
t
)
,ãäåâñèëó(5.10)
T
n
(
t
)=
A
n
cos
n
l
t
+
B
n
sin
n
l
t;
à
A
n
è
B
n
ïðîèçâîëüíûåäåéñòâèòåëüíûåïîñòîÿííûå.
Íàáîððåøåíèé
u
n
(
x;t
)=sin
n
l
x

A
n
cos
n
l
t
+
B
n
sin
n
l
t

;n
=1
;
2
:::;
(5.13)
óðàâíåíèÿ(5.1)ïîçâîëÿåòíàéòèðåøåíèåñëåäóþùåéîñíîâíîé
ñìåøàííîéçàäà÷è:òðåáóåòñÿîïðåäåëèòüðåãóëÿðíîåâïîëóïî-
ëîñå
0
xl;t&#x-277;&#xl-22;&#x;-33;t-2;眀
0
ðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(5.1),íåïðåðûâ-
íîåïðè
0
6
x
6
l;t

0
,óäîâëåòâîðÿþùååêðàåâûìóñëîâèÿì
(5.11)èíà÷àëüíûìóñëîâèÿì
u
(
x;
0)=
'
(
x
)
;
@u
(
x;t
)
@t



t
=0
=

(
x
)
;
(5.14)
ãäå
'
(
x
)
è

(
x
)
çàäàííûåäîñòàòî÷íîãëàäêèåäåéñòâèòåëüíûå
ôóíêöèè.
Ðåøåíèå
u
(
x;t
)
çàäà÷è(5.1),(5.11),(5.14)áóäåìèñêàòüâ
âèäåðÿäà
u
(
x;t
)=
1
X
n
=1
sin
n
l
x

A
n
cos
n
l
t
+
B
n
sin
n
l
t

:
(5.15)
Î÷åâèäíî,÷òîïðåäñòàâëåííàÿôîðìóëîé(5.15)ôóíêöèÿ
u
(
x;t
)
,âïðåäïîëîæåíèèðàâíîìåðíîéñõîäèìîñòèðÿäàâïðà-
âîéå¼÷àñòè,óäîâëåòâîðÿåòêðàåâûìóñëîâèÿì(5.11).Äëÿòî-
ãî÷òîáûîíàóäîâëåòâîðÿëàèíà÷àëüíûìóñëîâèÿì(5.14),ìû
äîëæíûèìåòü
1
X
n
=1
A
n
sin
n
l
x
=
'
(
x
)
;
1
X
n
=1
n
l
B
n
sin
n
l
x
=

(
x
)
;
(5.16)
138
îòêóäàíàõîäèì
A
n
=
2
l
l
Z
0
'
(
x
)sin
n
l
xdx;B
n
=
2
n
l
Z
0

(
x
)sin
n
l
xdx:
ÈçòåîðèèðÿäîâÔóðüåèçâåñòíî,÷òîíåïðåðûâíîñòüíàîò-
ðåçêå
0
6
x
6
l
ôóíêöèé
'
00
(
x
)
è

0
(
x
)
èâûïîëíåíèåóñëîâèé
'
(0)=
'
(
l
)=

(0)=

(
l
)=0
ãàðàíòèðóþòâîçìîæíîñòüïðåä-
ñòàâëåíèé(5.16)èðàâíîìåðíóþñõîäèìîñòüòðèãîíîìåòðè÷å-
ñêîãîðÿäàâïðàâîé÷àñòè(5.15).Êðîìåòîãî,âýòîìñëó÷àå
ñóììà
u
(
x;t
)
ðÿäà(5.15)áóäåòíåïðåðûâíîäèôôåðåíöèðóåìîé
ôóíêöèåéïðè
0
6
x
6
l;t

0
;
óäîâëåòâîðÿþùåéóñëîâèÿì
(5.11)è(5.14).
Åñëèäîïîëíèòåëüíîèçâåñòíî,÷òîôóíêöèè
'
(
x
)
è

(
x
)
íåïðåðûâíûíàîòðåçêå
0
6
x
6
l
âìåñòåñîñâîèìèïðîèçâîä-
íûìèäîòðåòüåãîèâòîðîãîïîðÿäêàñîîòâåòñòâåííî,ïðè÷åì
'
(0)=
'
00
(0)=
'
(
l
)=
'
00
(
l
)=0
;
(0)=

(
l
)=0
;
òîïðåäñòàâ-
ëåííàÿôîðìóëîé(5.15)ôóíêöèÿ
u
(
x;t
)
áóäåòîáëàäàòü÷àñòíû-
ìèïðîèçâîäíûìèäîâòîðîãîïîðÿäêàâêëþ÷èòåëüíî,êîòîðûå
ìîãóòáûòüâû÷èñëåíûäèôôåðåíöèðîâàíèåìïî÷ëåííîðÿäàâ
ïðàâîé÷àñòè(5.15).Î÷åâèäíî,÷òîâýòèõïðåäïîëîæåíèÿõñóì-
ìà
u
(
x;t
)
ðÿäà(5.15)áóäåòèñêîìûìðåøåíèåìîñíîâíîéñìå-
øàííîéçàäà÷è(5.1),(5.11),(5.14).Êàæäîåèçñëàãàåìûõ
sin
n
l
x

A
n
cos
n
l
t
+
B
n
sin
n
l
t

;n
=1
;
2
;:::;
âïðàâîé÷àñòè(5.15)âòåîðèèðàñïðîñòðàíåíèÿçâóêàíîñèò
íàçâàíèåñîáñòâåííîãîêîëåáàíèÿ(èëèãàðìîíèêè)ñòðóíûñçà-
êðåïëåííûìèâòî÷êàõ
(0
;
0)
;
(
l;
0)
êîíöàìè.
Ðàññìîòðèìòåïåðüíåîäíîðîäíîåóðàâíåíèå
u
tt
=
u
xx
+
f
(
x;t
)
;
(5.17)
ãäå
f
(
x;t
)
çàäàííàÿäåéñòâèòåëüíàÿíåïðåðûâíàÿôóíêöèÿ.
139
Ðåøåíèå
u
(
x;t
)
óðàâíåíèÿ(5.17)áóäåìèñêàòüââèäåðÿäà
u
(
x;t
)=
1
X
n
=1
sin
n
l
xT
n
(
t
)
.Ðàçëîæèìôóíêöèþ
f
(
x;t
)
âðÿä
Ôóðüåïîñèñòåìå

sin
n
l
x
o
1
n
=1
:
f
(
x;t
)=
1
X
n
=1
f
n
(
t
)sin
n
l
x;f
n
(
t
)=
2
l
l
Z
0
f
(
x;t
)sin
n
l
xdx:
Òîãäàäëÿîïðåäåëåíèÿ
T
n
(
t
)
èç(5.17)ïîëó÷àåìîáûêíîâåííîå
ëèíåéíîåäèôôåðåíöèàëüíîåóðàâíåíèå
T
00
n
(
t
)+

n
l

2
T
n
(
t
)=
f
n
(
t
)
;
îäíèìèç÷àñòíûõðåøåíèéêîòîðîãîÿâëÿåòñÿôóíêöèÿ
T
n
÷àñò
(
t
)=
l
n
t
Z
0
f
n
(

)sin
n
l
(
t


)
d:
Ñëåäîâàòåëüíî,îáùååðåøåíèåýòîãîóðàâíåíèÿèìååòâèä
T
n
(
t
)=
A
n
cos
n
l
t
+
B
n
sin
n
l
t
+
l
n
t
Z
0
f
n
(

)sin
n
l
(
t


)
d;
ãäå
A
n
è
B
n
ïðîèçâîëüíûåäåéñòâèòåëüíûåïîñòîÿííûå.
Íàáîððåøåíèé
u
n
(
x;t
)=sin
n
l
x
h
A
n
cos
n
l
t
+
+
B
n
sin
n
l
t
+
l
n
t
Z
0
f
n
(

)sin
n
l
(
t


)
d
i
;n
=1
;
2
;:::;
óðàâíåíèÿ(5.17)ïîçâîëÿåòèññëåäîâàòüñìåøàííóþçàäà÷ó
(5.17),(5.11),(5.14).Åñëèôóíêöèè
'
(
x
)
è

(
x
)
ïðåäñòàâëÿ-
þòñîáîéñóììûðÿäîâ(5.16),à
f
(
x;t
)=
1
X
n
=1
f
n
(
t
)sin
n
l
x
òî,
140
äîïóñêàÿâîçìîæíîñòüïî÷ëåííîãîèíòåãðèðîâàíèÿèäèôôå-
ðåíöèðîâàíèÿýòèõðÿäîâ,ðåøåíèå
u
(
x;t
)
ñìåøàííîéçàäà÷è
(5.17),(5.11),(5.14)ìîæíîâûïèñàòüââèäå
u
(
x;t
)=
1
X
n
=1
sin
n
l
x
h
A
n
cos
n
l
t
+
+
B
n
sin
n
l
t
+
l
n
t
Z
0
f
n
(

)sin
n
l
(
t


)
d
i
:
Çàìåòèì÷òî,êîãäàêðàåâûåóñëîâèÿíåîäíîðîäíû,ò.å.êî-
ãäàâìåñòî(5.11)èìååì
u
(0
;t
)=

(
t
)
;u
(
l;t
)=

(
t
)
;
ãäå

(
t
)
è

(
t
)
äâàæäûíåïðåðûâíîäèôôåðåíöèðóåìûåôóíêöèè,â
ðåçóëüòàòåçàìåíû
u
(
x;t
)=
v
(
x;t
)+

(
t
)+
x
l
h

(
t
)


(
t
)
i
èñêî-
ìîãîðåøåíèÿ
u
(
x;t
)
óðàâíåíèÿ(5.17),äëÿîïðåäåëåíèÿ
v
(
x;t
)
ïîëó÷àåìóðàâíåíèå
v
tt
=
v
xx
+
f
(
x;t
)


00
(
t
)

x
l
h

00
(
t
)


00
(
t
)
i
ñîäíîðîäíûìèêðàåâûìèóñëîâèÿìè
v
(0
;t
)=
v
(
l;t
)=0
èñîîò-
âåòñòâóþùèìîáðàçîìèçìåíåííûìèíà÷àëüíûìèóñëîâèÿìè.
5.2.Ðåøåíèåñìåøàííûõçàäà÷äëÿóðàâíåíèéïàðàáîëè÷åñêîãî
òèïàìåòîäîìðàçäåëåíèÿïåðåìåííûõ
Ïàðàáîëè÷åñêèåóðàâíåíèÿíàèáîëåå÷àñòîâñòðå÷àþòñÿ
ïðèèçó÷åíèèïðîöåññîâòåïëîïðîâîäíîñòèèäèôôóçèè,èïðî-
ñòåéøèìïðèìåðîìòàêèõóðàâíåíèéÿâëÿåòñÿîäíîìåðíîåóðàâ-
íåíèåòåïëîïðîâîäíîñòè
u
t
=
u
xx
+
f
(
x;t
)
:
(5.18)
Îñíîâíûåñâîéñòâàðåøåíèéýòîãîóðàâíåíèÿíåçàâèñÿòîò
u:
Íàïðèìåð,êïåðâîéêðàåâîéçàäà÷åäëÿóðàâíåíèÿòåïëî-
ïðîâîäíîñòèâïðÿìîóãîëüíèêå
D
=
f
0
xl;
0
tT
g
;
141
ïðèâîäèòçàäà÷àîíàõîæäåíèèòåìïåðàòóðû
u
(
x;t
)
âòåïëîèçî-
ëèðîâàííîìñòåðæíå,åñëèèçâåñòíàåãîíà÷àëüíàÿòåìïåðàòóðà
íàêîíöàõñòåðæíÿâïîñëåäóþùååâðåìÿ.
Ðàññìîòðèìñëåäóþùóþçàäà÷ó
u
t
=
u
xx
+
f
(
x;t
)
;
(5.19)
u
(0
;t
)=

(
t
)
;u
(
l;t
)=

(
t
)
;
(5.20)
u
(
x;
0)=
g
(
x
)
;
(5.21)
ïðåäïîëàãàÿ,÷òîôóíêöèè

(
t
)
;
(
t
)
;g
(
x
)
íåïðåðûâíûè
g
(0)=

(0)
;g
(
l
)=

(0)
:
Ïðèðåøåíèèýòîéçàäà÷èñóùåñòâåííî,÷òîðåøåíèåèùåòñÿ
ïðè
t�
0
;
àíàëîãè÷íàÿçàäà÷àäëÿîòðèöàòåëüíûõçíà÷åíèé
t;
âîîáùåãîâîðÿ,íåèìååòðåøåíèÿ.Óðàâíåíèåòåïëîïðîâîäíî-
ñòèñóùåñòâåííîìåíÿåòñÿïðèçàìåíå
t
íà

t:
Ýòîòèïè÷íîå
óðàâíåíèåíåîáðàòèìîãîïðîöåññà.
ÏîñêîëüêóìåòîäÔóðüåïðèìåíèìëèøüêçàäà÷åñîäíîðîä-
íûìèêðàåâûìèóñëîâèÿìè,òîïðåäñòàâèì
u
(
x;t
)
ââèäåñóììû
v
(
x;t
)
è
W
(
x;t
)
;
ïðè÷åì
W
(
x;t
)
òàêàÿôóíêöèÿ,÷òî
W
(0
;t
)=

(
t
)
;W
(
l;t
)=

(
t
)
:
(5.22)
Òîãäàíåòðóäíîâèäåòü,÷òîðåøåíèåçàäà÷è(5.19)(5.21)
ñâîäèòñÿêðåøåíèþçàäà÷èñîäíîðîäíûìèêðàåâûìèóñëîâèÿ-
ìè
@v
@t
=
@
2
v
@x
2
+
e
f
(
x;t
)
;
(5.23)
v
(0
;t
)=
v
(
l;t
)=0
;
(5.24)
v
(
x;
0)=~
g
(
x
)
;
(5.25)
ãäå
~
f
(
x;t
)=
f
(
x;t
)+
@
2
W
@x
2

@W
@t
;
142
~
g
(
x
)=
g
(
x
)

W
(
x;
0)
;
ïðè÷åì
~
g
(
x
)
2
C
1
(
D
)
;
~
g
(0)=~
g
(
l
)=0
:
Åñëèèñêàòüôóíêöèþ
W
(
x;t
)
ââèäå
W
(
x;t
)=(

1
+

2
x
)

(
t
)+(

1
+

2
x
)

(
t
)
;
òîíåòðóäíîâèäåòü,÷òîôóíêöèÿ
W
(
x;t
)=
h
1

x
l
i

(
t
)+
x
l

(
t
)
(5.26)
óäîâëåòâîðÿåòóñëîâèÿì(5.22).
Àíàëîãè÷íîàëãîðèòìóðåøåíèÿñìåøàííîéçàäà÷èäëÿ
óðàâíåíèÿãèïåðáîëè÷åñêîãîòèïàôóíêöèþ
v
(
x;t
)
áóäåìèñ-
êàòüââèäåðÿäà
v
(
x;t
)=
1
X
k
=1
X
k
(
x
)

T
k
(
t
)
;
óêîòîðîãîêàæäûé
÷ëåíóäîâëåòâîðÿåòóðàâíåíèþòåïëîïðîâîäíîñòèèîáðàùàåòñÿ
âíóëüïðè
x
=0
è
x
=
l
.Äëÿýòîãîäîëæíîáûòü
X
00
k
(
x
)+
X
k
(
x
)=0
;
(5.27)
X
k
(0)=
X
k
(
l
)=0
:
(5.28)
Èçóðàâíåíèÿ(5.27)èóñëîâèé(5.28)ñëåäóåò,÷òî
X
k
(
x
)=
C
1
sin
p

k
x
+
C
2
cos
p

k
x;C
2
=0
;

k
=

k
l

2
;k
=1
;
2
;:::
X
k
íîðìèðóåìòàê,÷òîáû
k
X
k
k
2
=
l
2
:
Âýòîìñëó÷àå
C
1

1
è
ñîáñòâåííûåôóíêöèèïðèíèìàþòâèä
X
k
(
x
)=sin
k
l
x:
143
Ôóíêöèþ
~
f
(
x;t
)
ðàçëîæèìâðÿäïî
X
k
:
~
f
(
x;t
)=
1
X
k
=1
f
k
(
t
)sin
k
l
x;f
k
(
t
)=
1
k
X
k
k
2
l
Z
0
X
k
(
x
)
~
f
(
x;t
)
dx:
Äëÿôóíêöèè
T
k
(
t
)
èìååìóðàâíåíèå
T
0
k
+
k
l
T
k
=
f
k
;
ðåøàÿêîòîðîåíàõîäèì
T
k
(
t
)=
0
@
t
Z
0
f
k
(

)
e
(
k
l
)
2

d
+
A
k
1
A
e

(
k
l
)
2
t
:
Ñëåäîâàòåëüíî,ôóíêöèÿ
v
(
x;t
)=
1
X
k
=1
sin

l
x
0
@
t
Z
0
f
k
(

)
e
(
k
l
)
2

d
+
A
k
1
A
e

(
k
l
)
2
t
(5.29)
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ(5.23)èóäîâëåòâîðÿåòêðàåâûì
óñëîâèÿì(5.24).
Ïîñòîÿííûå
A
k
îïðåäåëÿþòñÿèçíà÷àëüíîãîóñëîâèÿ
v
(
x;
0)=
~
g
(
x
)=
1
X
k
=1
A
k
sin
k
l
x:
(5.30)
Òàêêàêìûïðåäïîëîæèëè,÷òî
~
g
(
x
)
ÿâëÿåòñÿíåïðåðûâíî-
äèôôåðåíöèðóåìîéôóíêöèåéèîáðàùàåòñÿâíóëüïðè
x
=0
,
x
=
l
,òîôóíêöèÿ
~
g
(
x
)
ìîæåòáûòüðàçëîæåíàâðÿäÔóðüå,à
êîýôôèöèåíòû
A
k
îïðåäåëÿþòñÿïîôîðìóëàìÔóðüå:
A
k
=
2
l
l
Z
0
~
g
(
x
)sin
k
l
xdx:
(5.31)
144
Èòàê,ðåøåíèåçàäà÷è(5.23)-(5.25)èìååòâèä
v
(
x;t
)=
1
X
k
=1
sin
k
l
xe

(
k
l
)
2
t

(5.32)

0
@
2
l
l
Z
0
~
g
(
x
)sin
k
l
xdx
+
t
Z
0
f
k
(

)
e
(
k
l
)
2

d
1
A
:
Ðåøåíèåìçàäà÷è(5.19)-(5.21)ÿâëÿåòñÿôóíêöèÿ
u
(
x;t
)=
W
(
x;t
)+
v
(
x;t
)
;
ãäå
W
(
x;t
)
îïðåäåëÿåòñÿïî
ôîðìóëå(5.26),à
v
(
x;t
)
ïîôîðìóëå(5.32).
Ïîñêîëüêóïðè
t

0
0
e

(
k
l
)
2
t
6
1
;
òîðÿä(5.32)ñõîäèòñÿàáñîëþòíîèðàâíîìåðíî,ïîýòîìóôóíê-
öèÿ
u
(
x;t
)
îïðåäåëÿåìàÿðÿäîì(5.32),íåïðåðûâíàâïðÿìî-
óãîëüíèêå

D
=
f
0
6
x
6
l;
0
6
t
6
T
g
èïðèíèìàåòçàäàííûå
çíà÷åíèÿíàíèæíåìîñíîâàíèèèáîêîâûõåãîñòîðîíàõ.
Îñòàåòñÿïîêàçàòü,÷òîâíóòðè
D
èíàåãîâåðõíåìîñíî-
âàíèèôóíêöèÿ
u
(
x;t
)
óäîâëåòâîðÿåòóðàâíåíèþòåïëîïðîâîä-
íîñòè.Äëÿýòîãîäîñòàòî÷íîïîêàçàòü,÷òîðÿäû,ïîëó÷åííûå
èç(5.32)ïî÷ëåííûìîäíîêðàòíûìäèôôåðåíöèðîâàíèåìïî
t
èäâóêðàòíûìäèôôåðåíöèðîâàíèåìïî
x
òàêæåàáñîëþòíîè
ðàâíîìåðíîñõîäÿòñÿ.Àýòîóòâåðæäåíèåñëåäóåòèçòîãî,÷òî
ïðèâñÿêîìïîëîæèòåëüíîì
t

k
l

2
e

(
k
l
)
2
t

1
;
åñëè
k
äîñòàòî÷íîâåëèêî.
Ñîâåðøåííîòàêæåìîæíîïîêàçàòüñóùåñòâîâàíèåóôóíê-
öèè
u
(
x;t
)
âíóòðè
D
èíàåãîâåðõíåìîñíîâàíèèíåïðåðûâíûõ
ïðîèçâîäíûõâñåõïîðÿäêîâïî
x
è
t
.
145
5.3.Ðåøåíèåêðàåâûõçàäà÷äëÿóðàâíåíèéýëëèïòè÷åñêîãî
òèïàìåòîäîìðàçäåëåíèÿïåðåìåííûõ
5.3.1.Ïîñòðîåíèåðåøåíèéêðàåâûõçàäà÷âïðÿìîóãîëüíûõîáëàñòÿõ
Âïðÿìîóãîëüíîéîáëàñòè
D
=
f
0
xp;
0
ys
g
äëÿ
óðàâíåíèÿËàïëàñàðàññìîòðèìñëåäóþùóþçàäà÷ó
u
xx
+
u
yy
=0
;
(5.33)
u
(0
;y
)=
u
(
p;y
)=0
;
(5.34)
u
(
x;
0)=
f
(
x
)
;u
(
x;s
)=
g
(
x
)
:
(5.35)
Áóäåìèñêàòüå¼ðåøåíèåââèäå
u
(
x;y
)=
X
(
x
)

Y
(
y
)
:
(5.36)
Ïîäñòàâëÿÿ(5.36)âóðàâíåíèå(5.33),ïîñëåðàçäåëåíèÿïåðå-
ìåííûõïîëó÷àåì
X
00
(
x
)
X
(
x
)
=

Y
00
(
y
)
Y
(
y
)
=

:
Âñèëóóñëîâèÿ(5.34)äëÿôóíêöèè
X
èìååìçàäà÷óØòóðìà-
Ëèóâèëëÿ

X
00
+
X
=0
;
X
(0)=
X
(
p
)=0
;
(5.37)
àôóíêöèÿ
Y
ÿâëÿåòñÿðåøåíèåìóðàâíåíèÿ
Y
00

Y
=0
:
(5.38)
Ðåøåíèåìçàäà÷è(5.37)ÿâëÿåòñÿôóíêöèÿ
X
k
(
x
)=sin
p

k
x;
k
=

k
p

2
;
k
X
k
k
2
=
p
2
;k
=1
;
2
;:::
Èçóðàâíåíèÿ(5.38)íàõîäèì
Y
(
y
)=
A
k
e
p

k
y
+
B
k
e

p

k
y
146
Ïîñêîëüêó
u
(
x;y
)=
1
X
k
=1
X
k
Y
k
;
òî
u
(
x;y
)=
1
X
k
=1
sin
p

k
x
h
A
k
e
p

k
y
+B
k
e

p

k
y
i
:
(5.39)
Óäîâëåòâîðÿåì(5.39)óñëîâèÿì(5.35),ïîëó÷àåì
1
X
k
=1
sin
p

k
x
[A
k
+B
k
]=
f
(
x
)
;
1
X
k
=1
sin
p

k
x
h
A
k
e
p

k
s
+B
k
e

p

k
s
i
=
g
(
x
)
:
Ðàçëàãàÿôóíêöèè
f
(
x
)
,
g
(
x
)
âðÿäÔóðüåïî
X
k
,
k
=1
;
2
;:::
èìååì
f
(
x
)=
1
X
k
=1
f
k
X
k
;g
(
x
)=
1
X
k
=1
g
k
X
k
;
ãäå
f
k
;g
k
êîýôôèöèåíòûÔóðüå.Âñèëóíåçàâèñèìîñòèñîá-
ñòâåííûõôóíêöèé
X
k
äëÿîïðåäåëåíèÿ
A
k
;B
k
ïîëó÷àåìñè-
ñòåìóàëãåáðàè÷åñêèõóðàâíåíèé
A
k
+
B
k
=
1
k
X
k
k
2
p
Z
0
f
(
x
)X
k
(
x
)
dx
=
f
k
;
A
k
e
p

k
s
+
B
k
e

p

k
s
=
1
k
X
k
k
2
p
Z
0
g
(
x
)X
k
(
x
)
dx
=
g
k
;
ðåøàÿêîòîðóþíàõîäèì
A
k
=
f
k
e

p

k
s

g
k
e

p

k
s

e
p

k
s
;
(5.40)
B
k
=
g
k

f
k
e
p

k
s
e

p

k
s

e
p

k
s
:
(5.41)
147
Ðåøåíèåçàäà÷è(5.33)(5.35)äàåòñÿôîðìóëîé(5.39),ãäå
A
k
;B
k
âû÷èñëÿþòñÿïîôîðìóëàì(5.40),(5.41).
Ðàññìîòðèìòåïåðüâïðÿìîóãîëüíîéîáëàñòè
D
=
f
0
xa;
0
yb
g
çàäà÷ó:
u
xx
+
u
yy
=
w
(
x;y
)
;
(5.42)
u
(0
;y
)=

(
y
)
;u
(
a;y
)=

(
y
)
;
(5.43)
u
(
x;
0)=
f
(
x
)
;u
(
x;b
)=
g
(
x
)
:
(5.44)
Ôóíêöèè

,

,
f
,
g
óäîâëåòâîðÿþòóñëîâèÿì

(0)=
f
(0)
;
(0)=
g
(0)
:
Ìåòîäðàçäåëåíèÿïåðåìåííûõýôôåêòèâåí,êîãäàíàâñåé
ãðàíèöåèëè÷àñòèå¼êðàåâûåóñëîâèÿíóëåâûå.Ïðåîáðàçóåì
èñêîìóþôóíêöèþòàê,÷òîáûäëÿíîâîéíåèçâåñòíîéôóíêöèè
ïðè
x
=0
,
x
=
a
êðàåâûåóñëîâèÿáûëèáûíóëåâûìè:
u
(
x;y
)=
v
(
x;y
)+
W
(
x;y
)
:
Ôóíêöèþ
W
(
x;y
)
íóæíîâûáðàòüòàêîé,÷òîáû
W
(0
;y
)=

(
y
)
;W
(
a;y
)=

(
y
)
:
Â÷àñòíîñòè,
W
(
x;y
)
ìîæíîâçÿòüââèäå
W
(
x;y
)=
x
a
h

(
y
)


(
y
)
i
+

(
y
)
:
Íîâîéíåèçâåñòíîéôóíêöèåéáóäåìñ÷èòàòü
v
(
x;y
)
;
äëÿ
êîòîðîéïîëó÷àåìñëåäóþùóþçàäà÷óÄèðèõëå:
v
xx
+
v
yy
=~
w
(
x;y
)
;
(5.45)
v
(0
;y
)=0
;v
(
a;y
)=0
;
(5.46)
v
(
x;
0)=
~
f
(
x
)
;v
(
x;b
)=~
g
(
x
)
;
(5.47)
ãäå
~
w
(
x;y
)=
w
(
x;y
)

W
xx
(
x;y
)

W
yy
(
x;y
)
;
148
~
f
(
x
)=
f
(
x
)

W
(
x;
0)
;
~
g
(
x
)=
g
(
x
)

W
(
x;b
)
:
Ðàññìîòðèìîäíîðîäíîåóðàâíåíèå
v
xx
+
v
yy
=0
:
Ðåøåíèåýòîãîóðàâíåíèÿïðåäñòàâèìââèäåïðîèçâåäåíèÿ:
v
(
x;y
)=
X
(
x
)
Y
(
y
)
:
Ïîñëåïîäñòàíîâêèâóðàâíåíèåèäåëåíèÿîáåèõ÷àñòåéíà
X
(
x
)
Y
(
y
)
;
ïîëó÷èì
X
00
X
=

Y
00
Y
=

:
Äëÿîïðåäåëåíèÿôóíêöèè
X
(
x
)
ïîëó÷àåìñïåêòðàëüíóþçàäà-
÷ó
X
00
+
X
=0
;X
(0)=0
;X
(
a
)=0
:
(5.48)
Ïðè

=

k
=

k
a

2
;k
=1
;
2
;:::
èìååìíåòðèâèàëüíûåðåøåíèÿ
X
k
(
x
)=sin
p

k
x;
k
X
k
k
2
=
a
Z
0
sin
2
p

k
xdx
=
a
2
:
Ñèñòåìàôóíêöèé
X
k
(
x
)
ïîëíàèîðòîãîíàëüíàâïðîñòðàíñòâå
ôóíêöèé
L
2
(0
;a
)
:
Ïóñòüðåøåíèåçàäà÷è(5.45)(5.47)ïðåäñòàâèìîââèäå
v
(
x;y
)=
1
X
k
=1
sin
p

k
xY
k
(
y
)
:
(5.49)
149
Ðàçëîæèìôóíêöèè
~
f
(
x
)
;
~
g
(
x
)
;
~
w
(
x;y
)
âðÿäûÔóðüåïî
X
k
(
x
)
;k
=1
;
2
;:::
~
f
(
x
)=
1
X
k
=1
f
k
X
k
;
~
g
(
x
)=
1
X
k
=1
g
k
X
k
;
(5.50)
~
w
(
x;y
)=
1
X
k
=1
w
k
(
y
)
X
k
;
(5.51)
ãäå
f
k
=
1
k
X
k
k
2
p
Z
0
f
(
x
)
X
k
(
x
)
dx;
g
k
=
1
k
X
k
k
2
p
Z
0
g
(
x
)
X
k
(
x
)
dx;
w
k
(
y
)=
1
k
X
k
k
2
p
Z
0
~
w
(
x;y
)
X
k
(
x
)
dx:
Ïîäñòàâëÿåì(5.49),(5.51)âíåîäíîðîäíîåóðàâíåíèå(5.45)
ñó÷åòîìðàâåíñòâà
X
00
k
(
x
)=


k
X
k
(
x
)
ïðèðàâíèâàÿêîýôôè-
öèåíòûðàçëîæåíèéâðÿäÔóðüå,ïîëó÷àåìáåñêîíå÷íîå÷èñëî
äèôôåðåíöèàëüíûõóðàâíåíèéâòîðîãîïîðÿäêà
Y
00
k


k
Y
k
=
w
k
(
y
)
;
ñîâîêóïíîñòüðåøåíèéêîòîðûõîïèñûâàåòñÿôîðìóëîé
Y
k
(
y
)=
A
k
e
p

k
y
+
B
k
e

p

k
y
+
1
p

k
y
Z
0
w
k
(
s
)
sh
p

k
(
y

s
)
ds;
âêîòîðîéïåðâûåäâàñëàãàåìûåýòîîáùååðåøåíèåîäíîðîä-
íîãîóðàâíåíèÿ,ïîñëåäíåå÷àñòíîåðåøåíèåíåîäíîðîäíîãî
150
óðàâíåíèÿ.Òîãäà
v
(
x;y
)=
1
X
k
=1
sin
p

k
x
h
A
k
e
p

k
y
+B
k
e

p

k
y
+
+
1
p

k
y
Z
0
w
k
(
s
)
sh
p

k
(
y

s
)
ds
i
:
(5.52)
×òîáûîïðåäåëèòüêîýôôèöèåíòû
A
k
;B
k
,íåîáõîäèìîóäîâëå-
òâîðèòüýòîðåøåíèåóñëîâèÿì(5.47).Ñó÷åòîì(5.50),ïîëó÷àåì
1
X
k
=1
sin
p

k
x
[
A
k
+
B
k
]=
~
f
(
x
)=
1
X
k
=1
f
k
X
k
;
1
X
k
=1
sin
p

k
x
h
A
k
e
p

k
b
+
B
k
e

p

k
b
+
+
1
p

k
b
Z
0
w
k
(
s
)
sh
p

k
(
b

s
)
ds
i
=
~
g
(
x
)=
1
X
k
=1
g
k
X
k
:
ÏðèðàâíèâàÿêîýôôèöèåíòûðàçëîæåíèéâðÿäÔóðüå,ïîëó÷à-
åìñèñòåìóàëãåáðàè÷åñêèõóðàâíåíèéäëÿîïðåäåëåíèÿïîñòî-
ÿííûõ
A
k
,
B
k
A
k
+
B
k
=
f
k
;
A
k
e
p

k
b
+
B
k
e

p

k
b
=
g
k

1
p

k
b
Z
0
w
k
(
s
)
sh
p

k
(
b

s
)
ds:
Ïîäñòàâëÿÿíàéäåííûåçíà÷åíèÿ
A
k
,
B
k
â(5.52),ïîëó÷àåìðå-
øåíèåçàäà÷è(5.45)(5.47).Çàòåì,ñóììèðóÿñ
W
(
x;y
)
ïîëó÷à-
åìðåøåíèåçàäà÷è(5.42)(5.44).
151
5.3.2.Ïîñòðîåíèåðåøåíèéêðàåâûõçàäà÷âêðóãîâûõ
îáëàñòÿõ
ÄâóìåðíîåóðàâíåíèåËàïëàñàâïîëÿðíûõêîîðäèíàòàõ
x
=
r
cos
';y
=
r
sin
'
èìååòâèä
u
rr
+
1
r
2
u
''
+
1
r
u
r
=0
:
(5.53)
Íàéäåìîáùååðåøåíèåýòîãîóðàâíåíèÿïðèóñëîâèè,÷òîîíî
ÿâëÿåòñÿïåðèîäè÷åñêèì,ñïåðèîäîìðàâíûì
2

ïîïåðåìåííîé
':
Èùåìðåøåíèåââèäåïðîèçâåäåíèÿäâóõôóíêöèé
u
(
r;'
)=
R
(
r
)(
'
)
:
Ïîäñòàâëÿåì
u
âóðàâíåíèå(5.53),ïîëó÷àåì
R
00
+
1
r
2
R

00
+
1
r
R
0
=0
:
Óìíîæàÿýòîóðàâíåíèåíà
1

R
,ïîñëåðàçäåëåíèÿïåðåìåííûõ
èìååìäâàóðàâíåíèÿ

00
+

=0
;
(5.54)
r
2
R
00
+
rR
0

R
=0
:
(5.55)
Çàìåòèì,÷òîïðè

0
ôóíêöèÿ
(
'
)
íåÿâëÿåòñÿïåðèîäè÷å-
ñêîé,ïðè
�
0
ðåøåíèåìóðàâíåíèÿ(5.54)ÿâëÿåòñÿôóíêöèÿ

k
=
A
k
cos
p
'
+
B
k
sin
p
';
êîòîðàÿÿâëÿåòñÿïåðèîäè÷åñêîéïðè

=
k
2
,
k
=1
;
2
;:::
Òîãäà

k
=
A
k
cos
k'
+
B
k
sin
k':
Ïðè

=0
óðàâíåíèå(5.54)èìååòâèä

00
=0
.Åãîðåøåíè-
åìÿâëÿåòñÿìíîãî÷ëåíïåðâîéñòåïåíè.Ïîñêîëüêóêîíñòàíòà
152
ÿâëÿåòñÿïåðèîäè÷åñêîéôóíêöèåé,òîñîáñòâåííîìóçíà÷åíèþ

=0
ñîîòâåòñòâóåòñîáñòâåííàÿôóíêöèÿ

0
=
const
.Âîçüì¼ì
ýòóêîíñòàíòóðàâíîéåäèíèöå,òîãäà
jj

0
jj
2
=
2

Z
0
d'
=2
:
Óðàâíåíèå(5.55)ÿâëÿåòñÿóðàâíåíèåìÝéëåðà.Ïðîèçâåä¼ìçà-
ìåíóïåðåìåííûõ
r
=
e
t
,
t
=ln
r
.Âðåçóëüòàòåïîëó÷èìóðàâ-
íåíèå
R
tt

k
2
R
=0
:
Íåòðóäíîóâèäåòü,÷òîôóíêöèè
R
k
(
t
)=

k
e
kt
+

k
e

kt
ÿâëÿþòñÿåãîðåøåíèåì.Ïîñëåîáðàòíîéçàìåíû,íàõîäèì
R
k
(
r
)=

k
r
k
+

k
r

k
:
Ïðè

=0
èìååìóðàâíåíèå
R
tt
=0
.Åãîðåøåíèåìÿâëÿåòñÿ
ìíîãî÷ëåí
a
0
+
b
0
t
.Ïîñëåîáðàòíîéçàìåíû
t
=ln
r
,íàõîäèì
R
0
(
r
)=
a
0
+
b
0
ln
r:
Èòàê,îáùååðåøåíèåóðàâíåíèÿËàïëàñà(5.53)ïðåäñòàâëÿåòñÿ
ââèäåðÿäà
u
(
r;'
)=
1
X
k
=0

k
(
'
)
R
k
(
r
)=
a
0
+
b
0
ln
r
+
1
X
k
=1
sin
k'
(
A
k
r
k
+
B
k
r

k
)+
+
1
X
k
=1
cos
k'
(
C
k
r
k
+
D
k
r

k
)
;
(5.56)
ãäå
a
0
,
b
0
,
A
k
,
B
k
,
C
k
,
D
k
íåèçâåñòíûåïîñòîÿííûå.
153
Ðåøåíèåêðàåâûõçàäà÷äëÿóðàâíåíèÿËàïëàñàâêîëüöå
Âêîëüöå
D
=
f
(
r;'
):
r ;
0
6
'
6
2

g
ðàññìîòðèì
çàäà÷óÄèðèõëåäëÿóðàâíåíèÿËàïëàñà
8









:
u
rr
+
1
r
2
u
''
+
1
r
u
r
=0
;
u
(
;'
)=
f
(
'
)
;
u
(
;'
)=
g
(
'
)
:
(5.57)
Óäîâëåòâîðÿåìðåøåíèå(5.56)êðàåâûìóñëîâèÿì(5.57),ïîëó-
÷àåì
u
(
;'
)=
a
0
+
b
0
ln

+
1
X
k
=1
sin
k'
(
A
k

k
+
B
k


k
)+
+
1
X
k
=1
cos
k'
(
C
k

k
+
D
k


k
)=
f
(
'
)
;
u
(
;'
)=
a
0
+
b
0
ln

+
1
X
k
=1
sin
k'
(
A
k

k
+
B
k


k
)+
+
1
X
k
=1
cos
k'
(
C
k

k
+
D
k


k
)=
g
(
'
)
Ôóíêöèè
f
(
'
)
,
g
(
'
)
ðàçëàãàåìâðÿäÔóðüåïîòðèãîíîìåòðè-
÷åñêîéñèñòåìå
f
1
;
sin
k';
cos
k'
g
f
(
'
)=
f
0
+
1
X
k
=1
f
1
k
sin
k'
+
1
X
k
=1
f
2
k
cos
k';
g
(
'
)=
g
0
+
1
X
k
=1
g
1
k
sin
k'
+
1
X
k
=1
g
2
k
cos
k';
(5.58)
ãäå
154
f
0
=
1
2

2

Z
0
f
(
'
)
d';
f
1
k
=
1

2

Z
0
f
(
'
)sin
k'd';
(5.59)
f
2
k
=
1

2

Z
0
f
(
'
)cos
k'd';
g
0
=
1
2

2

Z
2

g
(
'
)
d';
g
1
k
=
1

2

Z
0
g
(
'
)sin
k'd';
(5.60)
g
2
k
=
1

2

Z
0
g
(
'
)cos
k'd':
Òîãäàäëÿîïðåäåëåíèÿíåèçâåñòíûõêîýôôèöèåíòîâ
a
0
,
b
0
,
A
k
,
B
k
,
C
k
,
D
k
ïîëó÷àåìñèñòåìûàëãåáðàè÷åñêèõóðàâíåíèé

a
0
+
b
0
ln

=
f
0
;
a
0
+
b
0
ln

=
g
0
;

A
k

k
+
B
k


k
=
f
1
k
;
A
k

k
+
B
k


k
=
g
1
k
;

C
k

k
+
D
k


k
=
f
2
k
;
C
k

k
+
D
k


k
=
g
2
k
;
155
ðàçðåøàÿêîòîðûåíàõîäèì
a
0
,
b
0
,
A
k
,
B
k
,
C
k
,
D
k
.Äàëåå,ïîä-
ñòàâëÿåìýòèêîýôôèöèåíòûâ(5.56),ïîëó÷àåìãàðìîíè÷åñêóþ
ôóíêöèþ,óäîâëåòâîðÿþùóþêðàåâûìóñëîâèÿì(5.57).
Ðåøåíèåêðàåâûõçàäà÷äëÿóðàâíåíèÿËàïëàñàâêðóãå
Íóæíîíàéòèâíóòðèêðóãàðàäèóñà
R
ãàðìîíè÷åñêóþ
ôóíêöèþ,óäîâëåòâîðÿþùóþêðàåâîìóóñëîâèþ
u
(
R;'
)=
f
(
'
)
:
(5.61)
Çàìåòèì,ïîñêîëüêóðåøåíèåäàííîéçàäà÷èäîëæíîáûòüîãðà-
íè÷åííûì,òîâôîðìóëå(5.56)êîíñòàíòû
b
0
,
B
k
,
D
k
íóæíî
âçÿòüðàâíûìèíóëþ.ÒîãäàîáùååðåøåíèåóðàâíåíèÿËàïëàñà
âíóòðèêðóãàèìååòâèä
u
(
r;'
)=
a
0
+
1
X
k
=1
sin
k'A
k
r
k
+
1
X
k
=1
cos
k'C
k
r
k
:
(5.62)
Óäîâëåòâîðÿåìåãîêðàåâîìóóñëîâèþïðè
r
=
R
,ïîëó÷àåì
a
0
+
1
X
k
=1
sin
k'A
k
R
k
+
1
X
k
=1
cos
k'C
k
R
k
=
f
(
'
)
:
Ðàçëàãàåìôóíêöèþ
f
(
'
)
âðÿäÔóðüå,ïîñèñòåìå
f
1
;
sin
k';
cos
k'
g
,ïîëó÷àåìðÿä
f
(
'
)=
f
0
+
1
X
k
=1
f
1
k
sin
k'
+
1
X
k
=1
f
2
k
cos
k';
ãäåêîýôôèöèåíòû
f
0
;
f
1
k
;f
2
k
îïðåäåëÿþòñÿïîôîðìóëàì(5.59).
Òîãäà
a
0
=
f
0
;A
k
R
k
=
f
1
k
;C
k
R
k
=
f
2
k
:
Íàõîäèì
a
0
,
A
k
,
C
k
,ïîäñòàâëÿåì(5.62)èïîëó÷àåìðåøåíèå
156
óðàâíåíèåËàïëàñàâíóòðèêðóãà
u
(
r;'
)=
1
2

2

Z
0
f
(
'
)
d'
+
1

1
X
k
=1
sin
k'
2

Z
0
f
(
'
)sin
k'd'

r
R

k
+
+
1

1
X
k
=1
cos
k'
2

Z
0
f
(
'
)cos
k'd'

r
R

k
;
(5.63)
óäîâëåòâîðÿþùååíàãðàíèöå
r
=
R
êðàåâîìóóñëîâèþ(5.61).
Ðåøåíèåêðàåâûõçàäà÷äëÿóðàâíåíèÿËàïëàñàâíåêðóãà
Íóæíîíàéòèâíåêðóãàðàäèóñà
R
ãàðìîíè÷åñêóþôóíê-
öèþ,óäîâëåòâîðÿþùóþêðàåâîìóóñëîâèþ
u
(
R;'
)=
f
(
'
)
:
(5.64)
Ïîñêîëüêóðåøåíèåäàííîéçàäà÷èäîëæíîáûòüîãðàíè÷åííûì
íàáåñêîíå÷íîñòè,òîâôîðìóëå(5.56)êîíñòàíòû
b
0
,
A
k
,
C
k
íóæíîâçÿòüðàâíûìèíóëþ.Òîãäàîáùååðåøåíèåóðàâíåíèå
Ëàïëàññàâíåêðóãàèìååòâèä
u
(
r;'
)=
a
0
+
1
X
k
=1
sin
k'B
k
r

k
+
1
X
k
=1
cos
k'D
k
r

k
:
(5.65)
Óäîâëåòâîðÿåìåãîêðàåâîìóóñëîâèþïðè
r
=
R
,ïîëó÷àåì
a
0
+
1
X
k
=1
sin
k'B
k
R

k
+
1
X
k
=1
cos
k'D
k
R

k
=
f
(
'
)
:
Ðàçëàãàåìôóíêöèþ
f
(
'
)
âðÿäÔóðüåïîñèñòåìå
f
1
;
sin
k';
cos
k'
g
,ïðèðàâíèâàåìêîýôôèöèåíòûðàçëîæå-
íèé,èìååì
a
0
=
f
0
;B
k
R

k
=
f
1
k
;D
k
R

k
=
f
2
k
:
157
f
0
;f
1
k
;f
2
k
îïðåäåëÿþòñÿïîôîðìóëàì(5.59).Íàéäÿ
a
0
;B
k
;D
k
,
ïîäñòàâèìâ(5.65)èïîëó÷èìðåøåíèåóðàâíåíèÿËàïëàñàâíå
êðóãà
u
(
r;'
)=
1
2

2

Z
0
f
(
'
)
d'
+
1

1
X
k
=1
sin
k'
2

Z
0
f
(
'
)sin
k'd'

R
r

k
+
+
1

1
X
k
=1
cos
k'
2

Z
0
f
(
'
)cos
k'd'

R
r

k
;
(5.66)
óäîâëåòâîðÿþùååíàãðàíèöå
r
=
R
êðàåâîìóóñëîâèþ(5.64).
5.4.Âîïðîñûèçàäà÷è
1.Ïîêàæèòå,÷òîôóíêöèÿâèäà
u
(
x;t
)=cos(
at
)(
A
sin(
x
)+
B
cos(
x
))
;
óäîâëåòâîðÿåòóðàâíåíèþ
u
t
=
a
2
u
xx
ïðèïðîèçâîëüíûõçíà÷å-
íèÿõäåéñòâèòåëüíûõïîñòîÿííûõ
A
,
B
è

.
2.Ïîêàæèòå,÷òîïðèëþáûõíàòóðàëüíûõ
n
è
m
1
Z
0
sin(
nx
)sin(
mx
)
dx
=
8



:
0
;m
6
=
n;
1
2
;m
=
n;
1
Z
0
cos(
nx
)cos(
mx
)
dx
=
8



:
0
;m
6
=
n;
1
2
;m
=
n:
158
3.Íàéäèòåðåøåíèåñìåøàííîéçàäà÷è
8











:
u
tt
=
a
2
u
xx
+
x;
0
x
1
;
0
t
1
;
u
(0
;t
)=0
;
0
t
1
;
u
(1
;t
)=0
;
0
t
1
;
u
(
x;
0)=sin(
x
)
;
0
6
x
6
1
;
u
t
(
x;
0)=0
;
0
6
x
6
1
:
4.Íàéäèòåðåøåíèåñìåøàííîéçàäà÷è
8







:
u
tt
=4
u
xx
+
u;
0
x
1
;
0
t
1
;
u
j
x
=0
=0
;u
j
x
=1
=0
;;
0
t
1
;
u
j
t
=0
=0
;u
t
j
t
=0
=1
:;
0
6
x
6
1
:
5.ÍàéäèòåðàçëîæåíèåâðÿäÔóðüåôóíêöèè
'
(
x
)=
x
ïî
òðèãîíîìåòðè÷åñêîéñèñòåìå
f
sin(
nx
)
g
1
n
=1
íàîòðåçêå
[0
;
1]
.
6.Èñïîëüçóÿðåçóëüòàòûçàäà÷è5,íàéäèòåðåøåíèåñìåøàííîé
çàäà÷è
8







:
u
t
=
a
2
u
xx
;
0
x
1
;
0
t
1
;
u
(0
;t
)=0
;
0
t
1
;
u
(1
;t
)=1
;
0
t
1
;
u
(
x;
0)=
x;
0
6
x
6
1
:
7.Íàéäèòåðåøåíèåñìåøàííîéçàäà÷è
8











:
u
t
=
a
2
u
xx
;
0
x
1
;
0
t
1
;
u
(0
;t
)=0
;
0
t
1
;
u
(1
;t
)=0
;
0
t
1
;
u
(
x;
0)=sin(2
x
)+
1
3
sin(4
x
)+
1
5
sin(6
x
)
;
0
6
x
6
1
:
159
8.Íàéäèòåðåøåíèåñìåøàííîéçàäà÷è
8







:
u
t
=
a
2
u
xx
+
x
cos
t;
0
x
1
;
0
t
1
;
u
(0
;t
)=1
;
0
t
1
;
u
(1
;t
)=1
;
0
t
1
;
u
(
x;
0)=sin(
x
)+1
;
0
6
x
6
1
:
9.Âêðóãå
0
6
r
1
íàéäèòåãàðìîíè÷åñêèåôóíêöèè,óäîâëå-
òâîðÿþùèåñîîòâåòñòâåííîçíà÷åíèÿì:
à)
u
(1
;'
)=1+sin
'
+
1
2
cos
';
0
6
'
6
2

;
á)
u
(1
;'
)=5
;
0
6
'
6
2

;
â)
u
(1
;'
)=sin
';
0
6
'
6
2

;
ã)
u
(1
;'
)=sin3
';
0
6
'
6
2
:
10.Âíåêðóãà
0
6
r
6
R
íàéäèòåãàðìîíè÷åñêóþôóíêöèþ,
óäîâëåòâîðÿþùóþãðàíè÷íîìóçíà÷åíèþ
u
(
R;'
)=
'
sin
';
0
6
'
6
2
:
160
Ñïèñîêëèòåðàòóðû
1.
Áèöàäçå,À.Â.Ñáîðíèêçàäà÷ïîóðàâíåíèÿììàòåìàòè÷å-
ñêîéôèçèêè/À.Â.Áèöàäçå,Ä.Ô.Êàëèíè÷åíêî.Ì.:
Íàóêà,1985.
2.
Áèöàäçå,À.Â.Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè/
À.Â.Áèöàäçå.Ì.:Íàóêà;ÔÈÇÌÀÒËÈÒ,1982.
3.
Âàñèëüåâà,À.Á.Äèôôåðåíöèàëüíûåèèíòåãðàëüíûåóðàâ-
íåíèÿ,âàðèàöèîííîåèñ÷èñëåíèåâïðèìåðàõèçàäà÷àõ/
À.Á.Âàñèëüåâà,Ã.Í.Ìåäâåäåâ,Í.À.Òèõîíîâ,Ò.À.Óðàç-
ãèëüäèíà.ÑÏá.:Ëàíü,2010.
4.
Âëàäèìèðîâ,Â.Ñ.Ñáîðíèêçàäà÷ïîóðàâíåíèÿììàòåìà-
òè÷åñêîéôèçèêè/Â.Ñ.Âëàäèìèðîâ.Ì.:Íàóêà;ÔÈÇ-
ÌÀÒËÈÒ,2001.
5.
Âëàäèìèðîâ,Â.Ñ.Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè/
Â.Ñ.Âëàäèìèðîâ,Â.Â.Æàðèíîâ.Ì.:ÔÈÇÌÀÒËÈÒ,
2004.
6.
Êîøëÿêîâ,Í.Ñ.Óðàâíåíèÿâ÷àñòíûõïðîèçâîäíûõìà-
òåìàòè÷åñêîéôèçèêè/Í.Ñ.Êîøëÿêîâ,Ý.Á.Ãëèíåð,
Ì.Ì.Ñìèðíîâ.Ì.:Âûñøàÿøêîëà,1970.
7.
Ìèõàéëîâ,Â.Ï.Äèôôåðåíöèàëüíûåóðàâíåíèÿâ÷àñòíûõ
ïðîèçâîäíûõ/Â.Ï.Ìèõàéëîâ.Ì.:Íàóêà,1983.
8.
Ìîãèëåâñêèé,È.Ø.Ñáîðíèêçàäà÷ïîóðàâíåíèÿìñ÷àñò-
íûìèïðîèçâîäíûìè/È.Ø.Ìîãèëåâñêèé,Ã.Ñ.Øàðîâ.
Òâåðñêîéãîñ.óí-ò.Òâåðü:Èçä-âîÒâåðÃÓ,2004.
9.
Ïåòðîâñêèé,È.Ã.Ëåêöèèîáóðàâíåíèÿõñ÷àñòíûìèïðî-
èçâîäíûìè/È.Ã.Ïåòðîâñêèé.Ì.:ÔÈÇÌÀÒËÈÒ,2009.
161
10.
Ðîìàíêî,Â.Ê.Êóðñäèôôåðåíöèàëüíûõóðàâíåíèéèâàðè-
àöèîííîãîèñ÷èñëåíèÿ/Â.Ê.Ðîìàíêî.Ì.:Ëàáîðàòîðèÿ
áàçîâûõçíàíèé,2001.
11.
Òèõîíîâ,À.Í.Óðàâíåíèÿìàòåìàòè÷åñêîéôèçèêè/
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162
Ó÷åáíîåèçäàíèå
ÏðîêóäèíÄìèòðèéÀëåêñååâè÷,
ÃëóõàðåâàÒàòüÿíàÂëàäèìèðîâíà,
Êàçà÷åíêîÈðèíàÂàëåðüåâíà
ÓÐÀÂÍÅÍÈßÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ
ÔÈÇÈÊÈ
ÐåäàêòîðË.Ã.Áàðàøêîâà
Òåõíè÷åñêèéðåäàêòîðÂ.Ï.Äîëãèõ
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84
1
=
16
.
Ïå÷àòüîôñåòíàÿ.Áóìàãàîôñåòíàÿ1.Ïå÷.ë.10.
Òèðàæ100ýêç.Çàêàç59.
Êåìåðîâñêèéãîñóäàðñòâåííûéóíèâåðñèòåò
650043,ã.Êåìåðîâî,óë.Êðàñíàÿ,6
Îòïå÷àòàíîâòèïîãðàôèè"Ïå÷àòíûéäâîðÊóçáàññà",650000,ã.Êåìåðîâî,
óë.Ìè÷óðèíà,56,òåë.8(384-2)76-58-88.

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