классификация уравнений с двумя независимыми переменными

Ìèíèñòåðñòâî îáðàçîâàíèÿ è íàóêè Ðîññèéñêîé Ôåäåðàöèè
Ôåäåðàëüíîå àãåíòñòâî ïî îáðàçîâàíèþ
Äàëüíåâîñòî÷íûé ãîñóäàðñòâåííûé óíèâåðñèòåò
Ë.À. ÌÎË×ÀÍÎÂÀ
ÊËÀÑÑÈÔÈÊÀÖÈß ÓÐÀÂÍÅÍÈÉ
Ñ ÄÂÓÌß ÍÅÇÀÂÈÑÈÌÛÌÈ
ÏÅÐÅÌÅÍÍÛÌÈ
Ìåòîäè÷åñêèå óêàçàíèÿ äëÿ còóäåíòîâ
ìàòåìàòè÷åñêèõ cïåöèàëüíîcòåé
Âëàäèâîñòîê
Èçäàòåëüñòâî Äàëüíåâîñòî÷íîãî óíèâåðñèòåòà
2006
ÁÁÊ 32.97
Ì75
Ðåöåíçåíòû:
À.Ã. Êîëîáîâ, ê.ô.-ì.í. (ÈÌÊÍ ÄÂÃÓ);
Ì.À Êíÿçåâà, ê.ò.í. (ÈÀÏÓ ÄÂÎ ÐÀÍ).
Ìîë÷àíîâà Ë.À.
Êëàññèôèêàöèÿ óðàâíåíèé ñ äâóìÿ íåçàâèñèìûìè ïåðåìåííûìè. Ó÷åáíî-ìåòîäè÷åñêîå ïîñîáèå. -
M75
Âëàäèâîñòîê: Èçä-âî Äàëüíåâîñò. óí-òà, 2006. - 16 ñ.
Ìåòîäè÷åñêèå óêàçàíèÿ ðàçðàáîòàíû äëÿ ñòóäåíòîâ Èíñòèòóòà ìàòåìàòèêè è êîìïüþòåðíûõ íàóê ÄÂÃÓ. Â
íèõ äàåòñÿ òåîðåòè÷åñêèé ìàòåðèàë, ïîçâîëÿþùèé ñòóäåíòàì
èñïîëüçîâàòü ñðåäñòâà ìàòåìàòè÷åñêîãî ïàêåòà Maple â
ñâîåé ïðàêòè÷åñêîé äåÿòåëüíîñòè ïðè âûïîëíåíèè çàäàíèé
ïî Ïðàêòèêóìó íà ÝÂÌ è ñïåöêóðñó Ïàêåòû ïðèêëàäíûõ
ïðîãðàìì.
Äëÿ ñòóäåíòîâ ìàòåìàòè÷åñêèõ ñïåöèàëüíîñòåé.
2405000000
Ì 180(03)−2006
ÁÁÊ 32.97
c Ìîë÷àíîâà Ë.À., 2006
°
c ÈÌÊÍ ÄÂÃÓ, 2006
°
Ó÷åáíîå èçäàíèå
Ëèëèÿ Àëåêñàíäðîâíà Ìîë÷àíîâà
ÊËÀÑÑÈÔÈÊÀÖÈß ÓÐÀÂÍÅÍÈÉ Ñ ÄÂÓÌß
ÍÅÇÀÂÈÑÈÌÛÌÈ ÏÅÐÅÌÅÍÍÛÌÈ
Ìåòîäè÷åñêèå óêàçàíèÿ äëÿ còóäåíòîâ
ìàòåìàòè÷åñêèõ cïåöèàëüíîcòåé
 àâòîðñêîé ðåäàêöèè
Òåõíè÷åñêèé ðåäàêòîð Ë.Ì. Ãóðîâà
Êîìïüþòåðíûé íàáîð è âåðñòêà àâòîðà
Ïîäïèñàíî â ïå÷àòü 31.01.06
Ôîðìàò 60 × 84 1/16. Óñë. ïå÷. ë. 1,1. Ó÷.-èçä. ë. 0,9.
Òèðàæ 25 ýêç.
Èçäàòåëüñòâî Äàëüíåâîñòî÷íîãî óíèâåðñèòåòà
690950, Âëàäèâîñòîê, óë. Îêòÿáðüñêàÿ, 27.
Îòïå÷àòàíî â ëàáîðàòîðèè
êàôåäðû êîìïüþòåðíûõ íàóê ÈÌÊÍ ÄÂÃÓ
690950, Âëàäèâîñòîê, óë. Îêòÿáðüñêàÿ, 27, ê. 132.
Êëàññèôèêàöèÿ óðàâíåíèé ñ äâóìÿ íåçàâèñèìûìè ïåðåìåííûìè
Ðàññìîòðèì êâàçèëèíåéíîå óðàâíåíèå âòîðîãî ïîðÿäêà ñ äâóìÿ íåçàâèñèìûìè ïåðåìåííûìè
A(x, y)
∂2u
∂2u
∂2u
∂u ∂u
+
2B(x,
y)
+
C(x,
y)
+ Φ(x, y, u,
,
) = 0,
2
2
∂x
∂x∂y
∂y
∂x ∂y
(1)
ãäå êîýôôèöèåíòû A, B è C ñóòü ôóíêöèè îò x è y , èìåþùèå íåïðåðûâíûå
ïðîèçâîäíûå äî âòîðîãî ïîðÿäêà âêëþ÷èòåëüíî. Áóäåì ïðåäïîëàãàòü, ÷òî
A, B è C íå îáðàùàþòñÿ îäíîâðåìåííî â íóëü. Óðàâíåíèþ (1) ñîîòâåòñòâóåò
êâàäðàòè÷íàÿ ôîðìà
g = AX 2 + 2BXY + CY 2 ,
îïðåäåëèòåëü êîòîðîé åñòü
¯
¯A
∆ = ¯¯
B
¯
B ¯¯
= AC − B 2 .
C¯
Ïóñòü ∆ = AC − B 2 > 0 ⇒ A 6= 0. Òîãäà ìîæíî çàïèñàòü
g=
1
[(AX + BY )2 + (AC − B 2 )Y 2 ].
A
(2)
Ôîðìà (2) î÷åâèäíî, îïðåäåëåííàÿ. Ñëåäîâàòåëüíî, óðàâíåíèå ïðèíàäëåæèò ýëëèïòè÷åñêîìó òèïó.
Ïóñòü ∆ = AC − B 2 < 0. Âîçìîæíû äâà ñëó÷àÿ:
1. A 6= 0, òîãäà èìååò ìåñòî ðàâåíñòâî (2), íî â çàâèñèìîñòè îò Y çíàê
g ìåíÿåòñÿ; ôîðìà íåîïðåäåëåííàÿ. Ñëåäîâàòåëüíî, óðàâíåíèå (1) - ãèïåðáîëè÷åñêîãî òèïà.
2. A = 0, òîãäà çàïèøåì g â âèäå g = (2BX + CY )Y . ßñíî, ÷òî g íåîïðåäåëåííàÿ ôîðìà, òèï óðàâíåíèÿ - ãèïåðáîëè÷åñêèé.
Ïóñòü ∆ = AC − B 2 = 0. Òèï óðàâíåíèÿ - ïàðàáîëè÷åñêèé.
Òàêèì îáðàçîì, îêîí÷àòåëüíî ïîëó÷àåì: åñëè ∆ > 0 , òî óðàâíåíèå
ýëëèïòè÷åñêîãî òèïà; åñëè ∆ < 0, òî óðàâíåíèå ãèïåðáîëè÷åñêîãî òèïà;
åñëè ∆ = 0, òî óðàâíåíèå ïàðàáîëè÷åñêîãî òèïà.
3
Ïðåîáðàçîâàíèå óðàâíåíèé âòîðîãî ïîðÿäêà ñ ïîìîùüþ
çàìåíû ïåðåìåííûõ
Âñÿêîå ëèíåéíîå äèôôåðåíöèàëüíîå óðàâíåíèå âòîðîãî ïîðÿäêà ñ äâóìÿ íåçàâèñèìûìè ïåðåìåííûìè ìîæåò áûòü çàïèñàíî â âèäå
A
∂2u
∂2u
∂2u
∂u
∂u
+ 2B
+C 2 +D
+E
+ F u + G = 0,
2
∂x
∂x∂y
∂y
∂x
∂y
(3)
ãäå A, B, C, D, E, F, G - çàäàííûå ôóíêöèè îò x è y (â ÷àñòíîì ñëó÷àå,
ïîñòîÿííûå).
Ïîïûòàåìñÿ óïðîñòèòü ýòî óðàâíåíèå ñ ïîìîùüþ çàìåíû ïåðåìåííûõ
(
ξ = ϕ(x, y);
(4)
η = ψ(x, y).
Çäåñü ξ, η - íîâûå íåçàâèñèìûå ïåðåìåííûå. Ôóíêöèè ϕ è ψ , ñâÿçûâàþùèå íîâûå ïåðåìåííûå ñî ñòàðûìè ïåðåìåííûìè, áóäóò ïîäîáðàíû ïîçäíåå. Ñ÷èòàåì, ÷òî îòîáðàæåíèå (4) ÿâëÿåòñÿ âçàèìíî îäíîçíà÷íûì. Ñäåëàåì òðåáóåìóþ çàìåíó ïåðåìåííûõ:
∂u
∂u ∂ξ
∂u ∂η
=
+
;
∂x
∂ξ ∂x ∂η ∂x
∂u
∂u ∂ξ
∂u ∂η
=
+
;
∂y
∂ξ ∂y
∂η ∂y
∂2u
∂u ∂ 2 ξ
∂u ∂ 2 η h ∂ 2 u ∂ξ 2
∂ 2 u ∂ξ ∂η
∂ 2 u ∂η 2 i
=
+
+
(
)
+
2
+
( ) ;
∂x2
∂ξ ∂x2
∂η ∂x2
∂ξ 2 ∂x
∂ξ∂η ∂x ∂x ∂η 2 ∂x
∂ 2 u ∂ξ ∂η ∂ 2 u ∂η 2 i
∂ 2 u ∂u ∂ 2 ξ ∂u ∂ 2 η h ∂ 2 u ∂ξ 2
=
+
+
(
)
]
+
2
+
( ) ;
∂y 2 ∂ξ ∂y 2 ∂η ∂y 2
∂ξ 2 ∂y
∂ξ∂η ∂y ∂y ∂η 2 ∂y
∂ 2 u ∂u ∂ 2 ξ ∂u ∂ 2 η
=
+
+
∂x∂y ∂ξ ∂x∂y ∂η ∂x∂y
h ∂ 2 u ∂ξ ∂ξ ∂ 2 u ³ ∂ξ ∂η ∂ξ ∂η ´ ∂ 2 u ∂η ∂η i
+
+
+
+ 2
.
∂ξ 2 ∂x ∂y ∂x∂y ∂x ∂y ∂y ∂x
∂η ∂x ∂y
(5)
(6)
(7)
(8)
Ïðàâûå ÷àñòè ôîðìóë (5)-(8) ïðåäñòàâëÿþò ñîáîé ëèíåéíûå ôóíêöèè îòíîñèòåëüíî ÷àñòíûõ ïðîèçâîäíûõ u0ξ , u0η , u00ξξ , u00ηη , u00ξη . Ïîäñòàâëÿÿ u0x ,u0y ,...
èç ýòèõ ôîðìóë â óðàâíåíèå (3), ìû ïîëó÷èì ñíîâà ëèíåéíîå óðàâíåíèå
âòîðîãî ïîðÿäêà ñ íåèçâåñòíîé ôóíêöèåé u è íåçàâèñèìûìè ïåðåìåííûìè
ξèη
∂2u
∂2u
∂2u
∂u ∂u
Ā 2 + 2B̄
+ C̄ 2 + Φ(ξ, η, u,
,
) = 0,
(9)
∂ξ
∂ξ∂η
∂η
∂ξ ∂η
4
ãäå
Ā = A(
B̄ = A
∂ξ 2
∂ξ ∂ξ
∂ξ
) + 2B
+ C( )2 ;
∂x
∂x ∂y
∂y
∂ξ ∂η
∂η ∂ξ
∂ξ ∂η
∂ξ ∂η
+ B(
+
)+C
;
∂x ∂x
∂x ∂y
∂x ∂y
∂y ∂y
C̄ = A(
∂η 2
∂η ∂η
∂η
) + 2B
+ C( )2 ,
∂x
∂x ∂y
∂y
à ôóíêöèÿ Φ ëèíåéíà îòíîñèòåëüíî u, u0ξ , u0η .
Óðàâíåíèå (9) ñòàíîâèòñÿ îñîáåííî ïðîñòûì, åñëè êîýôôèöèåíòû Ā è C̄
îêàæóòñÿ ðàâíûìè íóëþ. Äëÿ òîãî ÷òîáû ïåðâîíà÷àëüíî çàäàííîå óðàâíåíèå (3) ìîæíî áûëî ïðèâåñòè ê òàêîìó ïðîñòîìó âèäó, íàäî â íåì ñäåëàòü
çàìåíó ïåðåìåííûõ (4), ïîäîáðàâ ôóíêöèè ϕ è ψ òàê, ÷òîáû îíè ÿâëÿëèñü
ðåøåíèÿìè óðàâíåíèÿ
A(
∂z 2
∂z ∂z
∂z
) + 2B
+ C( )2 = 0.
∂x
∂x ∂y
∂y
(10)
Ýòî óðàâíåíèå ÿâëÿåòñÿ íåëèíåéíûì óðàâíåíèåì â ÷àñòíûõ ïðîèçâîäíûõ
ïåðâîãî ïîðÿäêà. Ñëåäóþùàÿ òåîðåìà ïîêàçûâàåò, êàê ñâÿçàíû åãî ðåøåíèÿ ñ îáùèì ðåøåíèåì íåêîòîðîãî îáûêíîâåííîãî óðàâíåíèÿ.
Òåîðåìà. Äëÿ òîãî ÷òîáû ôóíêöèÿ z = f (x, y) âî âñåõ òî÷êàõ îáëàñòè Ω óäîâëåòâîðÿëà óðàâíåíèþ (10), íåîáõîäèìî è äîñòàòî÷íî, ÷òîáû
ñåìåéñòâî f (x, y) = const áûëî îáùèì èíòåãðàëîì óðàâíåíèÿ
A(dy)2 − 2Bdxdy + C(dx)2 = 0
(11)
â òîé æå îáëàñòè Ω.
Òåîðåìà îòêðûâàåò ïóòü äëÿ óïðîùåíèÿ èñõîäíîãî óðàâíåíèÿ (3). Äëÿ
ýòîãî ñíà÷àëà ñîñòàâëÿåì âñïîìîãàòåëüíîå óðàâíåíèå (11); îíî íàçûâàåòñÿ
õàðàêòåðèñòè÷åñêèì óðàâíåíèåì äëÿ äàííîãî óðàâíåíèÿ (3). Õàðàêòåðèñòè÷åñêîå óðàâíåíèå åñòü îáûêíîâåííîå äèôôåðåíöèàëüíîå óðàâíåíèå ïåðâîãî ïîðÿäêà, íî âòîðîé ñòåïåíè. Ðàçðåøàÿ åãî îòíîñèòåëüíî ïðîèçâîäíîé
y 0 , ïîëó÷èì äâà óðàâíåíèÿ
√
B + B 2 − AC
0
y =
;
(12)
A
√
B − B 2 − AC
0
y =
;
(13)
A
Åñëè îáùèé èíòåãðàë óðàâíåíèÿ (12) èìååò âèä ϕ(x, y) = const, òî, ïîëàãàÿ
ξ = ϕ(x, y), ìû îáðàùàåì â íóëü êîýôôèöèåíò ïðè ïðîèçâîäíîé u00ξξ . Åñëè
5
ψ(x, y) = const ÿâëÿåòñÿ îáùèì èíòåãðàëîì óðàâíåíèÿ (13), íåçàâèñèìûì
îò èíòåãðàëà ϕ(x, y) = const, òî, ïîëàãàÿ η = ψ(x, y), ìû îáðàòèì â íóëü
òàêæå è êîýôôèöèåíò ïðè ïðîèçâîäíîé u00ηη .
Èíòåãðàëüíûå êðèâûå õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ, òî åñòü âñå êðèâûå, âõîäÿùèå â ñåìåéñòâî ϕ(x, y) = const, ψ(x, y) = const, íàçûâàþòñÿ
õàðàêòåðèñòèêàìè çàäàííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ (3). Â ñâÿçè ñ ýòèì ðàññìàòðèâàåìûé ìåòîä óïðîùåíèÿ óðàâíåíèÿ (3) íàçûâàåòñÿ
ìåòîäîì õàðàêòåðèñòèê.
Óðàâíåíèå ãèïåðáîëè÷åñêîãî òèïà
Ñåìåéñòâî ϕ(x, y) = const è ψ(x, y) = const ìîæíî ðàññìàòðèâàòü, êàê
îáùèå èíòåãðàëû óðàâíåíèÿ (11) - ýòî óðàâíåíèå ðàñïàäàåòñÿ íà äâà óðàâíåíèÿ (12) è (13). Ïðàâûå ÷àñòè óðàâíåíèé (12) è (13) äåéñòâèòåëüíû è ðàçëè÷íû. Ñëåäîâàòåëüíî, ñîãëàñíî óêàçàííîé òåîðåìå, ôóíêöèè z = ϕ(x, y) è
z = ψ(x, y) ÿâëÿþòñÿ ðåøåíèÿìè óðàâíåíèÿ â ÷àñòíûõ ïðîèçâîäíûõ (10).
Ôóíêöèè ϕ(x, y) è ψ(x, y) ëèíåéíî íåçàâèñèìû (ìîæíî äîêàçàòü, ÷òî îïðåäåëèòåëü Âðîíñêîãî îòëè÷åí îò íóëÿ, åñëè AC − B 2 < 0). Ïîýòîìó, âîçâðàùàÿñü ê óðàâíåíèþ (3), ìû ìîæåì ñäåëàòü â íåì çàìåíó ïåðåìåííûõ ïî
ôîðìóëàì (4). Òàê êàê ôóíêöèè ϕ(x, y) è ψ(x, y) óäîâëåòâîðÿþò óðàâíåíèþ
(11), òî â ðåçóëüòàòå ýòîé çàìåíû ïåðåìåííûõ îêàæåòñÿ Ā = 0 è C̄ = 0.
Ñëåäîâàòåëüíî, óðàâíåíèå (3) ïðåîáðàçóåòñÿ ê âèäó
2B̄
∂2u
∂u ∂u
+ Φ(ξ, η, u,
,
) = 0,
∂ξ∂η
∂ξ ∂η
èëè, ïîñëå äåëåíèÿ íà 2B̄ è ïåðåíîñà â äðóãóþ ÷àñòü ðàâåíñòâà, ê âèäó
∂2u
∂u ∂u
= Φ̄(ξ, η, u,
,
).
∂ξ∂η
∂ξ ∂η
(14)
Ïîëó÷åííîå óðàâíåíèå èìååò áîëåå ïðîñòîé âèä, ÷åì èñõîäíîå óðàâíåíèå
(3); åñëè ìû ñìîæåì åãî ðåøèòü, òî äëÿ òîãî, ÷òîáû íàéòè ðåøåíèå èñõîäíîãî óðàâíåíèÿ, äîñòàòî÷íî âåðíóòüñÿ ê ñòàðûì ïåðåìåííûì.
Óðàâíåíèå (14) ïðåäñòàâëÿåò ñîáîé êàíîíè÷åñêóþ ôîðìó óðàâíåíèÿ ãèïåðáîëè÷åñêîãî òèïà. Èíîãäà ïîëüçóþòñÿ äðóãîé êàíîíè÷åñêîé ôîðìîé ãèïåðáîëè÷åñêîãî òèïà. Ñäåëàåì â óðàâíåíèå (14) çàìåíó ïåðåìåííûõ ïî çàêîíó ξ = t + τ, η = t − τ , ãäå t è τ - íîâûå ïåðåìåííûå.  ðåçóëüòàòå ýòîãî
ïðåîáðàçîâàíèÿ óðàâíåíèå (14) ïðèìåò âèä
∂2u ∂2u
− 2 = 4Φ̄.
∂t2
∂τ
6
Ïðèìåð. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó óðàâíåíèå
x2
2
∂2u
2∂ u
−
y
= 0,
∂x2
∂y 2
x > 0,
y > 0,
è íàéòè åãî îáùåå ðåøåíèå.
Ðåøåíèå. Îïðåäåëèì òèï óðàâíåíèÿ. Èìååì
A = x2 ,
B = 0,
C = −y 2 ,
⇒ ∆ = AC − B 2 = −x2 y 2 < 0.
Ñëåäîâàòåëüíî, òèï óðàâíåíèÿ - ãèïåðáîëè÷åñêèé.
Ñîñòàâèì õàðàêòåðèñòè÷åñêîå óðàâíåíèå
x2 dy 2 − y 2 dx2 = 0.
Ýòî óðàâíåíèå ðàñïàäàåòñÿ íà äâà óðàâíåíèÿ ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè
xdy − ydx = 0, xdy + ydx = 0.
Îòñþäà íàõîäèì
y
= C2 .
x
Òàêèì îáðàçîì, ïðÿìûå y = C2 x è ãèïåðáîëû y = C1 /x ÿâëÿþòñÿ õàðàêòåðèñòèêàìè çàäàííîãî óðàâíåíèÿ.
Ââåäåì íîâûå ïåðåìåííûå ïî ôîðìóëàì
xy = C1 ,
ξ = xy,
η=
y
.
x
Èñïîëüçóÿ ôîðìóëû (6) è (7), íàéäåì ∂ 2 u/∂x2 è ∂ 2 u/∂y 2 :
2
∂2u
y2 ∂ 2 u
y2 ∂ 2 u
y ∂u
2∂ u
=
y
−
2
+
+2 3
;
∂x2
∂ξ 2
x2 ∂ξ∂η x4 ∂η 2
x ∂η
2
∂2u
∂2u
1 ∂2u
2∂ u
=
x
+
2
+
.
∂y 2
∂ξ 2
∂ξ∂η x2 ∂η 2
Ïîäñòàâëÿÿ ýòè çíà÷åíèÿ âòîðûõ ïðîèçâîäíûõ â èñõîäíîå óðàâíåíèå, îêîí÷àòåëüíî ïîëó÷èì
∂2u
1 ∂u
−
,
∂ξ∂η 2ξ ∂η
ξ > 0,
η > 0.
Ââåäåì òåïåðü íîâóþ íåèçâåñòíóþ ôóíêöèþ v(ξ, η) = ∂u/∂η. Áóäåì èìåòü
∂v
v
∂v
∂ξ
1
−
=0⇒
=
⇒ ln |v| = ln |ξ| + ln |f (η)|,
∂ξ
2ξ
v
2ξ
2
7
ãäå f (η) - ïðîèçâîëüíàÿ ôóíêöèÿ η . Ïîñëå ïîòåíöèðîâàíèÿ ïîëó÷èì
p
v = f (η) ξ.
Äàëåå
p
p Z
∂u
∂u
= v(ξ, η) ⇒
= f (η) ξ ⇒ u = ξ f (η)dη + ψ(ξ),
∂η
∂η
ãäå ψ(ξ) - ïðîèçâîëüíàÿ ôóíêöèÿ ξ . Îáîçíà÷èâ
Z
ϕ(η) = f (η)dη,
îêîí÷àòåëüíî ïîëó÷èì
u=
p
ξϕ(η) + ψ(ξ),
ãäå ψ(ξ) è ϕ(η) - ïðîèçâîëüíûå ôóíêöèè ξ è η ñîîòâåòñòâåííî.
Âîçâðàùàÿñü ê ñòàðûì ïåðåìåííûì x è y , ïîëó÷èì
u=
√
y
xyϕ( ) + ψ(xy).
x
Óðàâíåíèå ïàðàáîëè÷åñêîãî òèïà
 ýòîì ñëó÷àå óðàâíåíèÿ (12) è (13) ñîâïàäàþò, è ìû ïîëó÷àåì îäèí
îáùèé èíòåãðàë ϕ(x, y) = const, îïðåäåëÿþùèé îäíî ñåìåéñòâî õàðàêòåðèñòèê. Òîãäà ìîæíî ïðèíÿòü ξ = ϕ(x, y), η = ψ(x, y), ãäå ψ(x, y) - ëþáàÿ
ôóíêöèÿ, íåçàâèñèìàÿ îò ôóíêöèè ϕ(x, y), ëèøü áû îíà áûëà äèôôåðåíöèðóåìà íóæíîå ÷èñëî ðàç. Î÷åâèäíî, ïðè âûáðàííîé çàìåíå ïåðåìåííûõ
êîýôôèöèåíò Ā â óðàâíåíèè (9) îáðàùàåòñÿ â íóëü, òî åñòü
∂ξ 2
∂ξ ∂ξ
∂ξ
) + 2B
+ C( )2 = 0.
∂x
∂x ∂y
∂y
√ √
Ñ ó÷åòîì òîãî, ÷òî AC − B 2 = 0 èëè B = A C , ïîñëåäíåå óðàâíåíèå
ìîæíî ïåðåïèñàòü â âèäå
Ā = A(
A(
√ ∂ξ √ ∂ξ 2
∂ξ 2
∂ξ ∂ξ
∂ξ
) + 2B
+ C( )2 = ( A
+ C ) = 0.
∂x
∂x ∂y
∂y
∂x
∂y
Òîãäà B̄ = 0 è óðàâíåíèå (9) ïðèíèìàåò âèä
C̄
∂2u
∂u ∂u
+ Φ(ξ, η, u,
,
) = 0.
2
∂η
∂ξ ∂η
8
Ïîñëå äåëåíèÿ íà C̄ , îêîí÷àòåëüíî ïîëó÷èì
∂2u
∂u ∂u
,
).
= Φ̄(ξ, η, u,
∂η 2
∂ξ ∂η
(15)
Óðàâíåíèå (15) íàçûâàåòñÿ êàíîíè÷åñêîé ôîðìîé óðàâíåíèÿ ïàðàáîëè÷åñêîãî òèïà. Èíòåðåñíî îòìåòèòü, ÷òî åñëè ïðàâàÿ ÷àñòü óðàâíåíèÿ (15)
íå ñîäåðæèò ïðîèçâîäíîé ∂u/∂ξ , òî îíî ñòàíîâèòñÿ îáûêíîâåííûì äèôôåðåíöèàëüíûì óðàâíåíèåì, ãäå ðîëü ïàðàìåòðà èãðàåò ξ .
Ïðèìåð. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó óðàâíåíèå
x2
∂2u
∂2u
∂u
+ 2xy
+ y 2 2 = 0,
2
∂x
∂x∂y
∂y
x > 0,
è íàéòè åãî îáùåå ðåøåíèå.
Ðåøåíèå. Îïðåäåëèì òèï óðàâíåíèÿ. Èìååì
A = x2 ,
B = xy,
C = y2 ,
⇒ ∆ = AC − B 2 = x2 y 2 − x2 y 2 = 0.
Ñëåäîâàòåëüíî, òèï óðàâíåíèÿ - ïàðàáîëè÷åñêèé.
Ñîñòàâèì õàðàêòåðèñòè÷åñêîå óðàâíåíèå
x2 dy 2 − 2xydxdy + y 2 dx2 = 0 ⇒ (xdy − ydx)2 = 0 ⇒ xdy − ydx = 0.
Ðàçäåëÿÿ ïåðåìåííûå è èíòåãðèðóÿ, ïîëó÷èì îäíî ñåìåéñòâî õàðàêòåðèñòèê
y
= C.
x
Ýòî ñåìåéñòâî ïðÿìûõ. Íîâûå ïåðåìåííûå ââîäèì ïî ôîðìóëàì
ξ=
y
x
η = y.
Âûáîð âòîðîé ôóíêöèè ñäåëàí ñ ó÷åòîì òîãî, ÷òîáû îíà áûëà íàèáîëåå
ïðîñòîé, íî òàê, ÷òîáû ôóíêöèîíàëüíûé îïðåäåëèòåëü Îñòðîãðàäñêîãî äëÿ
ïðåîáðàçîâàíèÿ ïåðåìåííûõ áûë îòëè÷åí îò íóëÿ. Íàõîäèì òåïåðü
∂2u
y2 ∂ 2 u
y ∂u
= 4 2 +2 3
;
2
∂x
x ∂ξ
x ∂ξ
y ∂2u
y ∂2u
1 ∂u
∂2u
=− 3 2 − 2
−
;
∂x∂y
x ∂ξ
x ∂ξ∂η x2 ∂ξ
∂2u
1 ∂2u 2 ∂2u
∂2u
= 2 2 +
+ 2.
2
∂y
x ∂ξ
x ∂ξ∂η
∂η
9
Ïîäñòàâëÿÿ ïîëó÷åííûå âûðàæåíèÿ ïðîèçâîäíûõ â èñõîäíîå óðàâíåíèå,
îêîí÷àòåëüíî ïîëó÷èì êàíîíè÷åñêóþ ôîðìó ýòîãî óðàâíåíèÿ
∂2u
= 0.
∂η 2
Ïîëó÷åííîå óðàâíåíèå ëåãêî èíòåãðèðóåòñÿ. Áóäåì èìåòü
Z
∂u
= f (ξ) ⇒ u = f (ξ)dη + ϕ(ξ)
∂η
èëè
u = f (ξ)η + ϕ(ξ),
ãäå f è ϕ - ïðîèçâîëüíûå ôóíêöèè.
Âîçâðàùàÿñü ê ñòàðûì ïåðåìåííûì, ïîëó÷èì îáùåå ðåøåíèå çàäàííîãî
óðàâíåíèÿ
y
y
u = f ( )y + ϕ( ).
x
x
Óðàâíåíèå ýëëèïòè÷åñêîãî òèïà
 ýòîì ñëó÷àå ïðàâûå ÷àñòè óðàâíåíèé (12) è (13) êîìïëåêñíî ñîïðÿæåíû. Ïóñòü ϕ(x, y) - êîìïëåêñíûé èíòåãðàë óðàâíåíèÿ (12), à ϕ∗ (x, y)=const
- èíòåãðàë óðàâíåíèÿ (13), ãäå ϕ∗ (x, y) - ôóíêöèÿ, êîìïëåêñíî ñîïðÿæåííàÿ ñ ôóíêöèåé ϕ(x, y). Åñëè òåïåðü ïåðåéòè ê êîìïëåêñíûì ïåðåìåííûì
ξ = ϕ(x, y), η = ϕ∗ (x, y), òî, ñîãëàñíî îáùåé òåîðèè, óðàâíåíèå (3) ïðèâåäåòñÿ ê âèäó
∂2u
∂u ∂u
= Φ(ξ, η, u,
,
),
∂ξ∂η
∂ξ ∂η
òî åñòü òî÷íî ê òàêîìó æå âèäó, êàê è ãèïåðáîëè÷åñêîå óðàâíåíèå. ×òîáû
îñòàòüñÿ â äåéñòâèòåëüíîé îáëàñòè, ñäåëàåì åùå îäíó çàìåíó ïåðåìåííûõ:
α=
ϕ + ϕ∗
,
2
β=
√
ϕ − ϕ∗
, i = −1,
2i
ãäå α è β - íîâûå ïåðåìåííûå. Òîãäà ξ = α+iβ, η = α−iβ. Ïðè òàêîé çàìåíå
ïåðåìåííûõ Ā = C̄ ,B̄ = 0. Òàêèì îáðàçîì, óðàâíåíèå (3) ïðèâîäèòñÿ ê âèäó
∂2u
∂2u
∂u ∂u
+
= θ(α, β, u,
,
),
2
∂α
∂β 2
∂α ∂β
êîòîðîå íàçûâàåòñÿ êàíîíè÷åñêèì âèäîì óðàâíåíèÿ ýëëèïòè÷åñêîãî òèïà.
10
Ïðèìåð. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó óðàâíåíèå
∂2u
∂2u
∂2u
−
4
+
5
= 0.
∂x2
∂x∂y
∂y 2
Ðåøåíèå. Îïðåäåëèì òèï óðàâíåíèÿ. Èìååì
A = 1,
B = −2,
C=5
⇒ ∆ = AC − B 2 = 5 − 4 = 1 > 0.
Ñëåäîâàòåëüíî, òèï óðàâíåíèÿ - ýëëèïòè÷åñêèé.
Ñîñòàâèì õàðàêòåðèñòè÷åñêîå óðàâíåíèå
dy 2 + 4dxdy + 5dx2 = 0.
Ýòî óðàâíåíèå ðàñïàäàåòñÿ íà äâà ñëåäóþùèõ óðàâíåíèÿ
y 0 = −2 + i,
y 0 = −2 − i.
Èíòåãðèðóÿ èõ, íàéäåì äâà îáùèõ èíòåãðàëà õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ
y = (−2 + i)x + C1 , y = (−2 − i)x + C2 .
Ââåäåì îáîçíà÷åíèÿ
ϕ = y − (−2 + i)x;
ϕ∗ = y + (2 + i)x,
ϕ − ϕ∗
ϕ + ϕ∗
= y + 2x, β =
= −x.
2
2i
Ñîãëàñíî îáùåé òåîðèè, íóæíî ââåñòè íîâûå ïåðåìåííûå α è β ïî ôîðìóëàì
ξ = α + iβ = y + 2x − ix, η = α − iβ = y + 2x + ix.
α=
Âû÷èñëÿÿ ïðîèçâîäíûå ïî ôîðìóëàì (6)-(8), ïîñëå âñåõ ïðåîáðàçîâàíèé
îêîí÷àòåëüíî ïîëó÷èì êàíîíè÷åñêèé âèä çàäàííîãî óðàâíåíèÿ
∂2u ∂2u
+ 2 = 0.
∂ξ 2
∂η
11
Ëèòåðàòóðà
1. Ãîâîðóõèí Â.Í., Öèáóëèí Â.Ã. Êîìïüþòåð â ìàòåìàòè÷åñêîì èññëåäîâàíèè. Ó÷åáíûé êóðñ. - ÑÏá.: Ïèòåð, 2001.
2. Ãîëîñêîêîâ Ä. Ï. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ðåøåíèå çàäà÷
â ñèñòåìå Maple. Ó÷åáíèê äëÿ âóçîâ. -ÑÏá.: Ïèòåð, 2004.
3. Áèöàäçå À.Â., Êàëèíè÷åíêî Ä.Ô. Ñáîðíèê çàäà÷ ïî óðàâíåíèÿì ìàòåìàòè÷åñêîé ôèçèêè. Ì.: Íàóêà, 1977.
4. Áóäàê Á.Ì., Ñàìàðñêèé À.À., Òèõîíîâ À.Í. Ñáîðíèê çàäà÷ ïî ìàòåìàòè÷åñêîé ôèçèêå. Ì.: Íàóêà, 1972.
5. Êîâòàíþê À.Å. Êîíòðîëüíûå ðàáîòû ïî óðàâíåíèÿì ìàòåìàòè÷åñêîé
ôèçèêè. Âëàäèâîñòîê: Äàëüíàóêà, 1999.
6. Ñáîðíèê çàäà÷ ïî ìàòåìàòèêå äëÿ âòóçîâ. Ñïåöèàëüíûå êóðñû. Ì.:
Íàóêà, 1984.
7. Ñìèðíîâ Ì.Ì. Çàäà÷è ïî óðàâíåíèÿì ìàòåìàòè÷åñêîé ôèçèêè. Ì.:
Íàóêà, 1975.
Çàäàíèå
1. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó è íàéòè îáùåå ðåøåíèå:
1. 3uxx + 2uxy − uyy + 2ux + 3uy = 0;
2. uxx + 4uxy + 5uyy + ux + 2uy = 0;
3. uxx + 2uxy + uyy + 3ux − 5uy + 4u = 0;
4. uxx − 2uxy + uyy + 2ux + uy + 4u
5. uxx + 4uxy + 3uyy + 5ux + uy + 4u = 0
6. 2uxx + 2uxy + uyy + 4ux + 4uy + u = 0;
7. uxx − 2uxy − 3uyy = 0;
8. 3uxx − 5uxy − 2uyy + 3ux + uy = 2;
9. uxx − 4uxy + 5uyy = 0;
10. uxx + 2uxy − 3uyy + 2ux + 6uy = 0;
11. uxx + 4uxy + 5uyy + ux + 2uy = 0;
12. uxx − 2uxy + uyy + αux + βuy + cu = 0;
13. 2uxx + 3uxy + uyy + 7ux + 4uy = 0;
14. uxx + 2uxy + 5uyy − 32uy = 0;
15. uxx − 2uxy + uyy + 9ux + 9uy = 0.
12
2. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó:
1. y 2 uxx + x2 uyy = 0;
2. x2 uxx + y 2 uyy = 0;
3. sgn(y)uxx + 2uxy + uyy = 0, x > 0, y < 0;
4. yuxx + xyuyy = 0, x > 0, y > 0;
5. yuxx + xuyy = 0, x < 0, y > 0;
6. yuxx + xuyy = 0, x > 0, y > 0;
7. yuxx + xuyy = 0, x < 0, y > 0;
8. uxx + yuyy + 0.5uy = 0, y > 0;
9. uxx + yuyy = 0, y > 0;
10. uxx + xuyy = 0, x > 0;
11. uxx − 2 sin(x)uxy − cos2 (x)uyy − cos(x)uy = 0;
12. x2 uxx − y 2 uyy − 2yuy = 0;
13. y 2 uxx + 2xyuxy + 2x2 uyy + yuy = 0;
14. x2 uxx + 2xyuxy + y 2 uyy = 0;
15. yuxx − xuyy = 0, x < 0, y < 0.
Ïðèìåðû ðåøåíèÿ òèïîâûõ çàäà÷ â Maple
Ïðèìåð 1. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó óðàâíåíèå
x2 uxx − y 2 uyy = 0, x > 0, y > 0,.
Ðåøåíèå. Áóäåì ðàññìàòðèâàòü óðàâíåíèå îáùåãî âèäà:
a1 uxx + a2 uxy + a3 uyy + a4 ux + a5 uy + a6 u + a7 = 0.
Çàäàäèì êîýôôèöèåíòû íàøåãî óðàâíåíèÿ
>restart:with(linalg):
> a:=x^2,0,-y^2,0,0,0,0:assume(x>0,y>0):
è ñàìî óðàâíåíèå:
>equ:=a[1]*diff(u(x,y),x,x)+a[2]*diff(u(x,y),x,y)+
>a[3]*diff(u(x,y),y,y)+a[4]*diff(u(x,y),x)+
>a[5]*diff(u(x,y),y)+a[6]*u(x,y)+a[7]=0;
equ := x ˜2 (
∂2
∂2
2
u(x ˜, y˜)) = 0
2 u(x ˜, y˜)) − y˜ (
∂x ˜
∂y˜2
Âû÷èñëÿåì ìàòðèöó ñòàðøèõ êîýôôèöèåíòîâ è åå îïðåäåëèòåëü:
>eq:=lhs(equ): >A:=matrix(2,2,[coeff(eq,diff(u(x,y),x,x)),
13
>coeff(eq,diff(u(x,y),x,y))/2,coeff(eq,diff(u(x,y),x,y))/2,
>coeff(eq,diff(u(x,y),y,y))]);
>Delta:=simplify(det(A));b:=signum(%);
·
¸
x ˜2
0
A :=
0
−y˜2
∆ := −x˜2 y˜2
Îïðåäåëÿåì òèï óðàâíåíèÿ è ïå÷àòàåì åãî:
>op(b): eq_type:="ele": if b=-1 then eq_type:='gip':
>elif b=0 then eq_type:="par":
>else b:=1:
>end if:
>print(eq_type,b);
gip, -1
Ôîðìèðóåì õàðàêòåðèñòè÷åñêîå óðàâíåíèå è ðåøàåì åãî:
>P:=A[1,1]*z^2-2*A[1,2]*z+A[2,2]:res1:=solve(P=0,z):
>simplify({solve(P,z)}):simplify(%,power):
>res1:=subs(y=y(x),%):
>if b=-1 then
>res2:={seq(dsolve(diff(y(x),x)=res1[i],y(x)),i=1..nops(%))}:
>r:={seq(solve(res2[i],_C1),i=1..nops(res2))}:
>subs(y(x)=y,r):
>itr:={xi=(%[2]),eta=(%[1])}:
>end if:
>if b=1 then mu:=1/select(has,%[1],y):
>{seq(int(expand((diff(y(x),x)+%%[i])*mu),x)=_C1,i=1..nops(%%))}:
>r:={seq(solve(%[i],_C1),i=1..nops(res2))}:
>subs(y(x)=y,r):
>itr:={xi=coeff(%[1],I),eta=%[1]-coeff(%[1],I)*I}:
>end if:
>if b=0 then
>if has(res1[1],_C1)=false then subs(y(x)=y,res1[1]):
>else
>subs(y=y(x),res1[1]):
>es2:=dsolve(diff(y(x),x)=%,y(x)): r:=solve(res2,_C1):
>subs(y(x)=y,r): end if:
>itr:={xi=%,eta=y}:
14
>end if:
y˜
, η = y˜ x ˜}
x˜
Ïðèâîäèì çàäàííîå óðàâíåíèå ê êàíîíè÷åñêîé ôîðìå:
>simplify(itr);with(PDEtools):tr:=solve(itr,{x,y}):
>if has(%,RootOf) then tr:=allvalues(%)[1] end if:
>dchange(tr,eq,itr,[eta,xi],simplify)=0;
{ξ =
−2 ξ (2 (
∂2
∂
u(η, ξ)) η − ( u(η, ξ))) = 0
∂ξ ∂η
∂ξ
Ïðèìåð 2. Ïðèâåñòè ê êàíîíè÷åñêîìó âèäó óðàâíåíèå
uxx − 2uxy − 3uyy = 0
è íàéòè åãî îáùåå ðåøåíèå.
Ðåøåíèå. Âîñïîëüçóåìñÿ ñòàíäàðòíîé ïðîãðàììîé mapde(eq,canom) èç
ïàêåòà PDTools.
>restart:a:=1,-2,-3,0,0,0:
Çàäàäèì óðàâíåíèå:
>equ:=a[1]*diff(u(x,y),x,x)+a[2]*diff(u(x,y),x,y)+
> a[3]*diff(u(x,y),y,y)+
>a[4]*diff(u(x,y),x)+a[5]*diff(u(x,y),y)+a[6]*u(x,y)=0;
equ := (
∂2
∂2
∂2
u(x, y)) − 2 (
u(x, y)) − 3 ( 2 u(x, y)) = 0
2
∂x
∂y ∂x
∂y
>with(PDEtools):p1:=mapde(equ,canom);
p1 :=
√
16 (
∂2
x y
u(_ξ1, _ξ2)) &where {_ξ1 = 3 x + y, _ξ2 = − }
∂ _ξ2 ∂ _ξ1
4
4
>op(%);
√
16 (
∂2
x y
u(_ξ1, _ξ2)), {_ξ1 = 3 x + y, _ξ2 = − }
∂ _ξ2 ∂ _ξ1
4
4
Íàéäåì îáùåå ðåøåíèå:
>pdsolve(%[1]);
u(_ξ1, _ξ2) = _F 2(_ξ1) + _F 1(_ξ2)
Âåðíåìñÿ ê ñòàðûì ïåðåìåííûì:
>sol:=u(x,y)=subs(%%[2],rhs(%));
sol := u(x, y) = _F 2(3x + y) + _F 1( x4 − y4 )
Ïðîâåðèì íàéäåííîå ðåøåíèå:
>simplify(subs(sol,equ)):simplify(lhs(%));
0
15