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.  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).  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