Диссертация (1136178), страница 38
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Òîãäà èç ôîðìóë (4.128) (4.135), (4.142) âûòåêàåòÒåîðåìà 4.4.Àñèìïòîòè÷åñêèé ýéðè-ïîëÿðîíðÿåò óðàâíåíèþ(0.45)(0.49)óäîâëåòâî-c òî÷íîñòüþ O(ξ −3/2 ) ïðè ξ → +∞.Òàêèì îáðàçîì, äëÿ êàæäîãî ðåøåíèÿt, ρçàäà÷è íà ñîáñòâåí-íûå çíà÷åíèÿ (0.46), (0.47) ïîñòðîåíî îäíîïàðàìåòðè÷åñêîå ñåìåéñòâî (ïàðàìåòðîì ÿâëÿåòñÿk)àñèìïòîòè÷åñêèõ ðåøåíèéGóðàâ-íåíèÿ (0.45) äëÿ ýéðè-ïîëÿðîíà. Íàëè÷èå ñâîáîäíîãî ïàðàìåòðàñâÿçàíî ñ òåì, ÷òî íà ôóíêöèþGkçäåñü íå íàêëàäûâàëîñü óñëîâèåíîðìèðîâêè.1.6.Àñèìïòîòè÷åñêèå ðàçëîæåíèÿ ýéðè-ïîëÿðîíà ïðèξ → −∞Ïåðåéäåì ê íàõîæäåíèþ àñèìïòîòèêè äëÿ ðàññìîòðåííîãî âûøå ñåìåéñòâà ðåøåíèé óðàâíåíèÿ (0.45) â îáëàñòèξ → −∞,ãäå0<ε<1(4.143) ïðîèçâîëüíàÿ êîíñòàíòà.ÏîñêîëüêóZZη = O(|ξ|1−ε ),G(ξ, η)∞ln−∞ýêñïîíåíöèàëüíî óáûâàåò ïðèh (ξ − ξ 0 )2 + (η − η 0 )2 i(ξ 0 )2ξ → −∞,G2 (ξ 0 , η 0 ) dη 0 dξ 0 =òî328∞ZZ ∞ Z ∞|ξ| ∞ 2 0 0(η − η 0 )2=2 ln 1 + 0 G (ξ , η ) dη 0 dξ 0 +×0 )2ξ(|ξ|+ξα−∞−∞−|ξ|−∞ZZ ∞∞(η − η 0 )4 2 0 0 0 0 2 0 00 0G (ξ , η )dη dξ , ξ → −∞.×G (ξ , η )dη dξ + O0 4−|ξ|α −∞ (|ξ| + ξ )Z(4.144)Çäåñü0 < α < 1.
Ðàçëîæåíèÿ G ïðè ξ → +∞ ïîçâîëÿþò âû÷èñëèòüàñèìïòîòèêó âõîäÿùèõ â (4.144) èíòåãðàëîâ.Òàê êàêZ0∞ √x√ ∞ πdx√ =+ arctg x = ,01+x2(1 + x)2 xòî â ñèëó (4.15), (4.18), (4.32) èìååìZ∞−|ξ|α1(|ξ| + ξ 0 )2Z∞20000ZG (ξ , η ) dη dξ =−∞0∞k1√dξ 0 +020(|ξ| + ξ ) 2 ξ 1 Z |ξ|α1ln ξ 00+O+Odξ +|ξ|2(|ξ| + ξ 0 )2 (ξ 0 )3/21 1 Z ∞ln ξ 0kπ10dξ =+O,(4.145)+O0 20 3/2|ξ|24|ξ|3/2|ξ|α (|ξ| + ξ ) (ξ )Z ∞Z ∞ 1 10 2 0 00 0η G (ξ , η ) dη dξ = O,(4.146)0 2|ξ|2−|ξ|α (|ξ| + ξ )−∞Z ∞Z ∞1(η 0 )2 G2 (ξ 0 , η 0 ) dη 0 dξ 0 =02−|ξ|α (|ξ| + ξ )−∞Z ∞ 1 1 1k 1/30=Odξ + O=O, (4.147)(|ξ| + ξ 0 )2 2(ξ 0 )1/6|ξ|2|ξ|7/60Z ∞ Z ∞ η4 1 (η − η 0 )4 2 0 00 0G (ξ , η ) dη dξ = O+O.0 )47/217/6(|ξ|+ξα|ξ||ξ|−|ξ|−∞(4.148)Äàëåå âû÷èñëèì àñèìïòîòèêó èíòåãðàëàdefE =2Z|ξ| ∞ 2 0 0ln 1 + 0 G (ξ , η ) dη 0 dξ 0 .ξ−∞−∞Z∞329Ëåììà 4.19.Ïðè ξ → −∞ ñïðàâåäëèâî ðàçëîæåíèåE = U (|ξ|) + O(1/|ξ|),(4.149)ãäåU (|ξ|) = 2πkpln |ξ||ξ| + 2A−1 ln |ξ| − 2A0 + 2πkA−1 p +|ξ|A0 4πk6πρk 5/3 k 2 π 2 A−1 ln |ξ|p −++ A−1 −.2|ξ|5|ξ|5/6|ξ|(4.150)Äîêàçàòåëüñòâî.
Èíòåãðèðóÿ ïî ÷àñòÿì, à òàêæå èñïîëüçóÿ ðàâåíñòâà [21]∞Z0xβ−1πdx =,1+xsin βπèìååìZ0Z∞∞Z0 < β < 1,0ln 1 +∞ln xdx = 0,1 + x21 dx √ = 2π,x xZ ∞ln x dxln |1 + x| 3/2 = 4π,x0(4.151)dx= 2π,3/2x0Z ∞Z ∞dx12πdxln |1+x| 11/6 =,ln |1+x| 2 = ln |ξ|+O(1), ξ → −∞,5xx0|ξ|−1Z ∞ ξ 0 0ln 1 + χ(ξ , |ξ|α )W (ξ 0 ) dξ 0 = O(|ξ|−∞ ),ξ → −∞.|ξ|α|ξ|ln |1 + x|Ñëåäîâàòåëüíî,∞ξ 0 ln 1 + K(ξ 0 , |ξ|α , k) dξ 0 +|ξ|−∞ZpE = 2πk |ξ| + 2A−1 ln |ξ| − 2A0 + 24πln |ξ|2π6π ρk 2/3 i+k A−1 p + 2πA−1 p − A0 p −+5 |ξ|5/6|ξ||ξ||ξ|1k 2 π 2 A−1+ln |ξ| + O,ξ → −∞,|ξ||ξ|h330ãäå ôóíêöèÿKçàäàåòñÿ ôîðìóëîé (4.119).
Íàêîíåö, èñïîëüçóÿ(4.121), (4.122), (4.112), ïîëó÷àåì∞1ξ 0 0α0ln 1 + K(ξ , |ξ| , k) dξ = O.|ξ||ξ|−∞ZËåììà äîêàçàíà.Èç óðàâíåíèÿ (0.45) è ðàçëîæåíèé (4.144) (4.149) âûòåêàåò,÷òî â îáëàñòè (4.143) 1 πk η 2 η2 ∂ 2G n∂ 2G++ − |ξ| − U (|ξ|) + O−++O∂|ξ|2∂η 2|ξ|4 |ξ|3/2|ξ|2 η 4 oG = 0.+O|ξ|7/2(4.152)Àñèìïòîòè÷åñêèå ðåøåíèÿ óðàâíåíèÿ (4.152) áóäåì èñêàòü â âèäå η2 η 4 iγη 2G(ξ, η) = exp − Ω(|ξ|) ++O+O.|ξ||ξ|3|ξ|3/2hÇäåñüγ êîíñòàíòà, àÒåîðåìà 4.5.Ω(|ξ|) = 2|ξ|3/2 /3 + O(|ξ|), ξ → −∞. îáëàñòèòè÷åñêèå ðåøåíèÿ âèäà(4.153)(4.143)(4.153),óðàâíåíèå(0.45)èìååò àñèìïòî-ãäå γ = πk/8, à ôóíêöèÿ Ω çàäàåò-ñÿ ôîðìóëîépp2Ω(|ξ|) = |ξ|3/2 + πk|ξ| + 2A−1 |ξ|(ln |ξ| − 2) − (2A0 + π 2 k 2 ) |ξ|+31 π3k3 9πρk 5/3 A2−1+ 2πkA−1 + ++ p (ln2 |ξ|+4 ln |ξ|)−ln |ξ|+δ1 +1/3425|ξ||ξ| 1 ln |ξ|2 2−2(π k + A0 )A−1 p + O p,ξ → −∞.(4.154)|ξ||ξ|Çäåñü δ1 = δ1 (k) íå çàâèñèò îò ξ .Äîêàçàòåëüñòâî.
Äèôôåðåíöèðóÿ (4.153), èìååì: 1 o∂ 2G n η2 = O+OG,∂η 2|ξ|2|ξ|n2∂ 2G0=Ω(|ξ|)+∂|ξ|2331 η 4 Ω0 (|ξ|) o η 2 Ω0 (|ξ|) γη 200+2Ω (|ξ|) 2 − Ω (|ξ|) + O+OG.|ξ||ξ|4|ξ|5/20ÏîñêîëüêópΩ (|ξ|) = |ξ| + O(1),011Ω (|ξ|) = p + o,|ξ|2 |ξ|00ξ → −∞,òî óðàâíåíèå (4.152) ïðèâîäèò ê ðàâåíñòâó2πk η 210p2γ −−|ξ|−U(|ξ|)+Ω(|ξ|)−+4 |ξ|3/22 |ξ|1 η2 η4 = 0,+O+O+O|ξ||ξ|2|ξ|7/2îòêóäàγ = πk/8,à äëÿ îïðåäåëåíèÿΩ(|ξ|)ïîëó÷àåì óðàâíåíèå121pΩ (|ξ|) = |ξ| + U (|ξ|) ++O,|ξ|2 |ξ|0ãäå ôóíêöèÿU (|ξ|)(4.155)çàäàåòñÿ ôîðìóëîé (4.150).Èç (4.155) ïî ôîðìóëå Òåéëîðà íàõîäèìpπ2k2 1A−1 ln |ξ| p +− A0 +Ω (|ξ|) = |ξ| + πk + p2|ξ||ξ|01 π3k3 13πρk 5/3 A2−1 ln2 |ξ|+ 2πkA−1 + +−−+42 |ξ|5|ξ|4/32|ξ|3/2 1 ln |ξ|2 2+(π k + A0 )A−1 3/2 + O,ξ → −∞.(4.156)|ξ||ξ|3/2Íàêîíåö, èíòåãðèðóÿ (4.156), ïîëó÷àåì (4.154).
Òåîðåìà äîêàçàíà.Çàìå÷àíèå4.12. Ïîñëå ïîñòðîåíèÿ àñèìïòîòèêG(ξ, η)â îáëàñòÿõ(0.48) è (4.143) âîçíèêàåò èíòåðåñíàÿ ïðîáëåìà íàõîæäåíèÿ ôîðìóëñâÿçè ìåæäó êîíñòàíòàìè â ýòèõ ðàçëîæåíèÿõ.3321.7.Î ðàçðåøèìîñòè óðàâíåíèÿ â âàðèàöèÿõ äëÿîäíîìåðíîãî ïîëÿðîíà. Ôîðìóëà äëÿ ðåøåíèÿÐàññìîòðèì óðàâíåíèånbp = f,(4.157)f = f (τ ) íåïðåðûâíàÿ, ýêñïîíåíöèàëüíî óáûâàþùàÿ ïðè|τ | → ∞ ôóíêöèÿ, à îïåðàòîð nb çàäàí ôîðìóëîé (4.65). Âõîäÿùèå âîïåðàòîð nb ôóíêöèÿ t è ÷èñëî ρ ÿâëÿþòñÿ ðåøåíèåì çàäà÷è (0.46),ãäå(0.47) íà ñîáñòâåííûå çíà÷åíèÿ.Èçó÷èì ðàçðåøèìîñòü óðàâíåíèÿ (4.157).
Îãðàíè÷èìñÿ ëèøüôîðìóëèðîâêîé ðåçóëüòàòîâ, áîëüøèíñòâî èç êîòîðûõ äîêàçûâàåòñÿíåïîñðåäñòâåííûì äèôôåðåíöèðîâàíèåì.Îïðåäåëèìdefg(τ ) = 2πZ∞|τ − τ 0 |t(τ 0 )p(τ 0 ) dτ 0 .−∞Òîãäà ïîñëå çàìåíûp = g 00 /(4πt)(4.158)g ïîëó÷àåì ëèíåéíîå äèôôåðåíöèàëüíîå óðàâíåíèåäëÿ íàõîæäåíèÿ4-ãî ïîðÿäêà g 00 00 t00 g 00−− tg = f,4πt4πt2(4.159)[g − τ g 0 ](−∞) + [g − τ g 0 ](+∞) = 0,(4.160)g 0 (−∞) + g 0 (+∞) = 0.(4.161)à òàêæå óñëîâèÿÑîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå èìååò âèä g 00 00 t00 g 00−− tg = 0,4πt4πt2(4.162)333à ñîïðÿæåííîå ê (4.162) óðàâíåíèå t00 ϕ 00−− tϕ = 0.4πt 4πt2 ϕ00Îòìåòèì, ÷òî åñëègÿâëÿåòñÿ ðåøåíèåì (4.162), òî(4.163)ϕ = g 00 /(4πt)óäîâëåòâîðÿåò (4.163).Èç ñîîòíîøåíèÿ (4.34) âûòåêàåòËåììà 4.20.Ïóñòü g1 = (t00 /t)0 , g2 = τ (t00 /t)0 /2 + t00 /t, à g3 , g4 ëèíåéíî íåçàâèñèìûå ðåøåíèÿ óðàâíåíèÿhd3i[2πt4 − 3tt0 (t00 /t)0 /2]0 2000g −ln (t ) − t t g +g = 0.dτ2(t00 t − 3(t0 )2 /2)00(4.164)Òîãäà g1 , g2 , g3 , g4 îáðàçóþò ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèé(ô.ñ.ð.) óðàâíåíèÿ(4.162).Ñîîòâåòñòâåííî, åñëè ϕ1 = t0 , ϕ2 =t + τ t0 /2, à ϕ3 , ϕ4 ëèíåéíî íåçàâèñèìûå ðåøåíèÿ óðàâíåíèÿndh 3π 23 0 2 io 000ϕ −ln+ 4π t t − (t )ϕ+dτ2200+[−3π 2 t00 /(2t) − 8π(t00 )2 + 6πt0 t000 ] ϕ = 0,[3π 2 /2 + 4π t00 t − 3(t0 )2 /2 ]òî ϕ1 , ϕ2 , ϕ3 , ϕ4 îáðàçóþò ô.ñ.ð.
óðàâíåíèÿ(4.165)(4.163).Ââåäåì îïåðàòîðûb def=Kb defE=(t000 t − 2t00 t0 ) d(2πt3 − 3t0 (t00 /t)0 /2)−,t(t00 t − 3(t0 )2 /2) dτ(t00 t − 3(t0 )2 /2)(t000 t − 2t00 t0 )d(3t0 (t00 /t)0 /2 − (t00 )2 /t)−.t[3π/8 + tt00 − 3(t0 )2 /2] dτ[3π/8 + tt00 − 3(t0 )2 /2]Ñ èõ ïîìîùüþ óðàâíåíèÿ (4.164), (4.165) ìîãóò áûòü çàïèñàíû âñèììåòðè÷íîé ôîðìåËåììà 4.21.b Kg)b − 4πg = 0, K(b Eϕ)b − 4πϕ = 0.E(Åñëèg30 g4−g40 g33 0 232 00=t t − (t ) ,32334òî ôóíêöèè ϕ3 = g300 /(4πt), ϕ4 = g400 /(4πt) èìåþò âðîíñêèàíϕ04 ϕ3−ϕ03 ϕ43 0 28 3π00+ t t − (t )=3π 82è ñâÿçàíû ñ g3 , g4 ñîîòíîøåíèÿìèϕ3 =1 bKg3 ,4πÇàìå÷àíèåϕ4 =1 bKg4 ,4πb 3,g3 = Eϕb 4.g4 = Eϕ4.13.
Èç (4.34) âûòåêàåò íåðàâåíñòâît00 t − (t0 )2 < 0.(4.166)Ïîýòîìó êîýôôèöèåíòû óðàâíåíèÿ (4.165), à òàêæå îïåðàòîðàb,Kíåïðåðûâíû.Íåïîñðåäñòâåííûì äèôôåðåíöèðîâàíèåì ïðîâåðÿåòñÿ, ÷òî îáùåå ðåøåíèå óðàâíåíèÿ (4.159) èìååò âèäg(τ ) =4Xi=108n− g1 (τ )ci gi (τ ) +3π×f (τ ) dτ0oZZτ00− g3 (τ )τ0τ00Z0ci , i = 1, 2, 3, 4èτ0τϕ4 (τ )f (τ ) dτ + g4 (τ )τ0ãäåϕ1 (τ 0 )×ϕ2 (τ )f (τ )dτ + g2 (τ )τ0τZ0ϕ3 (τ 0 )f (τ 0 ) dτ 0 ,τ0 ïðîèçâîëüíûå êîíñòàíòû. Äàëåå, òàê êàê[g2 −τ g20 ](−∞)+[g2 −τ g20 ](+∞) = 2ρ 6= 0,g20 (−∞)+g20 (+∞) = 0,òî óñëîâèå (4.160) âûïîëíÿåòñÿ çà ñ÷åò âûáîðà êîíñòàíòûïîëíåíèå æå óñëîâèÿ (4.161) ñâÿçàíî ñ ïîâåäåíèåì íà±∞c2 .Âû-ðåøåíèéóðàâíåíèÿ (4.165). Çäåñü âîçìîæíû äâà ñëó÷àÿ â çàâèñèìîñòè îò âûáîðà âõîäÿùèõ â (4.165) ðåøåíèé (0.46), (0.47). ñëó÷àå 1 ýêñïîíåíöèàëüíî óáûâàþùåå íà−∞ϕ3+∞ (ϕ3 ∼ðåøåíèå∼ c− t, τ → −∞) ýêñïîíåíöèàëüíî âîçðàñòàåò íà√c+ /( τ t), τ → +∞).
Çäåñü c− , c+ íåêîòîðûå êîíñòàíòû. Òîãäà√√ϕ4 ýêñïîíåíöèàëüíî âîçðàñòàåò íà −∞ (ϕ4 ∼ −1/(2 πc− −τ t),(ϕ3335τ → −∞)τ → +∞).è ýêñïîíåíöèàëüíî óáûâàåò íàϕ3 ñëó÷àå 2 ðåøåíèå√+∞ (ϕ4 ∼ −t/(2 πc+ ),ýêñïîíåíöèàëüíî óáûâàåò íàýêñïîíåíöèàëüíî âîçðàñòàåò íà±∞,àϕ4±∞. ñëó÷àå 1 äëÿ âûïîëíåíèÿ (4.161) äîñòàòî÷íî îäíîãî óñëîâèÿðàçðåøèìîñòèZ∞t0 (τ )f (τ ) dτ = 0.(4.167)−∞Òîãäàg(τ ) = c1 g1 (τ ) + c2 g2 (τ )+Z τZ τo8n000000− g1 (τ )ϕ2 (τ )f (τ ) dτ + g2 (τ )ϕ1 (τ )f (τ ) dτ ++3π−∞−∞Z ∞Z τ000ϕ4 (τ )f (τ ) dτ + g4 (τ )ϕ3 (τ 0 )f (τ 0 ) dτ 0(4.168)+g3 (τ )−∞τóäîâëåòâîðÿåò çàäà÷å (4.159) (4.161).
Ïîäñòàâëÿÿ (4.168) â (4.158),íàõîäèì, ÷òîZp(τ ) = c1 ϕ1 (τ ) + c2 ϕ2 (τ )+Zτϕ2 (τ 0 )f (τ 0 ) dτ 0 + ϕ2 (τ )8n+− ϕ1 (τ )3π−∞Z ∞Z000+ϕ3 (τ )ϕ4 (τ )f (τ ) dτ + ϕ4 (τ )c1oϕ1 (τ )f (τ ) dτ +000−∞τϕ3 (τ 0 )f (τ 0 ) dτ 0 ,(4.169)−∞τãäåτ ïðîèçâîëüíàÿ êîíñòàíòà, àc2îïðåäåëÿåòñÿ èç óñëîâèÿ(4.160).Ëåììà 4.22.Åñëè óðàâíåíèå(4.165)íå èìååò ýêñïîíåíöèàëüíîóáûâàþùèõ íà ±∞ íåòðèâèàëüíûõ ðåøåíèé, òî äëÿ ðàçðåøèìîñòè óðàâíåíèÿâèÿ(4.167).(4.157)íåîáõîäèìî è äîñòàòî÷íî âûïîëíåíèÿ óñëî-Òîãäà äëÿ ðåøåíèÿ(4.157)ñïðàâåäëèâà ôîðìóëà(4.169).Àíàëîãè÷íî ðàññìàòðèâàåòñÿ è ñëó÷àé 2, êîãäà ïîëó÷àåì óæåäâà óñëîâèÿ ðàçðåøèìîñòè óðàâíåíèÿ (4.157)Z∞0Z∞t (τ )f (τ ) dτ = 0,−∞ϕ3 (τ )f (τ ) dτ = 0,−∞336à ôóíêöèÿp(τ ) = c1 ϕ1 (τ ) + c2 ϕ2 (τ ) + c3 ϕ3 (τ )+Z τZ τo8n000000+− ϕ1 (τ )ϕ2 (τ )f (τ ) dτ + ϕ2 (τ )ϕ1 (τ )f (τ ) dτ −3π−∞−∞Z τZ τ000−ϕ3 (τ )ϕ4 (τ )f (τ ) dτ + ϕ4 (τ )ϕ3 (τ 0 )f (τ 0 ) dτ 0−∞τ0ñîäåðæèò äâå ïðîèçâîëüíûå êîíñòàíòûc1 , c3 . çàêëþ÷åíèå, èçó÷èì ïîâåäåíèå ðåøåíèé óðàâíåíèÿ (4.165)ïðè óñëîâèè, ÷òîËåììà 4.23.t 6= 0.Åñëè t 6= 0, òî óðàâíåíèåíå èìååò ýêñïîíåí-(4.165)öèàëüíî óáûâàþùèõ íà ±∞ íåòðèâèàëüíûõ ðåøåíèé.Äîêàçàòåëüñòâî.
Ïðåäïîëîæèì, ÷òî èìååò ìåñòî ñëó÷àé 2. Òîãäà ôóíêöèèt2ψ1 = ,2òàêèå, ÷òîτ t23 t00 0ψ2 =+,48π tψi0 = tϕi , i = 1, 2, 3, 1 ψ 0 00ttZτψ3 =t(τ 0 )ϕ3 (τ 0 ) dτ 0 ,−∞óäîâëåòâîðÿþò óðàâíåíèþt00 ψ 0 0− 3− 4πψ = 0,têîòîðîå èìååò ïåðâûé èíòåãðàët3 h d (t0 )2 d ψ 0 t00 0 ψ 0 i+= c.t0 dτ t2 dτ tt0tt2Çäåñüc ïðîèçâîëüíàÿ êîíñòàíòà.ÏóñòüóðàâíåíèåÒàê êàêω=tR2ω = ψ + 4cψ2 /(3π).Òîãäà äëÿ íàõîæäåíèÿωïîëó÷àåìd (t0 )2 d ω 0 t00 0 ω 0+= 0.dτ t2 dτ tt0tt2ψ1 ÿâëÿåòñÿ ðåøåíèåì (4.170), òî ïîñëå çàìåíû(y/t) dτ ïðèõîäèì ê ëèíåéíîìó óðàâíåíèÿ âòîðîãîy 00 + U y = 0,(4.170)ïîðÿäêà(4.171)337ãäåU = 5t00 /t − 6(t0 /t)2 . ñèëó (4.166)U < 0.Èç ñóùåñòâîâàíèÿ ýêñïîíåíöèàëüíî óáûâàþùåé íàöèèϕ3±∞ôóíê-âûòåêàåò, ÷òî2t0 (τ )y = ϕ3 (τ ) − 2t (τ )Zτt(τ 0 )ϕ3 (τ 0 ) dτ 0−∞±∞ ðåøåíèåì óðàâíåíèÿóðàâíåíèå nbp = t ðàçðåøèìî, òîÿâëÿåòñÿ ýêñïîíåíöèàëüíî óáûâàþùèì íà(4.171).
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