Диссертация (1136178), страница 35
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ßäðî âõîäÿùåãîb ñîäåðæèò ãëàäêóþ, ýêñïîíåíöèàëüíî óáûâàNþùóþ ïðè |τ | → ∞ ôóíêöèþ ∂B0 /∂η . Äåéñòâèòåëüíî, äèôôåðåíb ∂B0 /∂η = 0. Ïðåäïîëîæèì,öèðóÿ ïî η óðàâíåíèå (4.28), èìååì Nâ (4.49) îïåðàòîðà÷òî âûáðàíî òàêîå ðåøåíèå çàäà÷è (4.28) (4.30), ïðè êîòîðîì ÿäðîîïåðàòîðàbNîäíîìåðíî. Òîãäà óñëîâèåì ðàçðåøèìîñòè óðàâíåíèÿ(4.49) áóäåò óñëîâèå îðòîãîíàëüíîñòè â ïðîñòðàíñòâåöèè∂B0 /∂ηL2 (R1 )ôóíê-ïðàâîé ÷àñòè (4.49) (ñì. ï. 1.7 èç 1 ãëàâû 4):Z∞h ∂ϕ 2i ∂B000 ∂ϕ0`1 ++ 2Sdη = 0.B0∂η∂ξ∂η−∞(4.53)301Åñëè â (4.53) âìåñòîϕ0ïîäñòàâèòü âûðàæåíèå (4.40) è ïðîèíòåãðè-ðîâàòü ïî ÷àñòÿì, òî óñëîâèå ðàçðåøèìîñòè (4.53) ïðèìåò âèä (S 00 )29S 0 S 000 +3Z∞ηB02 dη = 0.−∞(4.54)Ðàâåíñòâî (4.54) âûïîëíåíî â ñèëó (4.30).
Ïîýòîìó óðàâíåíèå (4.49)ðàçðåøèìî.Èçó÷èì äàëåå óñëîâèÿ (4.50), (4.51), ïåðâîå èç êîòîðûõ ïîçâîëÿåò íàéòè`1 (ξ).Ëåììà 4.6.Íàì ïîòðåáóåòñÿÏóñòüdefΨ(ξ, η) = B0 +η ∂B0.2 ∂ηÒîãäà ñïðàâåäëèâî ðàâåíñòâîΨb= B0 .N`0Çàìå÷àíèåçíà÷åíèÿáûρ ≥ 0,4.5. Ôóíêöèÿ(4.55)`0 (ξ) = ρ(k/S 0 )2/3 < 0, òàê êàê ñîáñòâåííûåρ çàäà÷è (0.46), (0.47) îòðèöàòåëüíûå.
Äåéñòâèòåëüíî, åñëèòî â ñèëó (0.46)t−1d2 t> 0.dτ 2Îäíàêî íå ñóùåñòâóåò íåïðåðûâíûõ, ýêñïîíåíöèàëüíî óáûâàþùèõïðèτ → ±∞ôóíêöèé, òàêèõ, ÷òîâûïóêëà ââåðõ ïðètâûïóêëà âíèç ïðèt > 0èt < 0.Äîêàçàòåëüñòâî ëåììû 4.6. Ñðàâíèâàÿ (4.28) è (4.47), èìååìb B0 = −2πNZ∞−∞|η − η0|B02 dη 0B0 .(4.56)302Òàê êàêbNb ∂B0 /∂η = 0,N η ∂B 02 ∂ηòî, èíòåãðèðóÿ ïî ÷àñòÿì, íàõîäèì1 n ∂ 2 B0∂ 3 B0=2 2 +η−π2∂η∂η 3Z∞∞Z|η − η−∞0|B02 dη 0 ∂B0−η∂η∂B0 odη B0 − `0 η=−2π|η − η∂η−∞Z ∞∂ 2 B0∂20 B000=−π=|η − η |(η − η) 0 (B0 ) dη∂η 2∂η2−∞Z ∞020= `0 B0 + 2π|η − η |B0 dη B0 .0∂B0|B0 η 0 0∂η0(4.57)−∞Íàêîíåö, ñêëàäûâàÿ (4.56), (4.57), ïðèõîäèì ê (4.55).
Ëåììà äîêàçàíà.Ïîëó÷èì ôîðìóëó äëÿdefσ = σi =Z`1 (ξ).Ââåäåì êîíñòàíòû∞−∞τ 2 t2i (τ ) dτ > 0,i = 1, 2, . . .(4.58)õàðàêòåðèçóþùèå ñîáñòâåííûå ôóíêöèè çàäà÷è (0.46), (0.47). ÑïðàâåäëèâàËåììà 4.7.Èìååò ìåñòî ðàâåíñòâî0`1 (ξ) = −σ S /k2/3(S 00 )2 + 3S 0 S 000 /27.(4.59)Äîêàçàòåëüñòâî.  ñèëó (4.50), (4.55), (4.49), (4.29)Z∞0 = `0ZB0 T1 dη = `0−∞Z∞∞bN−∞Ψ`0Z∞T1 dη =b T1 dη =ΨN−∞ ∂ϕ 2η ∂B0 00 ∂ϕ0=`1 ++ 2SB0 dη =B0 +2 ∂η∂η∂ξ−∞Z ∞Z(S 00 )2 + 3S 0 S 000 ∞ 2η ∂22= `1B0 +(B ) dη +B0 +4 ∂η 09−∞−∞00 20 000 Z ∞3k(S)+3SSη ∂(B02 ) η 2 dη = `1 0 +η 2 B02 dη. (4.60)+4 ∂η4S36−∞303Òàê êàêZ∞η−∞2B02 dηy2= 3βZ∞2 2τ t (τ ) dτ = k 1/3S0−∞σ,òî (4.59) âûòåêàåò èç (4.60). Ëåììà äîêàçàíà.Ðàññìîòðèì òåïåðü óñëîâèå (4.51). ÅñëèT1,0 íåêîòîðîå ÷àñò-íîå ðåøåíèå óðàâíåíèÿ (4.49), òî â ñèëó îäíîìåðíîñòè ÿäðà îïåðàòîðàbNîáùåå ðåøåíèå (4.49) èìååò âèäT1 = T1,0 + c1 (ξ)ãäåc1 (ξ)∂B0,∂η ïðîèçâîëüíàÿ ôóíêöèÿ.
Òàê êàêZ∞B0−∞∂B0dη = 0,∂ηòî (4.50) áóäåò âûïîëíåíî ïðè âñåõc1 (ξ).Ôóíêöèÿc1 (ξ)íàõîäèòñÿèç óñëîâèÿ (4.51). Ó÷èòûâàÿ, ÷òîZ∞0=ZηB0 T1 dη = c1−∞Z∞∂B0ηB0dη +∂η−∞∞1∂B0ηB0dη = −∂η2−∞èìååì2S 0 (ξ)c1 (ξ) =kZZZ∞ηB0 T1,0 dη,−∞∞−∞B02 dη = −k,2S 0∞ηB0 T1,0 dη.−∞Òàêèì îáðàçîì, çàäà÷à (4.49) (4.51) îäíîçíà÷íî îïðåäåëÿåòèT1 (ξ, η)`1 (ξ).Îòìåòèì, ÷òî ôóíêöèÿT1ïðåäñòàâèìà â âèäåT1 (ξ, η) = y1 (ξ)p(τ ),(4.61)2/3 0y1 (ξ) = S /k3S 0 S 000 + (S 00 )2 /9,(4.62)ãäå304τ = β(ξ)η ,p = p(τ )àÿâëÿåòñÿ ðåøåíèåì çàäà÷ènbp = (τ 2 − σ/3)t(τ ),Z(4.63)∞τ t(τ )p(τ ) dτ = 0.(4.64)−∞ÇäåñüZ ∞def d2 p(τ )−π|τ − τ 0 |t2 (τ 0 ) dτ 0 p(τ )−nbp =2dτ−∞Z ∞−2π|τ − τ 0 |t(τ 0 )p(τ 0 ) dτ 0 t(τ ) − ρp(τ ).(4.65)−∞Äåéñòâèòåëüíî, ïîäñòàâëÿÿ (4.61), (4.59), (4.40), (4.31) â (4.49), ïðèõîäèì ê (4.62).
Óñëîâèå (4.64) âûòåêàåò èç (4.51).Çàìå÷àíèå4.6. Äëÿ ôóíêöèèp(τ ) ïðè τ → ±∞ ñïðàâåäëèâà îöåíêàp(τ ) = O |τ |5/2 t(τ ) .Èòàê, äëÿB1ïîëó÷åíà ôîðìóëàB1 = y1 (ξ)p(τ ) − ϕ20 y(ξ)t(τ )/2,ãäåϕ0(4.66)èìååò âèä (4.40).B2 , I1 , `2 [40; 73].`2 (ξ) èç (4.21) (4.24)Âûâåäåì, íàêîíåö, ôîðìóëû äëÿ ïîïðàâîêÄëÿ íàõîæäåíèÿ ôóíêöèéB2 (ξ, η), I1 (ξ, η)èïîëó÷àåì ñëåäóþùóþ çàäà÷ó: Zb 1= πRI∞|η−η−∞0|(I02 +2B0 B1 ) dη 0 +`1∞ Zb B2 = πN|η − η−∞ Z+ π0|(I02|η − η−∞|(B12 + 2I0 I1 ) dη 0 + `2ZZ(4.67)+ 2B0 B1 ) dη + `1 B1 +∞0∂B1+S 00 B1 ,I0 +2S 0∂ξ0B0 − 2S 0∂I1− S 00 I1 ,∂ξ(4.68)∞(B12 + 2B0 B2 + 2I0 I1 ) dη = 0,(4.69)η(B12 + 2B0 B2 + 2I0 I1 ) dη = 0.(4.70)−∞∞−∞305Çäåñü îïåðàòîðûbèNbRçàäàíû ôîðìóëàìè (4.39), (4.47), à àðãóìåí-òû ó ôóíêöèé îïóùåíû â ñîîòâåòñòâèè ñ çàìå÷àíèåì 4.3.Áóäåì èñêàòüI1 (ξ, η)â âèäåI1 = Z − ϕ0 T1 + ϕ30 B0 /6.(4.71)ÑïðàâåäëèâàËåììà 4.8.Ôóíêöèÿ Z(ξ, η) óäîâëåòâîðÿåò óðàâíåíèþb = y 2 ψp(τ ),RZ(4.72)ãäåψ(ξ) = 2 S 0 2 hk4 00 3 1 0 2 IV iS S S + (S ) + (S ) S ,273000000τ = β(ξ)η.(4.73)(4.74)Äîêàçàòåëüñòâî.
Ïîäñòàâëÿÿ (4.71) â (4.67) è ó÷èòûâàÿ (4.28),(4.49), èìååìb = F1 .RZÇäåñü∂ 2 ϕ30 ∂T1 ∂ϕ0∂ 2 ϕ0∂B0 ∂ ϕ30 − B0 2+2+ T1 2 +F1 = −2∂η ∂η 6∂η 6∂η ∂η∂η+ϕ0h ∂ϕ 20∂ηiS 00 ϕ20 B00 ∂T10 ∂200+ 2SB0 + 2S− S (ϕ0 B0 ) + S T1 −.∂ξ∂ξ∂ξ20 ∂ϕ0Èñïîëüçóÿ (4.31), (4.40), (4.61), íàõîäèì, ÷òîF1 =n5(S 00 )3 4(S 00 )3 4β(S 00 )3 5 dt(τ )η−yη t(τ ) +yη t(τ )+−y108dτ216541 0 00 000 41S 0 (S 00 )2 0 4S S S yη t(τ ) − S 0 S 00 S 000 yη 4 t(τ ) −y η t(τ )−181836o n2S 0 (S 00 )2 0 5 dt(τ ) (S 00 )3 4dp(τ )−yβ η−yη t(τ ) +y1 βS 00 η+36dτ723dτoy1 000 000 dp(τ )00+ S p(τ ) + 2S y1 p(τ ) + 2S y1 β η+ S y1 p(τ ) .(4.75)3dτ+306Îñòàåòñÿ ïîäñòàâèòü â (4.75) âûðàæåíèÿ (4.32), (4.62) äëÿβ , y , y1èñîêðàòèòü ïîäîáíûå ÷ëåíû.  ñèëó ôîðìóë (4.43) ïðèõîäèì ê (4.72).Ëåììà äîêàçàíà.ÏóñòüZ(ξ, η) = y(ξ)ψ(ξ)z(τ ).(4.76)Òîãäà èç (4.72) âûòåêàåò, ÷òî ýêñïîíåíöèàëüíî óáûâàþùàÿ ïðè∞ôóíêöèÿz(τ )|τ | →äîëæíà óäîâëåòâîðÿòü óðàâíåíèþrbz = p(τ ),(4.77)ãäådef d2 z(τ )− πrbz =dτ 2àp(τ )∞Z0 200|τ − τ |t (τ ) dτ + ρ z(τ ),(4.78)−∞ ðåøåíèå çàäà÷è (4.63), (4.64).Óðàâíåíèå (4.77) ÿâëÿåòñÿ ëèíåéíûì íåîäíîðîäíûì îáûêíîâåííûì äèôôåðåíöèàëüíûì óðàâíåíèåì 2-ãî ïîðÿäêà.
Òàê êàê0,rbt =òî, ñîãëàñíî ìåòîäó âàðèàöèè ïîñòîÿííûõ, íà êàæäîì èíòåðâàëå,ãäåt(τ ) 6= 0,îáùåå ðåøåíèå (4.77) âûïèñûâàåòñÿ â êâàäðàòóðàõ. Èçàíàëèçà ôîðìóë äëÿËåììà 4.9.z(τ )âûòåêàåòÏóñòü ôóíêöèÿ p(τ ) ãëàäêàÿ è ýêñïîíåíöèàëüíî óáû-âàåò ïðè |τ | → ∞. Òîãäà íåîáõîäèìûì è äîñòàòî÷íûì óñëîâèåìðàçðåøèìîñòè óðàâíåíèÿ(4.77)â êëàññå ãëàäêèõ, ýêñïîíåíöèàëüíîóáûâàþùèõ ïðè |τ | → ∞ ôóíêöèé ÿâëÿåòñÿ ðàâåíñòâîZ∞p(τ )t(τ ) dτ = 0.(4.79)−∞ ñëó÷àå, åñëè âîëíîâàÿ ôóíêöèÿ ïîëÿðîíà t íå îáðàùàåòñÿ â íóëü,ðåøåíèÿ èç óêàçàííîãî êëàññà ïðåäñòàâèìû â âèäåZτz(τ ) = c1 +−20Zτ0t (τ )τi0p(x)t(x) dxdτ t(τ ),−∞−∞ < τ < +∞,307ãäå c1 ïðîèçâîëüíàÿ êîíñòàíòà. Åñëè æå t èìååò êîíå÷íîå ÷èñëîïðîñòûõ íóëåé α1 < α2 < · · · < αn , òî äëÿ ãëàäêèõ, ýêñïîíåíöèàëüíî óáûâàþùèõ ïðè |τ | → ∞ ðåøåíèéZτz(τ ) = ci +−2Z0(4.77)ñïðàâåäëèâû ôîðìóëûτ00t (τ )p(x)t(x) dxdτ t(τ ),−∞τiαi−1 < τ < αi ,i = 1, .
. . , n + 1.Çäåñü α0 = −∞, αn+1 = +∞, òî÷êè τi ∈ (αi−1 , αi ), c1 ïðîèçâîëüíàÿ êîíñòàíòà, à êîíñòàíòû ci ïðè i > 1 âûðàæàþòñÿ ÷åðåçc1 . (Ïðè ýòîì èñïîëüçóåòñÿ óñëîâèÿ íåïðåðûâíîñòè dz/dτ â íóëÿõt(τ ).) Êðîìå òîãî,Zαiz(αi ) = −p(τ )t(τ ) dτ. dt−∞dτ(αi ),Ðàññìîòðèì òåïåðü óðàâíåíèå (4.77), ãäåi = 1, .
. . , n.p(τ ) ðåøåíèå (4.63),(4.64).  ñèëó (4.50) ñîîòíîøåíèå (4.79) âûïîëíåíî. Ïîýòîìó óðàâíåíèå (4.77) â êëàññå ãëàäêèõ, ýêñïîíåíöèàëüíî óáûâàþùèõ ïðè∞ôóíêöèé ðàçðåøèìî. Îäíàêî (4.77) îïðåäåëÿåòíîñòüþ äî ñëàãàåìîãîc1 t(τ ),ãäåc1z(τ )|τ | →ëèøü ñ òî÷- ïðîèçâîëüíàÿ êîíñòàíòà.Äîïîëíèì (4.77) óñëîâèåìZ∞ τ dt(τ ) + t(τ ) z(τ ) dτ = 0,2dτ−∞(4.80)êîòîðîå, êàê ìû óâèäèì íèæå, íåîáõîäèìî äëÿ âûïîëíåíèÿ ñîîòíîøåíèÿ (4.69). Òàê êàêZ∞ τ dt3(τ ) + t(τ ) t(τ ) dτ = ,4−∞ 2 dτ(4.81)òî çàäà÷à (4.77), (4.80) â êëàññå ãëàäêèõ, ýêñïîíåíöèàëüíî óáûâàþùèõ ïðè|τ | → ∞ôóíêöèé áóäåò èìåòü óæå åäèíñòâåííîå ðåøåíèå.Òàêèì îáðàçîì,(4.71), (4.76), ãäåz(τ )I1 (ξ, η)ïîëíîñòüþ îïðåäåëÿåòñÿ ôîðìóëàìè ðåøåíèå çàäà÷è (4.77), (4.80).308Ïåðåéäåì ê íàõîæäåíèþB2 (ξ, η).ÔóíêöèþB2áóäåì èñêàòü ââèäåB2 = T2 − ϕ20 T1 /2 + ϕ0 Z + ϕ40 B0 /4!.(4.82)ÑïðàâåäëèâàËåììà 4.10.Ôóíêöèè T2 (ξ, η) è `2 (ξ) ÿâëÿþòñÿ ðåøåíèåì çàäà÷èb T2 = `1 T1 + `2 B0 + 2πN∞Z00|η − η |B0 T1 dη T1 +−∞+πZ∞|η − η−∞0|T12 dη 0ZZB0 +h ∂ϕ 20∂η+ 2S0 ∂ϕ0∂ξiT1 − 2S 0 yψ 0 z(τ ),(4.83)∞(2B0 T2 + T12 ) dη = 0,(4.84)η(2B0 T2 + T12 ) dη = 0.(4.85)−∞∞−∞b çàäàí ôîðìóëîéÇäåñü îïåðàòîð N(4.47).Äîêàçàòåëüñòâî.
Òàê êàê â ñèëó (4.41), (4.48), (4.71), (4.82)B12 +2B0 B2 +2I0 I1ϕ20ϕ40 ϕ20 2 = T1 − B0 +2 T2 − T1 +ϕ0 Z + B0 B0 +224!ϕ30 +2(−B0 ϕ0 ) Z − ϕ0 T1 + E0 = T12 + 2B0 T2 ,6òî óñëîâèÿ (4.84), (4.85) âûòåêàþò íåïîñðåäñòâåííî èç (4.69), (4.70).Ïîäñòàâèì òåïåðü (4.41), (4.48), (4.71), (4.82) â (4.68). Ó÷èòûâàÿ(4.28), (4.49), (4.72), ïîëó÷àåìb T2 = `1 T1 + `2 B0 + 2πNZ∞00|η − η |B0 T1 dη T1 +−∞+πZ∞−∞|η − η0|T12 dη 0B0 − F2,1 + F2,2 + F2,3 ,(4.86)ãäåF2,1∂ϕ0 ∂Z∂ 2 ϕ0∂Z=2+ Z 2 + 2S 0+ S 00 Z,∂η ∂η∂η∂ξF2,2 =h ∂ϕ 20∂η+309 ϕ3 2 ∂B0 ∂ϕ40 B0 ∂ 2 ϕ40ϕ20ϕ3000 ∂00+2SB0 −−−2SB−SB,00∂ξ24! ∂η ∂η4! ∂η 2∂ξ660 ∂ϕ0F2,3i∂ϕ20 ∂T1 T1 ∂ 2 ϕ2020 ∂+−ϕyψp(τ)+2S(T1 ϕ0 ) + S 00 T1 ϕ0 .=02∂η ∂η2 ∂η∂ξÓïðîñòèì âûðàæåíèÿ äëÿF2,1 , F2,2 , F2,3 .Èñïîëüçóÿ (4.40),(4.76), (4.43), èìååì2dz(τ ) y 00F2,1 = S 00 ηyψβ+ ψS z(τ ) + 2S 0 y 0 ψz(τ ) + 2S 0 yψ 0 z(τ )−3dτ32dz(τ )− S 00 yψηβ+ S 00 yψz(τ ) = 2S 0 yψ 0 z(τ ).3dτ(4.87)Äàëåå, â ñèëó (4.40), (4.31), (4.43),F2,2 =h (S 00 )29S 0 S 000 i (S 00 )2 η 64dt(τ )+yt(τ ) − 5 (S 00 )4 yβη 7−3726dτ142S 0 0 00 3 62S 0 0 00 3 7 dt(τ )00 4 6−− 5 y(S ) η t(τ ) − 4 y (S ) η t(τ ) − 4 yβ (S ) η666dτ(S 00 )4S0(4.88)− 3 y(S 00 )2 S 000 η 6 t(τ ) − 4 yη 6 t(τ ) = 0.66Íàêîíåö, ó÷èòûâàÿ (4.40), (4.61), (4.62), (4.73), (4.43), íàõîäèìF2,3(S 00 )2(S 00 )2 2S 00 2 23 dp(τ )=y1 βη+y1 η p(τ ) − y ψη p(τ )+9dτ66S 0 0 00 2S0(S 00 )2 20 00 3 dp(τ )+2ST1 + y1 S η p(τ ) + y1 β S η+y1 η p(τ ) =∂ξ33dτ6 S 0 2/3 n (S 00 )2S 00 h 0 00 0000 ∂ϕ00 00000 2T1 +[3S S + (S ) ] −SS S += 2S∂ξk2734 00 3 1 0 2 IV i2+ (S ) + (S ) S+ (S 00 )2 [3S 0 S 000 + (S 00 )2 ]+27381oh ∂ϕ 2iS 0 S 00000 0000 IV00 00020 ∂ϕ0+[3S S + 3S S + 2S S ] η p(τ ) =+ 2ST1 .27∂η∂ξ0 ∂ϕ0(4.89)Óðàâíåíèå (4.83) âûòåêàåò èç (4.86) (4.89).
Ëåììà äîêàçàíà.310Ðåøåíèå çàäà÷è (4.83) (4.85) áóäåì èñêàòü â âèäåτ(4.90)`2 (ξ) = −4(S 0 /k)2/3 y12 u/3,(4.91)t2,1 (τ ), t2,2 (τ ),ãäå ôóíêöèèëåíèþ;T2 (ξ, η) = y12 t2,1 (τ )/y + 2S 0 ψ 0 t2,2 (τ ),à òàêæå êîíñòàíòàuïîäëåæàò îïðåäå-çàäàíà ôîðìóëîé (4.74). Ïîäñòàâèì (4.90), (4.91) â (4.83) (4.85) è ó÷òåì (4.31), (4.32), (4.59), (4.61), (4.62).  ðåçóëüòàòå ïîëó÷àåì ñëåäóþùèå çàäà÷è äëÿ íàõîæäåíèÿnbt2,14σ= − p(τ ) − ut(τ ) + 2π33ZZt2,1 (τ )èt2,2 (τ )∞|τ − τ 0 |t(τ 0 )p(τ 0 ) dτ 0 p(τ )+−∞∞|τ − τ 0 |p2 (τ 0 ) dτ 0 t(τ ) + τ 2 p(τ ),+π(4.92)−∞Z∞τ 2t2,1 (τ )t(τ ) + p2 (τ ) dτ = 0;(4.93)−∞nbt2,2 = −z(τ ),Z(4.94)∞τ t2,2 (τ )t(τ ) dτ = 0;(4.95)−∞à òàêæå ðàâåíñòâàZ∞t2,2 (τ )t(τ ) dτ = 0,(4.96)−∞Z∞2t2,1 (τ )t(τ ) + p2 (τ ) dτ = 0.(4.97)−∞Çäåñü îïåðàòîð(4.64), àz(τ )nbçàäàí ôîðìóëîé (4.65),p(τ ) ðåøåíèå (4.63), ðåøåíèå (4.77), (4.80).Èçó÷èì ðàâåíñòâà (4.96), (4.97).
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