Диссертация (1136178), страница 42
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 ðåçóëüòàòå, óðàâíåíèå (4.264) ïðèíèìàåò âèäe−Znn2/3U (ex− ) − 2h1/3 h1/3 Ω− D0 (ek− ) +1 h2/3 ek−(ln |ξ|)H(ξ) dξ + ln √×32Ω−io Z xe+ −εek−k dxH(ξ) dξ +ln |x − xe− | 0+22S (x, h)1xe− +εZ e+ ehiok+2/31/3ex+ − xe− | D−1 (k+ ) ++h Ω+ ln |eH(ξ) dξ = r2 , h → 0,21e−Zh× D−1 (ek− ) +(4.265)ãäå1+ωr2 = O h h16/9 1 111/3 3/2+ O 5/3 ln+ O h ε ln +lnhhhε7/9+ω1+O hÀíàëîãè÷íî, çàìåíÿÿxe−íà√xe+ ,1ε ln .h(4.266)èìååì:Zhn2/3x+ − xe− | · D−1 (ek− ) +U (ex+ ) − 2h1/3 h1/3 Ω− ln |ee−1iek−H(ξ) dξ +2xe+ −εZ e+ enk dxk+1/3 2/3+ln |x − xe+ | 0+ h Ω+ D0 (ek+ ) +(ln |ξ|)×2S (x, h)2xe− +ε1Z e+ e h2/3 hiook+e×H(ξ) dξ + ln √H(ξ) dξ= r2 , h → 0. · D−1 (k+ ) +32Ω+1Z(4.267)Ïåðåéäåì ê ïðåîáðàçîâàíèþ ñîîòíîøåíèé (4.236), (4.240).Ëåììà 4.37.Ïðè h → 0 ñïðàâåäëèâû ðàâåíñòâà01/3Ω− = U (ex− ) + h∞h kθ(e1x+ − ε − x)−e− )S 0 (x, h)xe− +ε (x − xZ√ek− Ω− (x − xe− ) 3 Ω− i− 1/3 Hdx + r3 ,hh2/3(4.268)365xe+ −εh kθ(x − xe− + ε)1−Ω+ = −U (ex+ ) + h(ex+ − x)S 0 (x, h)−∞√ek+ Ω+ (ex+ − x) 3 Ω+ idx + r3 ,(4.269)− 1/3 Hhh2/301/3Zãäår3 = O(h4/9+ω11/3) + O(h√ε) + O h4/3+ωε3/2 h16/9 1ln+ O 8/3 .hεÄîêàçàòåëüñòâî.
Ïîäñòàâèì â (4.236) ôîðìóëû äëÿòîòèêèg.g−(4.270)è àñèìï-Ó÷èòûâàÿ (4.188), (4.251), èìååì:01/3ZΩ− = U (ex− ) + 2h∞(1 − χ∗− (x, ε)) nhk+(x − xe− )2S 0 (x, h)xe− +ε∞ S(x, h) iT 2 (x, τ1 , h)1/9+cos 2+ h ϕ(x, τ1 , h) dτ1 θ(ex+ − ε − x)−2h−∞Z ∞ 24/3 S (x, h)eT− (x, τ1 , h)k− Ω−−1/9−cos 2+ h ϕ− (x, τ1 , h) ×− 02S− (x, h)2h−∞Z ∞ Z ∞o4/9+ω11/32×dτ1 dx + O(h)+O hg+ (x, y) dydx . (4.271)Zxe+ −ε−∞Çäåñü â ñèëó (4.256)1/3Z∞Z∞hxe+ −ε−∞√2g+(x, y) dydx = O(h1/3 ε).Èíòåãðèðóÿ â (4.271) ïî ÷àñòÿì, íàõîäèì01/3Ω− = U (ex− ) + hZ∞(1 − χ∗− (x, ε)) h kθ(ex+ − ε − x)−(x − xe− )S 0 (x, h)xe− +ε4/3e√k− Ω− i− 0dx + O(h4/9+ω1 ) + O(h1/3 ε),S− (x, h)h → 0.(4.272)366Ðàâåíñòâà (4.260), (4.245) ïîçâîëÿþò îöåíèòüh1/3Zxe− +2εxe− +ε1/3= O(h√4/3eχ∗− (x, ε) h kk− Ω− idx =−(x − xe− ) S 0 (x, h) S−0 (x, h)ε) + O h4/3+ωε3/2 h7/9+ω1 1√+O,lnhε(4.273)à èç (4.249), (4.252) âûòåêàåò, ÷òîh1=√3Ω−Z1/3Z∞dx1=0e− ) S− (x, h)xe− +ε (x − x√ (x − x h16/9 1e− ) 3 Ω− Hdx + O 8/3 ,e− )h2/3εxe− +ε (x − x∞h → 0.(4.274)Íàêîíåö, ïîäñòàâëÿÿ (4.273), (4.274) â (4.272), ïðèõîäèì ê (4.268).Ôîðìóëà (4.269) ïðîâåðÿåòñÿ àíàëîãè÷íî.
Ëåììà äîêàçàíà.Ïðåîáðàçóåì, íàêîíåö, óðàâíåíèå (4.190). Ïîäñòàâèâ â ôîðìóëû (4.184), (4.185) äëÿäëÿg−èg+ ,Zj−èj+g , ñîîòâåòñòâåííî, âûðàæåíèÿâìåñòîèìååìxe− +ε00Z∞2 ln |x − x |χ− (x , ε)−∞−∞2g−(x0 , y 0 ) dy 0 dx0 =Zξ ∞ 2 0 0= 2hχln 1 − 0 G− (ξ , η ) dη 0 dξ 0 +e−ξ −∞−∞Z xe− +εZ ∞2+2χ− (x0 , ε) ln |x0 − xe− |g−(x0 , y 0 ) dy 0 dx0 .(4.275)1/3−∞2/3Ω−Ze− ξ0 −∞ÎïðåäåëèìZ ∞nZ ξZ ξhek1def±√ +D1 (ek± ) = A1 (ek± ) +H(z) dz −2 0z10Z ξ2/3 ieedz olnzA(k)ρk0±±2eee+A−1 (k± ) 3/2 − 3/2 − 11/6 dz −k± π A−1 (k± ) 2 dξ =zzz2z1367Z∞n=∞Z ξZ−−∞−∞0Ëåììà 4.38.ZZ ξoek±+ θ(ξ)H(z) dz dξ.21(4.276)Ïðè ξ > e− , e− → +∞ ñïðàâåäëèâî ðàâåíñòâîe−χ ξ0 e−−∞e−ZG2± (z, η) dηdzZξ ∞ 2 0 02 ln 1 − 0 G− (ξ , η ) dη 0 dξ 0 =ξ −∞ ξ0 ξ ek− ) ln ξ−=χln 1 − 0 k− H(ξ 0 ) dξ 0 + 2D−1 (ee−ξ1−5/3 −2D0 (ek− ) − 2D1 (ek− )/ξ + O (1 + ln |ξ/e− |)ε−.(4.277)Äîêàçàòåëüñòâî.
Èíòåãðèðóÿ ïî ÷àñòÿì â ñîäåðæàùåì áûñòðîîñöèëëèðóþùóþ ôóíêöèþ èíòåãðàëå è ó÷èòûâàÿ (4.250), ïîëó÷àåìZe−χ ξ0 −∞e−Zξ ∞ 2 0 02 ln 1 − 0 G− (ξ , η ) dη 0 dξ 0 =ξ −∞Z ∞ ξ eξ 00=ln 1 − 0 k− H(ξ ) dξ + 2ln 1 − 0 K(ξ 0 , e− ) dξ 0 −χe−ξξ−∞1Z ∞ ξ 0 ξ ek−0e1−χln 1 − 0 −− k− H(ξ ) dξ 0 + O(e−∞− ),00ee−ξ S (ξ )e− /2Ze− ξ0 (4.278)ãäåZ0∞K(ξ , e− ) =−∞G2− (ξ 0 , η 0 ) dη 0− 1−χ ξ 0 e−×∞ 0 θ(ξ 0 − 1)eTe2 (ξ 0 , η 0 )k− H(ξ 0 )00 0e×cos 2 S(ξ ) + ϕ(ξe , η ) dη −.22−∞ZÒàê êàê ïîñëå èíòåãðèðîâàíèÿ ïî ÷àñòÿìZ∞−∞ξ 0 K(ξ 0 , e− ) dξ 0 = D1 (ek− ) + O(e−∞− ),òî èìååì:Z∞ξ eln 1 − 0 K(ξ 0 , e− ) dξ 0 = ln |ξ|[D−1 (ek− ) + O(e−∞− )] − D0 (k− )+ξ−∞3681−∞e+O(e−∞− ) − [D1 (k− ) + O(e− )] +ξ∞Zξ 0 ξ 0 ln 1 − +K(ξ 0 , e− )dξ 0 .ξξ−∞(4.279)Íàêîíåö, â ñèëó (4.252) ïðè∞Z1 − χ− ξ 0 e− /2e−ξ > e− , e− → +∞ξ 10ln 1 − 0 − H(ξ ) dξ 0 =00ξ Se (ξ )∞ 1 + ln |ξ/e | ξ dξ 0 −=O,ln 1 − 0 0 8/3 = O5/3ξ (ξ )e− /2e−Z ∞ ξ 0 ξ 0 ln 1 − +K(ξ 0 , e− ) dξ 0 =ξξ−∞1 Z ∞ ξ 0 ξ 0 dξ 0 1 = O 2 +Oln 1− += O 5/3 .ξξξ (ξ 0 )8/3ξ1Z(4.280)(4.281)Ïîäñòàâëÿÿ (4.281) â (4.279), à çàòåì (4.279), (4.280) â (4.278), ïðèõîäèì ê (4.277).
Ëåììà äîêàçàíà.Èç ðàâåíñòâ (4.277), (4.263), (4.275) âûòåêàåò, ÷òî ïðèxe− +εZ00Z∞2 ln |x − x |χ− (x , ε)−∞x>xe− +ε−∞2g−(x0 , y 0 ) dy 0 dx0 =√ (x0 − x3e)Ω− 0−edx +√ χ− (x , ε) ln |x − x |k− Hh2/3xe− +h2/3 / 3 Ω−√ h13/9 1 32hΩ− D1 (ek− )1/3 2/3e+ O 5/3 ln .+2h Ω− D−1 (k− ) ln |x − xe− | −x−xe−hεΩ−= 1/3hZxe− +ε00(4.282)Àíàëîãè÷íî äîêàçûâàåòñÿ, ÷òî ïðèZ∞00x>xe+ − εZ∞2 ln |x − x |χ+ (x , ε)xe+ −ε−∞√Ω+= 1/3hZxe+ −h2/3 / 3 Ω+xe+ −ε2g+(x0 , y 0 ) dy 0 dx0 =√ (e3x−x)Ω+ 0+00 eχ+ (x , ε) ln |x − x |k+ Hdx +h2/33691/3+2h2/3k+ ) ln |ex+Ω+ D−1 (e√ h13/9 1 2h 3 Ω+ D1 (ek+ )− x| −+ O 5/3 ln .xe+ − xhε(4.283)Çäåñüh → 0.Ïîäñòàâèì òåïåðü ñîîòíîøåíèÿ (4.282), (4.283), (4.201) â óðàâíåíèå (4.190).
Ó÷èòûâàÿ (4.202), (4.33), èìååì:−(S 0 )2 + U (x)−√ (x0 − x3e)Ω− 0−edx +√ χ− (x , ε) ln |x − x |k− Hh2/3xe− +h2/3 / 3 Ω−Z xe+ −ε/2k dx0+h1/31 − χ− (x0 , ε) − χ+ (x0 , ε) ln |x − x0 | 0 0+S(x,h)xe− +ε/2√ oZ xe+ −h2/3 / √3 (eΩ+0 3Ω+x−x)+dx0 −+Ω+χ+ (x0 , ε) ln |x − x0 |ek+ H2/3hxe+ −ε 2/32/3k+ ) ln |ex+ − x| +k− ) ln |x − xe− | + Ω+ D−1 (e−2h2/3 Ω− D−1 (e√h√ k 2/333Ω− D1 (ek− )Ω+ D1 (ek+ ) i4/38/9+2h++h ρ 0+ h10/9 `1 (x)+x−xe−xe+ − xS h13/9 h13/9 h16/9 1 4/3+O+O= 0.+h `2 (x) + O 5/3 lnhε(x − xe− )7/6(ex+ − x)7/6n Z− Ω−xe− +ε00(4.284)Äàëåå âîñïîëüçóåìñÿ òåì, ÷òî ïåðåäìíîæèòåëèhµ ,ãäåµ > 1.`1 (x), `2 (x)â (4.284) ñòîÿòÌû çàìåíèì ýòè ôóíêöèè íà ãëàâíûå÷ëåíû àñèìïòîòèê. Îíè èìåþò îñîáåííîñòè ïðèx→xe± .
Îñòàëüíûåáîëåå ãëàäêèå ÷ëåíû ðàçëîæåíèé âîéäóò â îñòàòî÷íûé ÷ëåí.Èç (4.284) íàõîäèì(S 0 )2 = U 0 (ex− )(x − xe− ) + O h1/3 (x − xe− ) + O (x − xe− )2 +2/3+O h(x − xe− ) ln.h2/3370Ñëåäîâàòåëüíî, ïðèS0 =x→xe− + 0ppU 0 (ex− )(x − xe− ) + O h1/3 x − xe− + O (x − xe− )3/2 +(x − xe− ) h2/3pln.+Oh2/3x−xe−Àíàëîãè÷íî, ïðèS0 =p(4.285)x→xe+ − 0|U 0 (ex+ )|(ex+ − x) + O h1/3pxe+ − x + O (ex+ − x)3/2 +h2/3(ex+ − x) +O pln.h2/3xe+ − xÀñèìïòîòè÷åñêèå ðàçëîæåíèÿ äëÿ ïðîèçâîäíûõ(4.286)S0ïîëó÷àþòñÿïî÷ëåííûì äèôôåðåíöèðîâàíèåì (4.285), (4.286).
Ïîäñòàâëÿÿ çàòåìýòè àñèìïòîòèêè â (4.220), (4.230), ïðèõîäèì ê ñëåäóþùåé ëåììåËåììà 4.39.Ïðè x ∈ (ex− +ε, xe+ −ε), h → 0 ñïðàâåäëèâû ðàâåíñòâà4/3x− )σ h U 0 (e|U 0 (ex+ )|4/3 ih1/3`1 (x) =++O+54k 2/3 (x − xe− )2/3 (ex+ − x)2/3(x − xe− )7/6h1/3+ O(1),(4.287)(ex+ − x)7/63x− )|U 0 (ex+ )|3 oh1/9u n U 0 (e+`2 (x) = −++O243k 2x−xe−xe+ − x(x − xe− )7/6h1/9+O+ O(1).(4.288)(ex+ − x)7/6+OÄàëåå, èñïîëüçóÿ (4.252), (4.260), çàìåíèì â (4.284) ïðè(ex− + ε/2, xe− + ε)h1/3Zxe− +εxe− +ε/2ôóíêöèþ1/S 0 (x0 , h)íàx0 ∈H(ξ 0 ):h01 − χ− (x , ε) ln |x − x | O (x0 − xe− )1/2 +0 h4/9+ω1 ih1+ω1h13/9+O p+O+Olndx0 =03/208/30h(x − xe− )(x − xe− )x −xe−371 h4/9+ω 1 2 11/3 3/2ln= O h ε ln+O √+hhε h16/9 1 √17/9+ω1ε ln+O h+ O 5/3 ln , h → 0.(4.289)hhε0 0Òàêóþ æå, êàê â (4.289), ïîãðåøíîñòü âûçîâåò çàìåíà 1/S (x , h) íàH(ξ 0 ) â îáëàñòè x0 ∈ (ex+ − ε, xe+ − ε/2).
Íàêîíåö, ó÷òåì (4.289) èïîäñòàâèì (4.287), (4.288) â (4.284). ÄîêàçàíàÔóíêöèÿ S 0 (x, h) íà èíòåðâàëå (ex− + ε, xe+ − ε) óäî-Òåîðåìà 4.8.âëåòâîðÿåò óðàâíåíèþn Z−(S (x, h)) + U (x) − Ω−02xe− +εxe− +h2/3 /√3ln |x − x0 |×Ω−√Z xe+ −ε (x0 − x3e)Ωk dx0−−01/30e×k− Hdx + hln |x − x | 0 0+S (x , h)h2/3xe− +ε√ oZ xe+ −h2/3 / √3 (eΩ+0 3Ω+x−x)++Ω+dx0 −ln |x − x0 |ek+ H2/3hxe+ −ε 2/32/3−2h2/3 Ω− D−1 (ek− ) ln |x − xe− | + Ω+ D−1 (ek+ ) ln |ex+ − x| +4/32/310/9 h U 0 (ex)khσ|U 0 (ex+ )|4/3 i−8/9+h ρ 0+++(S (x, h))2/3 54k 2/3 (x − xe− )2/3 (ex+ − x)2/3√h√33eΩD(k)Ω+ D1 (ek+ ) i−1−4/3+2h+−x−xe−xe+ − x34/3 n U 0 (ex)h|U 0 (ex+ )|3 o−−+= R(x, h),(4.290)243k 2x−xe−xe+ − xãäå h13/9 h16/9 1 h13/9 R(x, h) = O+O+ O 5/3 ln +h(x − xe− )7/6(ex+ − x)7/6ε1/3 3/2+O hε h4/3+ω 1 2 √117/9+ω1lnln+O √+O hε ln +hhhε+O(h10/9 ),h → 0.372Ïîëîæèì â óðàâíåíèÿõ (4.290), (4.261), (4.265), (4.267) (4.269)R, r1 , r2 , r3 ðàâíûìè íóëþ, à êîíñòàíòû ek− , ek+ çà26/57äàäèì ðàâåíñòâàìè (4.245), (4.246).
Êðîìå òîãî, ïóñòü ε = h. (Òàêîé âûáîð ε îáåñïå÷èâàåò ìèíèìàëüíóþ ïîãðåøíîñòü.) Òàêèì îáðàçîì, ïîëó÷åíà ñèñòåìà óðàâíåíèé äëÿ íàõîæäåíèÿ ôóíêöèè S(x, h)ïðè x ∈ (ex− + ε, xe+ − ε), à òàêæå êîíñòàíò k , xe− , xe+ , Ω− , Ω+ . Ýòóîñòàòî÷íûå ÷ëåíûñèñòåìó áóäåì íàçûâàòü íèæå çàäà÷åé äëÿ ôàçû.Çàìå÷àíèåñòàíòûâáëèçè4.23. Îïðåäåëÿåìûå ïðè ðåøåíèè çàäà÷è äëÿ ôàçû êîí-Ω± , xe± , k íåîáõîäèìû äëÿòî÷åê xe± ñîîòâåòñòâåííî.Çàìå÷àíèå4.24. Åñëè(4.270) âûòåêàåò, ÷òîïîñòðîåíèÿ àñèìïòîòèêg± (x, y)ε = h26/57 , òî γ = 26/57 è èç ôîðìóë (4.266),ω = 1/57, ω1 = 20/171. Òîãäà äëÿ âõîäÿùèõâ çàäà÷ó äëÿ ôàçû óðàâíåíèé ïîëó÷àåì ñëåäóþùèå ïîãðåøíîñòè:r1 = O(h2/3+1/57 ); äëÿ óðàâíåíèé1+1/57(4.265), (4.267) r2 = O(hln 1/h); äëÿ óðàâíåíèé (4.268), (4.269)r3 = O(h4/9+20/171 ); äëÿ óðàâíåíèÿ (4.290)äëÿ óñëîâèÿ íîðìèðîâêè (4.261) h13/9 1h13/9 1+1/57+O+O hln .R(x, h) = Oh(x − xe− )7/6(ex+ − x)7/6(4.291)Çäåñüh → 0, x ∈ (ex− + ε, xe+ − ε).2.6.Ïðàâèëî êâàíòîâàíèÿ.
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