1611689230-c82f95de5ae5b8e93bc9da7770e8f930 (826746), страница 13
Текст из файла (страница 13)
Êðàåâûå óñëîâèÿ tA+e0−1T, −T0 0 G (t ) =00 T −1tA− T ,0È( )G tett< 0;> 0.îáëàäàåò ñâîéñòâàìè 1), 2), 3).Çàìåòèì, ÷òî äëÿ ìàòðèöû ðèíà ñïðàâåäëèâà îöåíêà:kG (t )k ≤ K (A)e − 2 |t |∀t ∈ R 1σ(8)ïðè óñëîâèè, ÷òî|Reτj (A)| ≥ σ > 0,j= 1,...,N .àññìîòðèì óíêöèþ( )=y tZ+∞( − s )f (s )ds = (G ∗ f )(t ).G t−∞Òàê êàê â ñèëó (8)kG (t − s )k ≤ K (A)e − 2 |t −s |σèkf (s )k ≤ F .Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿ(9)òî èíòåãðàë (9) ñõîäèòñÿ èky (t )k ≤4KFσ.àññìîòðèìãäåBy′By= Ay + f (t ),(0) = ϕ,(10)- ïðÿìîóãîëüíàÿ ìàòðèöà.Èòàê, íåîáõîäèìî íàéòè íåïðåðûâíóþ, îãðàíè÷åííóþ ïðè≤ t < +∞, íåïðåðûâíî - äèåðåíöèðóåìóþ óíêöèþ y (t )ïðè t > 0, åñëè âûïîëíåíî óñëîâèå, ÷òî f (t ) îãðàíè÷åíà èíåïðåðûâíà ïðè t ≥ 0.0Ïðè êàêîé ìàòðèöåëþáîéBçàäà÷à (10) ðàçðåøèìà îäíîçíà÷íî äëÿϕ?Ìîæíî ñ÷èòàòü, ÷òîf(t ) ≡ 0. ñàìîì äåëå: ïðîäîëæèì÷åòíûì îáðàçîì íà âñþ ïðÿìóþ⇒ f˜(t )f(t )íåïðåðûâíà èîãðàíè÷åíà íà âñåé ïðÿìîé.
Òîãäà ðàçíîñòü( ) = y (t ) − y0 (t ),z tÎãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿt≥0óäîâëåòâîðÿåò óðàâíåíèþ:ãäå óíêöèÿóðàâíåíèÿy′′(t ) = Az (t ), t > 0,Bz (0) = ϕ − By0 (0) = ϕ̃.z( ) - îãðàíè÷åííîå= Ay + f (t ).y0 tíà âñåé ïðÿìîé ðåøåíèåÒàê êàêtA z (0) = T −1z (t ) = e tA+e00etA−Tz(0) == T −1òî ïðèC+⇒C+6= 0tA+ C+e= 0.( )=Tz t−1íåîãðàíè÷åííî ðàñòåò ïðè0etA− C− tA+eC+tA− C ,et−→∞⇒ ñèëó ãðàíè÷íûõ óñëîâèé èìååìBT−10C−= ϕ̃.(11)Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿ⇒îäíîçíà÷íàÿ ðàçðåøèìîñòü (10)⇔îäíîçíà÷íàÿ.ðàçðåøèìîñòü (11).Çíà÷èò, rangBT −1= N− .Ïðåäñòàâèì ìàòðèöóBT−1BTâ âèäå:−1=D+ D−ÒîãäàBT−10C−= D− C− = ϕ̃detD−C−−1ϕ̃= D−è⇒6= 0.(12)−1k · kϕ̃k.kC− k ≤ kD−Äàëåå−1ke tA− C− k ≤ M ( )e − 2 t kD−k · kϕ̃kσÎãðàíè÷åííûå ðåøåíèÿ.
Êðàåâûå óñëîâèÿè−1k · kϕ̃k =kz (t )k ≤ kT −1 kM ( )e − 2 t kD−σkϕ̃k − σ t−1= kT −1 kM ( )kD−k · kB ke 2 .{z} kB k|L(13)îâîðÿò, ÷òî ãðàíè÷íûå óñëîâèÿ óäîâëåòâîðÿþò (12) - óñëîâèþËîïàòèíñêîãî, àL- ïîñòîÿííàÿ Ëîïàòèíñêîãî.15. Ëèíåéíîå óðàâíåíèå âòîðîãî ïîðÿäêà. Çàäà÷àØòóðìà - ËèóâèëëÿÐàññìîòèì óðàâíåíèå000(1)a0 (x)y + a1 (x)y + a2 (x)y = f (x),ãäåa0 (x) 6= 0íà[a, b].Çàäà÷à Êîøè000a0 (x)y + a1 (x)y + a2 (x)y = f (x), x ∈ [a, b];0y (x0 ) = y0 , y (x0 ) = y1 , x0 ∈ (a, b)ðàçðåøèìà åäèíñòâåííûì îáðàçîì, åñëè ôóíêöèèíåïðåðûâíû íà(2)a0,1,2 (x), f (x)[a, b].Çíà÷èò, îäíîðîäíîå óðàâíåíèå (f (x)≡ 0)ñ íóëåâûìèíà÷àëüíûìè äàííûìè èìååò òîëüêî òðèâèàëüíîå ðåøåíèå.Âìåñòî óðàâíåíèÿ (1), íå îãðàíè÷èâàÿ îáùíîñòè, ìîæíî (ïðèf (x) = 0)ðàññìàòðèâàòü óðàâíåíèå00y + Q(x)y = 0Çàäà÷à Øòóðìà - Ëèóâèëëÿ0(1 )èëè óðàâíåíèå â òàê íàçûâàåìîì ñàìîñîïðÿæåííîì âèäå:0000{P(x)y } + q(x)y = 0. ñàìîì äåëå, ïðèa0 (x) 6= 0y = exp{−12(1 )ïîñëå çàìåíûZxx0a1 (ξ)dξ}z(x)a0 (ξ)0óðàâíåíèå (1) ïðèâîäèòñÿ ê âèäó (1 ). ñâîþ î÷åðåäü, çàìåíà íåçàâèñèìîé ïåðåìåííîé0dξ =00ïðèâîäèò óðàâíåíèå (1 ) ê âèäó (1 ).Ïðèìåð 1.00y + a2 y = 000y − a 2 y = 0,a = const.Îáùèå ðåøåíèÿ:y = C1 sin(ax) + C2 cos(ax),Çàäà÷à Øòóðìà - Ëèóâèëëÿdxp(x)y = C1 e ax + C2 e −ax ,òî åñòü ó óðàâíåíèÿ âòîðîãî ïîðÿäêà åñòü ðåøåíèÿ, èìåþùèå∞íóëåé, à åñòü ðåøåíèÿ, íå îáðàùàþùèåñÿ â íîëü.Îïðåäåëåíèå 1.Ðåøåíèåy (x)óðàâíåíèÿ (1) íàçîâåìêîëåáëþùèìñÿíà[a, b],åñëè îíî áîëåå îäíîãî ðàçà îáðàùàåòñÿ â íóëü íà ýòîì îòðåçêå,èíà÷å -íåêîëåáëþùèìñÿ.Òåîðåìû îòíîñèòåëüíî íóëåé ðåøåíèÿ óðàâíåíèÿ00y + Q(x)y = 0.Òåîðåìà 1.Êîëåáëþùååñÿ ðåøåíèå íå ìîæåò èìåòü êîíå÷íóþ òî÷êóíàêîïëåíèÿ íóëåé.
Èíûìè ñëîâàìè, âñå íóëè ðåøåíèÿ0óðàâíåíèÿ (1 ) - èçîëèðîâàííûå òî÷êè.Äîêàçàòåëüñòâî.x0 - ïðåäåë {x1 , x2 , ..., xn , ...}, xi - íóëè= x0 .00= 0 ⇒ y1 (x0 ) = 0 ⇒ y1 (x0 ) = y1 (x0 ) = 0, òîÎò ïðîòèâíîãî: ïóñòüy1 (x),òî åñòü limn→∞ xnÒîãäày1 (xn )−y1 (x0 )xn −x0Çàäà÷à Øòóðìà - Ëèóâèëëÿ0(1 )åñòüy1 (x) ≡ 0,è â ñèëó åäèíñòâåííîñòè ðåøåíèÿ ïîëó÷àåìïðîòèâîðå÷èå.Òåîðåìà 2.ÏóñòüQ(x) ≤ 0, x ∈ [a, b].0Òîãäà âñå ðåøåíèÿ óðàâíåíèÿ (1 ) -íåêîëåáëþùèåñÿ.Äîêàçàòåëüñòâî.Äîïóñòèì, ÷òî íåêîòîðîå ðåøåíèåy1 (x) èìååò õîòÿ áû äâàx0 , x1 , x0 < x1 è ïóñòü â (x0 , x1 ) íåò äðóãèõ íóëåé y1 (x).Òîãäà y1 (x) ñîõðàíÿåò çíàê íà (x0 , x1 ).
Ïóñòü, íàïðèìåð,0y1 (x) > 0 ⇒ (ñì. ðèñ. 1) y1 (x0 ) > 0. Åñëè Q(x) ≤ 0, òî000y1 (x) ≥ 0 ïðè x ∈ (x0 , x1 ) ⇒ y1 (x) íå óáûâàåò, çíà÷èò0y1 (x1 ) ≥ y1 (x0 ) + y1 (x0 )(x1 − x0 ) > 0, ïðîòèâîðå÷èå.íóëÿ:×òî è òðåáîâàëîñü.Òåîðåìà 3. (Øòóðìà)Ìåæäó äâóìÿ ïîñëåäîâàòåëüíûìè íóëÿìè îäíîãî ðåøåíèÿóðàâíåíèÿ âòîðîãî ïîðÿäêà ëåæèò òî÷íî îäèí íóëü äðóãîãîðåøåíèÿ òîãî æå óðàâíåíèÿ, ëèíåéíî íåçàâèñèìîãî ñ ïåðâûì.Èíûìè ñëîâàìè, íóëè ëèíåéíî íåçàâèñèìûõ ðåøåíèéðàçäåëÿþò äðóã äðóãà.Çàäà÷à Øòóðìà - ËèóâèëëÿÄîêàçàòåëüñòâî.1)00y1 (x)y2 (x) − y2 (x)y1 (x) = W (x),(3)îïðåäåëèòåëü Âðîíñêîãî.Äîïóñòèì, ÷òî íàíåò íóëåé y2 (x).
Òîãäà â ñèëóy1 è y2 y2 (x0 ) 6= 0 è y2 (x1 ) 6= 0, òàêÇíà÷èò W (x) ñîõðàíÿåò çíàê, íàïðèìåð,(x0 , x1 )ëèíåéíîé íåçàâèñèìîñòèèíà÷åêàêW (x0 ) = 0.W (x) > 0.Ðàçäåëèì (3) íà y22 (x):W (x)d y1=( ) ⇒2y2dx y2Z x1y1 x1W (ξ)=dξ,y2 x0ξ2x0ïðîòèâîðå÷èå. Ñëåäîâàòåëüíî, åñòü õîòÿ áû îäèí íóëü. Åñëè èõäâà, òîy2 (x̄0 ) = y2 (x̄1 ) = 0,ïðîòèâîðå÷èå.Ñëåäñòâèå 1.Åñëè íà(a, b)îäíî ðåøåíèå èìååò áîëåå äâóõ íóëåé, òî âñåðåøåíèÿ êîëåáëþùèåñÿ.Çàäà÷à Øòóðìà - Ëèóâèëëÿyx0åñòü òàêàÿòî÷êàÐèñ.
1Òåîðåìà 4. (Òåîðåìà ñðàâíåíèÿ)Ïóñòü çàäàíû äâà óðàâíåíèÿ íà[a, b]:00y + Q1 (x)y = 0,00z + Q2 (x)z = 0Çàäà÷à Øòóðìà - Ëèóâèëëÿ(4)(5)èQ2 (x) ≥ Q1 (x)íà(a, b).Òîãäà ìåæäó äâóìÿ ïîñëåäîâàòåëüíûìè íóëÿìè ïåðâîãîóðàâíåíèÿ íàéäåòñÿ, ïî êðàéíåé ìåðå, îäèí íóëü âòîðîãîóðàâíåíèÿ. Èíûìè ñëîâàìè, ðåøåíèå âòîðîãî óðàâíåíèÿêîëåáëåòñÿ áûñòðåå (ñì. ðèñ. 2)baÐèñ. 2Äîêàçàòåëüñòâî.Ïóñòüx0 , x1- äâà ïîñëåäîâàòåëüíûõ íóëÿÎò ïðîòèâíîãî: ïóñòü íåò íóëåéȳ (x) > 0, z̄(x) > 0íàx0 , x1 ,ȳ (x)ðåøåíèÿ (4).z̄(x) ðåøåíèÿ (4).
Çíà÷èò,ȳ (x) ìîíîòîííî âîçðàñòàåòòî åñòüÇàäà÷à Øòóðìà - Ëèóâèëëÿâïðàâî îò0x0è ìîíîòîííî óáûâàåò ñëåâà îòȳ (x1 ) < 0.000x1 . ⇒ ȳ (x0 ) > 0,00ȳ z̄(x) − z̄ ȳ (x) = [Q2 − Q1 ]ȳ z̄,Z x1 00 x1ȳ z̄ − z̄ ȳ x =(Q2 − Q1 )ȳ z̄dx0ëåâàÿ ÷àñòü< 0,ïðàâàÿ(6)x0≥ 0,ïðîòèâîðå÷èå.Òåîðåìà äîêàçàíà.Åñëè äîïîëíèòåëüíîx ∈ (x0 , x1 )z̄(x) áóäåò ëåâåå x1 .çíà÷åíèÿQ2 (x) > Q1 (x) õîòÿ áû äëÿ íåêîòîðîãîz̄(x0 ) = 0, òî ñëåäóþùèé âïðàâî íóëüèÈòàê, äîêàçàíà ñëåäóþùàÿ òåîðåìà:Òåîðåìà 5.0Åñëè äëÿ óðàâíåíèÿ (1 )ȳ (x)èz̄(x)x0- îáùèé íóëü äâóõ ÷àñòíûõ ðåøåíèéêàæäîãî èç óðàâíåíèé è åñëè ìåæäóx0èñëåäóþùèì íóëåì x1 ðåøåíèÿ ȳ (x) ñóùåñòâóþò òî÷êèQ2 (x) > Q1 (x), à âñþäó Q2 (x) − Q1 (x) ≥ 0, òî áëèæàéøèéñïðàâà ê x0 íóëü z̄(x) ëåæèò ëåâåå x1 .Çàäà÷à Øòóðìà - ËèóâèëëÿÏðèìåð 3.Òåîðåìó 5 èñïîëüçóþò, áåðÿ â êà÷åñòâå îäíîãî óðàâíåíèÿ00y + a2 y = 0,a- const, à â êà÷åñòâå âòîðîãî00y + Q(x)y = 0,Q(x) íåïðåðûâíà íà [a, b].M = max Q(x); m = min Q(x).00M > m.
Ðàññìàòðèâàÿ óðàâíåíèå y + my = 0(7)ãäå ôóíêöèÿÏóñòüïîëó÷àåì, ÷òîπðàññòîÿíèå ìåæäó ñîñåäíèìè íóëÿìè óðàâíåíèÿ (7) ìåíüøå √πè áîëüøå √ , òî åñòü ïîëó÷àåì îöåíêó ñâåðõó è ñíèçómMðàññòîÿíèÿ ìåæäó ïîñëåäîâàòåëüíûìè íóëÿìè êîëåáëþùåãîñÿðåøåíèÿ.Çàäà÷à Øòóðìà - ËèóâèëëÿÏðèìåð 4. (Óðàâíåíèå Áåññåëÿ)000x 2 y + xy + (x 2 − n2 )y = 0.Çàìåíà1y = x−2 zïðèâîäèò óðàâíåíèå (8) ê âèäó00z + (1 −Ñðàâíèâàåì ñn2 − 41)z = 0.x200w +w =0 ⇒π < dist(xn , xn+1 )ïðèèdist(xn , xn+1 )Çíà÷èò, ïðè(8)x →∞< π,dist(xn , xn+1 )0n>121≤n< .2→ π.Ðàññìîòðèì çàäà÷ó00y + Q(x, λ)y = f (x),a < x < b,Çàäà÷à Øòóðìà - Ëèóâèëëÿ(9)óäîâëåòâîðÿþùóþ ãðàíè÷íûì óñëîâèÿì0y (a)l1 + y (a)l2 = 0,0y (b)r1 + y (b)r2 = 0,ãäåλ- íåêîòîðûé êîìïëåêñíûé ïàðàìåòð.Óðàâíåíèå (8) ìîæíî ïåðåïèñàòü â âèäå: 0 yy0=A 0 +,0f (x)yyA=01−Q0.Òî åñòü ýòî ÷àñòíûé ñëó÷àé êðàåâîé çàäà÷è èç 13 (ïðè(10)N = 2,L = (l1 , l2 ), R = r1 , r2 ).Ïóñòü â (8) f (x) ≡ 0, Q(x, λ) = λ − q(x), q(x) - íåïðåðûâíà íà[a, b], l1,2 , r1,2 íå çàâèñÿò îò λ è l12 + l22 6= 0, r12 + r22 6= 0.M = 1),ïðè ýòîìÏóñòülp 1= cos α,2l1 + l22r1√ 2= cos β,r1 + r22lp 2= sin α,2l1 + l22r2√ 2= cos β.r1 + r22Çàäà÷à Øòóðìà - ËèóâèëëÿÒîãäà ïîëó÷èì êðàåâóþ çàäà÷ó Øòóðìà - Ëèóâèëëÿ (Ç.
Ø. Ë.) 00 y + [λ − q(x)]y = 0, x ∈ (a, b);0y (a) cos α + y (a) sin α = 0,0y (b) cos β + y (b) sin β = 0,λ = λ0(11)- íàçûâàåòñÿ ñîñáñòâåííûìè çíà÷åíèÿìè Ç. Ø. - Ë.,åñëè ïðèλ = λ0 çàäà÷à èìååò íåòðèâèàëüíîåy = y (x, λ0 ) - ñîáñòâåííàÿ ôóíêöèÿ.ðåøåíèåÏðèìåð 5.00y + λy = 0,y (0) = y (b) = 0.q(x) ≡ 0, α = β = 0, a = 0.λ ≤ 0, òî âñå ðåøåíèÿ óðàâíåíèÿÇäåñüÅñëèíåêîëåáëþùèåñÿ è,ñëåäîâàòåëüíî, ãðàíè÷íûì óñëîâèÿì íå óäîâëåòâîðÿþò, òî åñòüλ ≤ 0 íåò ñîáñòâåííûõÏóñòü λ > 0.
Òîãäàïðèçíà÷åíèé.√√y (x) = C1 sin λx + C2 cos λxÇàäà÷à Øòóðìà - Ëèóâèëëÿ√y (b) = C1 sin λb = 0y (0) = C2 = 0,yk (x) = sinkπxb⇒- ñîáñòâåííûå ôóíêöèè.Íåêîòîðûå ñâîéñòâà ñîáñòâåííûõ ôóíêöèé è ñîáñòâåííûõçíà÷åíèé.1) Âñå ñîáñòâåííûå ôóíêöèè, îòâå÷àþùèå ñîáñòâåííîìóçíà÷åíèþλ0 ,ïðîïîðöèîíàëüíû, òî åñòü ïðåäñòàâëÿþòñÿ â âèäåC1 y (x, λ0 ). ñàìîì äåëå, åñëèy (x, λ0 )ôóíêöèÿòîæå óäîâëåòâîðÿåò óðàâíåíèþ èC1 y (x, λ0 )- ñîáñòâåííàÿ ôóíêöèÿ, òî ëþáàÿãðàíè÷íûì óñëîâèÿì.Ïóñòü íàøëàñü ñîáñòâåííàÿ ôóíêöèÿíåçàâèñèìàÿ ñy (x, λ0 ).w (x, λ0 ),ëèíåéíîÒîãäà ëþáîå ðåøåíèå óðàâíåíèÿ00y + [λ0 − q(x)]y = 0ïðåäñòàâèìî â âèäåC1 y (x, λ0 ) + C2 w (x, λ0 ),Çàäà÷à Øòóðìà - Ëèóâèëëÿòî åñòü ëþáîå ðåøåíèå óäîâëåòâîðÿåò äîïîëíèòåëüíûìãðàíè÷íûì óñëîâèÿì, ÷òî íåâîçìîæíî.2) Ñîáñòâåííûå ôóíêöèèñîáñòâåííûì çíà÷åíèÿìZy (x, λ1,2 ), ñîîòâåòñòâóþùèå ðàçíûìλ1 6= λ2 óäîâëåòâîðÿþò óñëîâèþby (x, λ1 ) · y (x, λ2 )dx = 0aòî åñòü óñëîâèþ îðòîãîíàëüíîñòè ôóíêöèé.Äåéñòâèòåëüíî,Zb0000y (x, λ1 )y (x, λ2 ) − y (x, λ2 )y (x, λ1 )]+a+(λ1 − λ2 )y (xλ1 )y (x, λ2 ) dx = 0Z(λ1 − λ2 )⇒by (x, λ1 )y (x, λ2 )dx = 0.a3) Âñå ñîáñòâåííûå çíà÷åíèÿ Ç.
Ø. - Ë. âåùåñòâåííû.Çàäà÷à Øòóðìà - ËèóâèëëÿÏóñòü ñóùåñòâóåòλ0- êîìïëåêñíîå ñîáñòâåííîå çíà÷åíèå, òîåñòü00y (x, λ0 ) + [λ0 − q]y (x, λ0 ) = 0.Íî òîãäà00[y (x, λ0 )] + [λ̄0 − q(x)]ȳ = 0ZbZby (x, λ0 )y (x, λ̄0 )dx = 0 =a⇒|y (x, λ0 )|2 dx,a÷òî è òðåáîâàëîñü. 13 áûëî äîêàçàíî, ÷òî âñå ñîáñòâåííûå çíà÷åíèÿ Ç. Ø. - Ë.- ýòî íóëè öåëîé àíàëèòè÷åñêîé ôóíêöèèLY (a)detRY (b)4(λ) =detY (a).Çàäà÷à Øòóðìà - Ëèóâèëëÿ íàøåì ñëó÷àå4(λ) = y1 (a, λ) cos α + y10 (a, λ) sin α y2 (a, λ) cos α + ... y1 (b, λ) cos β + y 0 (b, λ) sin β y2 (b, λ) cos β + ... 1= y1 (a, λ) y2 (a, λ) 0 y (a, λ) y 0 (a, λ) 12ãäåy1,2 (x, λ) - äâà ëèíåéíîy + [λ − q(x)]y = 0.00,íåçàâèñèìûõ ðåøåíèÿÁûëî äîêàçàíî, ÷òî åñëè ñîáñòâåííûå çíà÷åíèÿ ñóùåñòâóþò, òîèõ íå áîëåå, ÷åì ñ÷åòíîå ÷èñëî è îíè èçîëèðîâàíû.Äëÿ Ç.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
