В.П. Моденов - Дифференциальные уравнения (курс лекций), страница 10

ïðèðàù.|F (wn )| 6 K|wn |.|wn+1 | 6 |wn + hF (wn ) + hεn | 66 |wn | + hK|wn | + h|εn | == |wn |(1 + Kh) + h|εn |.Ñëåäîâàòåëüíî,|w1 | 6 |w0 |(1 + Kh) + h|ε0 |,|w2 | 6 |w1 |(1 + Kh) + h|ε1 | 66 |w0 |(1 + Kh)2 + h(1 + Kh)|ε0 | + h|ε1 |,···························Àíàëîãè÷íî,|wn | 6 |w0 |(1 + Kh)n + h(1 + Kh)n−1 |ε0 | + h(1 + Kh)n−2 |ε1 | + · · · + h|εn−1 |{z}|(*)n ñëàãàåìûõÓ÷èòûâàÿ, ÷òî n 6 N , h =H,NhN = H , 1 < (1 + Kh)n 6 (1 + Kh)N ,° °° °° °° °°(h)°°(h)°°° , |ε1 | 6 ° ε ° , ..., |εn−1 | 6 °(h)ε|w0 | = |ε0 | 6 °° ε °,° °° °èç íåðàâåíñòâà (*)ïîëó÷èì:=H° °z }| {°(h)°N°|wn | 6 °· · + h})° ε ° (1 + Kh) (1 + |h + ·{zN ñëàãàåìûõ° °°(h)°H°h⇐⇒ |wn | 6 °° ε ° (1 + H)(1 + Kh)° °°(h)°KH°=⇒ |wn | 6 °° ε ° (1 + H)e , ∀n.Ïîýòîìó, ∃h0 , ∀h 6 h0 :° °°(h)°(h)(h)(h)(h)KH°k w k < C1 °° ε ° , ãäå w = z − y , C1 = (1 + H)e , ÷.ò.ä.Óñòîé÷èâîñòü ñõåìû Ýéëåðà äîêàçàíà.Ñëåäñòâèå 7.1. Ïî Ò7.1 ïîðÿäîê ñõîäèìîñòè óñòîé÷èâîé ñõåìû ñîâïàäàåò ñïîðÿäêîì àïïðîêñèìàöèè, ò.å.

äëÿ ñõåìû Ýéëåðà, â ñèëó Óòâ.1, è àïïðîêñèìàöèÿè ñõîäèìîñòü ïåðâîãî ïîðÿäêà.767.5àçíîñòíàÿ êðàåâàÿ çàäà÷à.àññìîòðèì äèåðåíöèàëüíóþ çàäà÷ó′′y − q(x)y = f (x), q(x) > 0, 0 < x < l,y(0) = 0,y(l) = 0â îïåðàòîðíîì âèäå ′′  y − q(x)y   f (x)y(0)0Ly === ϕ. y(l)0Çàìåíÿÿ âûðàæåíèå äëÿ âòîðîé ïðîèçâîäíîé íà ñåòêå ðàçíîñòíûì îòíîøåíèåìyn+1 −ynh− yn −yhn−1yn+1 − 2yn + yn−1,=hh2ïîëó÷èì ðàçíîñòíóþ êðàåâóþ çàäà÷ó:  yn+1 −2yn +yn−1fn −qynn2h(h)(h)0 = ϕ=Lh y =y0 0yNy ′′ ≈ÑËÀÓ èç N + 1 óðàâíåíèé èìååò òðåõäèàãîíàëüíóþ ìàòðèöó .

Ñõåìàíåÿâíàÿ .7.6Ìåòîä ðàçíîñòíîé (àëãåáðàè÷åñêîé) ïðîãîíêè.àññìîòðèì ìåòîä ðåøåíèÿ ÑËÀÓ ñ òðåõäèàãîíàëüíîé ìàòðèöåé:An yn+1 − Cn yn + Bn yn−1 + Fn = 0,y0 = αy1 + β,yN = γyN −1 + δ,(n = 1, N − 1)(7.4)(7.5)(7.6)ãäå óñëîâèÿ (7.5) è (7.6) çàäàþò ãðàíè÷íûå óñëîâèÿ íà ëåâîì è ïðàâîì êîíöàõèíòåðâàëà èíòåãðèðîâàíèÿ äèåðåíöèàëüíîãî óðàâíåíèÿ. Ïðåäïîëîæèì:(*)An , Bn , Cn | > 0, Cn > An + Bn0 6 α < 1, 0 6 γ < 1,Äëÿ ðàçíîñòíîé ñõåìû êðàåâîé çàäà÷è ýòè óñëîâèÿ âûïîëíåíû.(h)Èùåì ðåøåíèå óðàâíåíèé (7.4) (çíà÷åíèÿ ñåòî÷íîé óíêöèè y ) â âèäå(èíäóöèðóåìîì ãðàíè÷íûì óñëîâèåì (7.5)):yn−1 = αn yn + βn ,(7.7)ãäå αn , βn - íåèçâåñòíûå êîýèöèåíòû, ïðè÷åìα1 = α,β1 = β.77(7.8)Ïîäñòàâëÿÿ (7.7) â (7.4), ïîëó÷èì ÑËÀÓ îòíîñèòåëüíî yn , yn+1 :(1 · yn − αn+1 · yn+1 = βn+1 ,(Bn αn − Cn ) · yn + An · yn+1 = −(Fn + Bn βn )Äëÿ ∀n èìååì òîæäåñòâà, êîãäà êîýèöèåíòû ïðîïîðöèîíàëüíû:1−αn+1βn+1==.Bn αn − CnAn−(Fn + Bn βn )Îòñþäà íàõîäèì "ïðîãîíî÷íûå"êîýèöèåíòûαn+1 =Fn + Bn βnAn, βn+1 =Cn − Bn αnCn − Bn αn(7.9)(n=1,N −1)Çàìå÷àíèå.

Äîêàæåì, ÷òî ∀n ñïðàâåäëèâî ñëåäóþùåå äâîéíîå íåðàâåíñòâî:(7.10)0 6 αn < 1 ñàìîì äåëå, èç (*) ñëåäóåò, ÷òîCn > An + Bn ⇐⇒ Cn = An + Bn + Dn , ãäå Dn > 0. ñèëó(7.9) αn+1 =AnAn⇐⇒ αn+1 =.Cn − Bn αnAn + [Bn (1 − αn ) + Dn ](**)Âîñïîëüçîâàâøèñü(7.8)è(∗) ïîëó÷èì:åñëèαn−1åñëèα1 < 1, òî[B1 (1 − α1 ) + D1 ] > 0 =⇒ α2 < 1,åñëèα2 < 1, òî[B2 (1 − α2 ) + D2 ] > 0 =⇒ α3 < 1,(∗∗)(∗∗)··········································< 1, òî [Bn−1 (1 − αn−1 ) + Dn−1 ] > 0 =⇒ αn < 1.(∗∗)Èòàê, ∀n : αn < 1. Ïðè ýòîì Cn − Bn αn > 0, òàê êàêCn − Bn αn > Cn − Bn > An > 0(ïî óñë. (*)).Ñëåäîâàòåëüíî∀n : αn > 0."Ïðÿìàÿ ïðîãîíêà". Çíàÿ êîýèöèåíòû αn è βn ïðè n = 1: α1 = α, β1 =β (7.8), íàõîäèì ïî îðìóëå (7.9) ýòè êîýèöèåíòû ïðè n = N − 1: αN , βN .Ñëåäîâàòåëüíî, â ñèëó (7.7) è (7.6) èìååì ñèñòåìó óðàâíåíèé:(yN −1 = αN yN + βN ,(7.11)yN = γyN −1 + δ.78Îòñþäà, èñêëþ÷àÿ yN −1 , íàõîäèì yN - ðåøåíèå íà ïðàâîì êîíöå èíòåðâàëàèíòåãðèðîâàíèÿ óðàâíåíèÿ:yN =γβN + δ,1 − γαN(7.12)ãäå 1 − γαN > 0, òàê êàê γ < 1 è αN < 1."Îáðàòíàÿ ïðîãîíêà".

Ïîëàãàÿ â îðìóëå (7.7)yn−1 = αn yn + βnïîñëåäîâàòåëüíî n = N, N − 1, · · · , 1, è çíàÿ "ïðîãîíî÷íûå"êîýèöèåíòû,íàõîäèì çíà÷åíèÿ â óçëàõ ñåòêè:yN −1 , yN −2 , · · · , y1 , y0 ,à, ñëåäîâàòåëüíî, y0 - ðåøåíèå íà ëåâîì êîíöå èíòåðâàëà èíòåãðèðîâàíèÿóðàâíåíèÿ.79ëàâà 8Ïîíÿòèå îá àñèìïòîòè÷åñêèõìåòîäàõ.8.1åãóëÿðíî âîçìóùåííàÿ çàäà÷à.àññìîòðèì íà÷àëüíóþ çàäà÷ó äëÿ íàõîæäåíèÿ óíêöèè(dy= f (y, x, µ),y(x, µ) : dxy(0, µ) = y 0 ,(8.1)ãäå µ - ìàëûé ïàðàìåòð.

Ïîëàãàÿ îðìàëüíî µ = 0, ïîëó÷èì çàäà÷ó äëÿîïðåäåëåíèÿ óíêöèè(dȳ= f (ȳ, x, 0),y(x, 0) = ȳ(x) : dx(8.2)ȳ(0) = y 0 .Çàäà÷ó (8.2) íàçûâàþò íåâîçìóùåííîé , à (8.1) - âîçìóùåííîé .Îïðåäåëåíèå8.1.Áóäåìíàçûâàòüóíêöèþ f (y, x, µ) ðåãóëÿðíî âîçìóùåííîé, åñëè â íåêîòîðîé îáëàñòè ÊÏ ïðîñòðàíñòâà (y, x, µ) îíà îáëàäàåò íåïðåðûâíûìè è ðàâíîìåðíî îãðàíè÷åííûìè÷àñòíûìè ïðîèçâîäíûìè ïî y è µ äî âòîðîãî ïîðÿäêà âêëþ÷èòåëüíî.Ëåììà 8.1. Åñëè y = y(x, µ) - ðåøåíèå (8.1), òî äèåðåíöèðóÿ (8.1) ïî µ,ïîëó÷èì íà÷àëüíóþ çàäà÷ó äëÿ óíêöèè ³ ´∂yd= ∂f·dx∂µ∂y∂y: ³ ´¯∂µ  ∂y ¯¯=0∂µ∂y∂µ+∂f∂µx=0Òåîðåìà 8.1. Ïóñòü â íåêîòîðîé îáëàñòè ÊÏ - ïðîñòðàíñòâà (y, x, µ) óíêöèÿf (y, x, µ) ÿâëÿåòñÿ ðåãóëÿðíî âîçìóùåííîé.Òîãäà ñóùåñòâóåò ñåãìåíò 0 6 x 6 H , íà êîòîðîì äëÿ ðåøåíèÿ y(x, µ) çàäà÷è(8.1) èìååò ìåñòî àñèìïòîòè÷åñêîå ïðåäñòàâëåíèåy(x, µ) = ȳ(x) + µ∂y(x, 0) + O(µ2 ),∂µãäå ȳ(x) = y(x, 0) - ðåøåíèå ñîîòâåòñòâóþùåé íåâîçìóùåííîé çàäà÷è (8.2).80(*)Äîêàçàòåëüñòâî àçëîæèì y â ðÿä ïî ïàðàìåòðó µ:y = y0 (x) + µy1 (x) + · · ·Ïîäñòàâèì ýòî ðàçëîæåíèå â (8.1) è ðàçëîæèì ïðàâóþ ÷àñòü óðàâíåíèÿ:(∂fdy01+ µ dy+ · · · = f (y0 , x, 0) + ∂f(y0 , x, 0)(µy1 + · · · ) + ∂µ(y0 , x, 0)µ + · · · ,dxdx∂yy0 (0) + µy1 (0) + · · · = y 0 .(8.3)Ïðèðàâíèâàåì ÷ëåíû ïðè îäèíàêîâûõ ñòåïåíÿõ µ:(dy0= f (y0 , x, 0),0ïðè µ : dxy0 (0) = y 0 ;ïðè µ1 :(dy1dx= ∂f(y0 , x, 0)y1 +∂yy1 (0) = 0.∂f(y , x, 0),∂µ 0Ïåðâàÿ çàäà÷à ñîâïàäàåò ñ íà÷àëüíîé çàäà÷åé (8.2) è â ñèëó òåîðåìûåäèíñòâåííîñòèy0 (x) = ȳ(x).Âòîðàÿ çàäà÷à â ñèëó ëåììû äàåòy1 (x) =∂y(x, 0).∂µÑëåäîâàòåëüíî ðàçëîæåíèå ïî ïàðàìåòðó èìååò âèäy(x, µ) = ȳ(x) + µ8.2∂y(x, 0) + O(µ2 ), ÷.ò.ä.∂µÑèíãóëÿðíî âîçìóùåííàÿ çàäà÷à.àññìîòðèì ñèñòåìó óðàâíåíèé, ñîäåðæàùóþ ìàëûé ïàðàìåòð ïðè ïðîèçâîäíîé.Òðåáóåòñÿ íàéòè óíêöèè z = z(x, µ), y = y(x, µ)ïðè ðåøåíèè çàäà÷è dzµ dx = F (z, y, x), dy = f (z, y, x),dx(8.4)0z(0,µ)=z,y(0, µ) = y 0 ,dzãäå 0 < µ - ìàëûé ïàðàìåòð.

Ïðàâàÿ ÷àñòü óðàâíåíèÿ dx= µ1 F (z, y, x) íåÿâëÿåòñÿ ðåãóëÿðíî âîçìóùåííîé. Áóäåì íàçûâàòü ýòó ñèñòåìó ñèíãóëÿðíîâîçìóùåííîé . Ïîëàãàÿ îðìàëüíî µ = 0, ïîëó÷èì íåâîçìóùåííóþ ñèñòåìó(âûðîæäåííóþ ñèñòåìó):(0 = F (z, y, x),(8.5)dy= f (z, y, x),dx81ãäå ïåðâîå óðàâíåíèå ñèñòåìû - êîíå÷íîå îòíîñèòåëüíî z .Ïðåäïîëîæèì, ÷òî ýòî óðàâíåíèå èìååò äåéñòâèòåëüíûå èçîëèðîâàííûå êîðíè z =ϕ(y, x).Ïóñòü z = ϕ(ȳ, x) - îäèí èç êîðíåé êîíå÷íîãî óðàâíåíèÿ.Òîãäà ïîëó÷èì âûðîæäåííóþ çàäà÷ó(dȳ= f (ϕ(ȳ, x), ȳ, x),dx(8.6)ȳ(0) = y 0 .Òåîðåìà 8.2 (Òåîðåìà Òèõîíîâà).

Ïóñòü:10 . F (z, y, x), f (z, y, x), Fz′ , Fy′ , fz′ , fy′ - íåïðåðûâíû â íåêîòîðîé îáëàñòè òðåõïåðåìåííûõ (z, y, x) : G = D̄ × Z, (y, x) ∈ D̄, z ∈ Z .20 . ϕ(y, x), ϕ′y - íåïðåðûâíû â D̄.30 . ∃ȳ(x) - ðåøåíèå âûðîæäåííîé çàäà÷è (8.6) íà ñåãìåíòå 0 6 x 6 H .40 . ϕ(ȳ, x) - óñòîé÷èâûé êîðåíü â D̄ ⇐⇒Def¯∂F ¯¯< 0, âD̄∂z ¯z=ϕ(ȳ,x)50 . Íà÷àëüíîå çíà÷åíèå z 0 ïðèíàäëåæèò îáëàñòè âëèÿíèÿ óñòîé÷èâîãî êîðíÿϕ(y 0 , 0) óðàâíåíèÿ F (z 0 , y 0 , 0) = 0 ⇐⇒ ò.å. åñëè ϕ1 è ϕ2 - äâà áëèæàéøèõDefê ϕ êîðíÿ ñîîòâåòñòâåííî ñíèçó è ñâåðõó, òî íåîáõîäèìî, ÷òîáû z 0 ëåæàëîâ îáëàñòè (ϕ1 , ϕ2 ), íàçûâàåìîéîáëàñòüþ âëèÿíèÿ (èëè îáëàñòüþïðèòÿæåíèÿ) êîðíÿ.Òîãäà çàäà÷à (8.6) èìååò ðåøåíèå z(x, µ), y(x, µ) íà ñåãìåíòå 0 6 x 6 H .Ñïðàâåäëèâ ïðåäåëüíûé ïåðåõîälimy(x, µ) = ȳ(x) ïðè 0 6 x 6 Hµ→0limz(x, µ) = ϕ(ȳ(x), x) ïðè 0 6 x 6 H,µ→0ãäå ȳ(x) - ðåøåíèå âûðîæäåííîé çàäà÷è.82Ëèòåðàòóðà[1℄ Òèõîíîâ À.Í., Âàñèëüåâàóðàâíåíèÿ.

Ì.: Íàóêà, 1998.À.Á.,ÑâåøíèêîâÀ..Äèåðåíöèàëüíûå[2℄ Ýëüñãîëüö Ë.Ý. Äèåðåíöèàëüíûå óðàâíåíèÿ è âàðèàöèîííîå èñ÷èñëåíèå.Ì.: Íàóêà, 1965.[3℄ Ñòåïàíîâ Â.Â. Êóðñ äèåðåíöèàëüíûõ óðàâíåíèé. Ì.: ÎÈÇ. îñòåõèçäàò,1945.[4℄ Ôèëèïïîâ À.Ô. Ñáîðíèê çàäà÷ ïî äèåðåíöèàëüíûì óðàâíåíèÿì. Ì.: Íàóêà,1970.83.