И.О. Арушунян - Конспект лекций (1160471), страница 9

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ïîð.?) ïðàêòè÷åñêèõ çàäà÷àõ, êàê ïðàâèëî, íà íà÷àëüíûõ îòðåçêàõ ðåøåíèå îñöèëëèðóåò,ïîòîì ñòàáèëèçèðóåòñÿ. Ìîæíî ïðèìåíèòü ðàññìîòðåííîå ïðàâèëî äëÿ óâåëè÷åíèÿøàãà. Ñðàâíèâàåì ïîãðåøíîñòü ñ çàäàííîé òî÷íîñòüþ ε, è åñëè òî÷íîñòü "ñ çàïàñîì" ,óâåëè÷èâàåì øàã â 2 ðàçà.Ìåòîäû Àäàìñà.½y 0 (x) = f (x, y)y(x0 ) = y0|x0|x1|...||-Ñåòêà äîëæíà áûòü ðàâíîìåðíîé, àäàïòèâíûå àëãîðèòìû íå ïîäõîäÿò.yn+1 =kXai yn−i + hi=0mXbi f (xn−i , yn−i )i=−sÊîýôôèöèåíòû ai , bi âûáèðàþòñÿ òàê, ÷òîáû ëîêàëüíàÿ ïîãðåøíîñòü áûëà áû ïîìåíüøå.Áóäåì ñ÷èòàòü k = 0, s = 0 ëèáî 1.µ¶s = 0 ⇒ - ìåòîä ÿâíûés = 1 ⇒ - ìåòîä íåÿâíûéßâíûå ìåòîäû Àäàìñà (ìåòîäû ïðîãíîçà)yn+1 = yn + hmXi=072bi f (xn−i , yn−i ) íà÷àëüíûõ òî÷êàõ x0 , x1 , ..., xm íóæíî çíàòü yi .

Èõ ìîæíî âû÷èñëèòü, íàïðèìåð,ìåòîäîì Ðóíãå-Êóòòà (ìåòîä ðàçãîíà).Ñ÷èòàåì yn−i = y(xn−i ). Òîãäà ìîæíî ïîäîáðàòü bi òàê, ÷òî âûïîëíÿåòñÿy(xn+1 ) − y(xn ) − hmXbi f (xn−i , y(xn−i )) = O(hm+2 )i=0Äîêàçàòåëüñòâî. Èíäóêöèÿ: m = 0. Ëîêàëüíàÿ ïîãðåøíîñòü äîëæíà áûòü O(h2 )y(xn+1 ) − y(xn ) − hb0 f (xn , y(xn )) = y(xn+1 ) − y(xn ) − hb0 y 0 (xn ) = y(xn ) + hy 0 (xn ) +O(h2 ) − hb0 y 0 (xn ) = O(h2 ).

⇒ b0 = 1 - ìåòîä Ýéëåðà.m=1h2y(xn+1 ) − y(xn ) − hb0 f (xn , y(xn )) − hb1 f (xn−1 , y(xn−1 )) = y(xn ) + hy 0 (xn ) + y 00 (xn ) +22h... − y(xn ) − hb0 y 0 (xn ) − hb1 (y(xn ) − hy 0 (xn ) + y 00 (xn ) − ...) = h(1 − b0 − b1 )y 0 (xn ) +2b1 0002 1003 14h ( + b1 )y (xn ) + h ( − )y (xn ) + O(h )262Õîòèì O(h3 ) ⇒y 0 (xn ) − b0 y 0 (xn ) − b1 y 0 (xn ) = 0y 00 (xn )+ b1 y 00 (xn ) = 02Ïîëó÷àåì ñèñòåìó óðàâíåíèé(11 − b0 − b1 = 0b1 = −21⇔3+ b1 = 0b0 =22è ðàñ÷åòíóþ ôîðìóëóhyn+1 = yn + (3f (xn , yn ) − f (xn−1 , yn−1 ))25Ëîêàëüíàÿ ïîãðåøíîñòü en = y 000 (xn )h3 +O(h4 ).

Ñëåäîâàòåëüíî ãëîáàëüíàÿ ïîãðåøíîñòü12èìååò 2-îé ïîðÿäîê.m = 2 (fn = f (xn , yn ))yn+1 = yn +h(23fn − 16fn−1 + 5fn−2 )12! Óíèâåðñàëüíàÿ ñèñòåìà íà bi äëÿ ïðîèçâîëüíîãî m.18Ëåêöèÿ 18(5).½y 0 = f (x, y)y(x0 ) = y073yn+1 = yn + hmX·bi f (xn−i yn−i ) ,s=i=−s0 − ÿâíûé1 − íåÿâíûéìåòîäûÐàññìîòðèì ñëó÷àé s = 1 :m = −1y(xn+1 )−y(xn )−h f (xn+1 , y(xn+1 )) = y(xn )+hy 0 (xn )+...−y(xn )−hb−1 (y 0 (xn )+hy 0 (xn )+|{z}y 0 (xn+1 )...) == h(1 − b−1 )y 0 (xn ) + h2 y 00 (xn )( 12 − b−1 ) + O(h3 )⇓b−1 =1Ëîê.

îøèáêà O(h2 ), à ãëîáàëüíàÿ O(h).(Ýòî íåÿâíûé ìåòîä Ýéëåðà) yn+1 = yn + f (xn+1 , yn+1 ).m=0yn+1 = yn + h(b−1 y 0 (xn + h) − hb0 y 0 (xn )) = ... = h(1 − b−1 − b0 )y 0 (xn ) + h2 ( 21 − b−1 )y 00 (xn ) +h3 000h3+ y (xn ) − b−1 y 00 (xn ) + O(h4 )62⇓(1 − b−1 − b0 = 0h1⇒ Íåÿâíûé ìåòîä òðàïåöèé yn+1 = yn + (f (xn , yn )+f (xn+1 , yn+1 ))b−1 =22ëîê.îø. e1 = O(h3 )ãë.îø. En = O(h2 )(Áåç âûâîäà)hyn+1 = yn + (5fn+1 + 8fn − fn−1 ) , En = O(h3 )12m=2hyn+1 = yn + (9fn+1 + 19fn − 5fn−1 − fn−2 )24Îáùèé ñëó÷àé: mXbi = 1 i=1mP1, j = 1, . .

. , m + 1bi (−1)j =j+1i=−1j = 1, . . . , m äëÿ s = 0m=1Âñïîìíèì, ÷òî ...÷òî-íèáóäü î÷åíü õîðîøååÄëÿ ðåøåíèÿ íåÿâíîé çàäà÷è ïðèìåíÿþò èòåðàö. ìåòîä. Íàïðèìåð, ïðè m = 1(k+1)yn+1 = yn +h(k)(5f (xn+1 , yn+1 ) + 8fn − fn−1 ),1274ãäå(0)yn+1 = yn +h(23fn − 16fn−1 + 5fn−2 ) (ÿâíûé ìåòîä Àäàìñà ïðè m = 2)12Ïðèìåðy 0 = λy= yn + hλyn+1 - ðàáîòàåò õîðîøî (ðàññì. ðàíåå)yλ < 0, /, |λ| À 1 n+1Îáùèé ñë.: æåñòêàÿ çàäà÷ày 0 = f (x, y)∂f2| |À1⇒ óñëîâèå h <∂y|λ|∂f<0∂y½ 0yn+1 = yn + hf (xn+1 , yn+1 )1 M > 0, M À 1y = f (x, y)., ïðè÷åì h <(k+1)(k)y(0)=1yn+1 = yn + hf (xn+1 , yn+1 )M (óñë-å íà æåñòêîñòü)Âîïðîñ: íà îòðåçîê êàêîé äëèíû X ìîæåì óéòè, ÷òîáû óñë-å æåñòêîñòè âûïîëíÿëîñü?X X1NÅñëè ñäåëàëè N øàãîâ, òî (h = )<èëè X <...?N NMMÏóñòü øàãè áóäóò ðàçíûìè:â çàäà÷å y 0 = λy(yn = (1 + hλ)n )yk+1 = yk + hk λyk ⇒ yn =n−1Y(1 + hk λ) y0k=0|{z}≤1N2M(Ýòî ìåòîä Ëåáåäåâà îí â êóðñ ýêçàìåíà íå âõîäèò!)Îêàçûâàåòñÿ, ÷òî â ýòîì ñëó÷àå X <00Êðàåâàÿ çàäà÷à0y = f (x, y, y )1.y(a) = αy(b) = β2.y 0 (a) = αy 0 (b) = β3.p1 y(a) + q1 y 0 (a) = αp2 y(b) + q2 y 0 (b) = β(x ∈ [a, b])àíàëîã çàäà÷è Äèðèõëå75Çàäà÷ó äëÿ óðàâíåíèÿ âòîðîãî ïîðÿäêà âñåãäà ìîæíî ñâåñòè ê ñèñòåìå óðàâíåíèéïåðâîãî ïîðÿäêà êàêîãî-òî âèäày10 = f1 (x, y1 , y2 )y20 = f2 (x, y1 , y2 )y1 (a) = α, y2 (b) = βµ~y =y1y2¶Áóäåì ðàññìàòðèâàòü ëèíåéíóþ çàäà÷ó 00y − p(x)y(x) = f (x), p(x) ≥ 0x ∈ [0, X]y(0) = ay(X) = by(xn+1 ) − 2y(xn ) + y(xn−1 )−p(xn )y(xn ) = f (xn ) + O(h2 )2h{z} |- çàìåíèëè y 00 íà ðàçíîñòí.

îòíîøåíèån = 1, ..., N − 1 y(0) = ay(X) = byn+1 − 2yn + yn−1− p n y n = fnh2y = a, yN = b 0n = 1, ..., N− 1 y1 ..~] Y = . , AY~ = FyN −1| | || | . . . . . . . . xNx021an = 1 : (− 2 − p1 )y1 + 2 y2 = f1 − 2 = F1hhh121n=2:y1 + (− 2 − p2 )y2 + 2 y3 = f2h2hh···ÑëåäîâàòåëüíîA=21− 2 − p10 ... hh2121−−p02h2h2h21.. .. .. 0h2...0Ðåøàåì ìåòîäîì ïðîãîíêè: ïðè ýòîì íåîáõîäèìî äèàãîíàëüíîå ïðåîáëàäàíèå (ñì.ïðåä. ñåìåñòð), à ýòî âûï-ñÿ, ò.å. pi ≥ 0 (ñì.í.ó.)Âîïðîñ óñòîé÷èâîñòè? ?76Ââåäåì äèôô. îïåð-ð L : Ly = y 00 (x) − p(x)y(x)yn−1 − 2yn + yn+1Íà ïð-âå ñåò.

ô-öèé ââåäåì lyn =−pn ynh2Ëåììà 1. Ïóñòü p(x) ≥ 0, l(zn ) ≤ 0, z0 ≥ 0, zN ≥ 0, òîãäà| | || | . . . . . . . . xNx0zn ≥ 0 ∀ n = 1, ..., N − 1¤ d := min0≤n≤NÏðåäïîëîæèì d < 0 , d 6= z0 è zN . Ïóñòü k − min öåëîå, ò.÷. zk = d, òîãäàzk−1 > d , zk+1 ≥ d.≥0>0(zk+1 − d) + (zk−1 − d)Ïðèìåíèì lzk =− pk d > 0h2≤0Ëåììà 2. Åñëè p(x) ≥ 0, òî ∀ {zn }äëèíà èñõ. îòðåçêà, à M = max |l(zn )|.∅¢max |zn | ≤ max{|z0 |, |ZN |} + M0≤n≤NX2, ãäå X 80<n<Nnhnhnh(X − nh)¤ ωn := |z0 |(1 −) + |zN |+M, òîãäàX{zX} ||{z2}(1)(2)ωn ≥ 0(X = N h)ωn+1 − 2ωn + ωn−1= −MÄ/Ç : äîê-òüh2Ñëåä. l(ωn ) = −M − pn ωn ≤ Ml(ωn ± zn ) ≤ −M ± l(zn ) ≤ 0, êðîìå òîãî ω0 ± z0 = |z0 | ± z0 ≥ 0 è ωN ± zN = ... ≥ 0Ò.î.

ωn ± zn óäîâë. óñë-ÿì Ëåììû 1 ⇒⇒ ωn ± zn ≥ 0 ⇒ |zn | ≤ |ωn | ≤ max |ωn |Íàéäåì max |ωn | (îöåíêó)nnnh nh(1) ≤ max{|z0 |, |zN |}(1 −+) = max{|z0 |, |zN |},XX22X.+MXnh(X + nh) ≤⇒ max |ωn | ≤ .....n48Âåðíåìñÿ ê íàøåé çàäà÷åp, f ∈ C 2 ⇒lyn = fn n=1,...,N−1y =a 0yN = b 00 y − py = f, p ≥ 0y(0) = ay(X) = by ∈ C 4 (áóäåì òàê ñ÷èòàòü)rn{z }|y (4) (ξn )h2l(y(xn )) = f +12y(x)=a0 y(x ) = bN77÷òî è òð.ä.¢zn := y(xn ) − yn½l(zn ) = rny(z0 ) = 0, y(zN ) = 0−2ïðèìåíÿåì Ëåììó 2:X,M = max |l(zn )|,n82M4 X 2|y(xn ) − yn | ≤h96M4 := max |y (4) | ⇒|y(xn ) − yn | = |zn | ≤ M⇒19Ëåêöèÿ 19(6).y 00 = f (x, y)y(a) = αy(b) = βx0pa(x ∈ [a, b])hxNpbyn+1 − 2yn + yn−1= f (xn , yn )h2y0 = α y =βNn=1,...,N −1Ñâåëè (ñì.

ïðåä. ëåêöèþ) ê Aȳ = F̄ (ȳ)Çàäàäèì íà÷. ïðèáëèæåíèå ȳ 0 , Aȳ k+1 = F̄ (ȳ (k) )Ñèñòåìà ðåøàåòñÿ ìåòîäîì ïðîãîíêè çà 8n + O(1) îïåðàöèé.Óñëîæíèì çàäà÷óy 00 = f (x, y, y 0 )y(a) = α(x ∈ [a, b])y(b) = β 0 g = f (x, y, g)y 0 = g(x)y 0 (x) := g(x), òîãäày(a) = α, y(b) = β 0y = f (x, y, g) 10y2 = y1y1 (a) = t − îòëè÷àåò îò èñõîäíîé çàäà÷èy2 (a) = αÝòî çàäà÷à Êîøè ñ íà÷. êðàåâûìè óñëîâèÿìè ⇒ y1 (b, t), y1 (b, t)Îòêóäà íàéòè t ?Õîòèì, ÷òîáû y2 (b, t) = βÒ.å. èìååì çàäà÷ó ϕ(t) = 0 (ϕ(t) = y2 (b, t) − βÅñëè íàøëè t1 , t2 : ϕ(t1 )ϕ(t2 ) < 0, òî ïðîáëåì íåò78%%%%%%%•%%t2% t3 t1%Ìåòîä ñåêóùèõ:A(x) (n × n), (Aij ) = aij (x)~~y 0 = A(x)~y (x) + f (x)~y (a) = ~y 0 - òàêóþ çàäà÷ó ðåøàòü óìååì! (ïðè óñëîâèè æåñòêîñòè)ppabB : (n − k) × n, 1 ≤ k ≤ nÏóñòü åñòü ìàòðèöûC (k × n)Ñôîðìóëèðóåì êðàåâóþ çàäà÷ó: 0 ȳ = A(x)ȳ(x) + f¯(x)B ȳ(a) = ᾱC ȳ(b) = β̄] ñíà÷àëàB ȳ = 0C ȳ = ᾱb11 ···bn−k 1ñèñòåìó:(Ñ÷èòàåì rankB = n − k). .

. b1n··· ···. . . bn−k n¯¯¯¯¯¯α1...αn−k¯. . . 0 ¯¯ b∗11. . . b∗1n äèàã. ¯.··· ··· −→  · · · . . · · · ¯ · · ·âèä¯ ∗∗0. . . 1 ¯ bn−k 1 . . . bn−k n1Åñëè ïîñëåäíèå êîìïîíåíòû ïîëîæèì = 0, òî ïîëó÷èì ÷àñòíîå∗ðåøåíèå: ȳ2 = (α1∗ , ..., αn−k, 0, ..., 0)Ôóíä.

ðåøåíèå B ȳ = 0y11.ȳ2 : ..ȳ1 :  y12 ... ... y1 n−k  − ïîäñò-åì 1 0 . ..079¯¯¯¯¯¯α1∗...∗αn−kÏîëó÷èì ȳ÷ ,À èìåííîȳ1 , ..., ȳkðåø-ÿ îäíîð. óð.ȳ1 = (−b∗1,n−k+1 , ..., −b∗n−k,n−k+1 , 1, 0, ..., 0)ȳk = (−b∗1,n , ..., −b∗n−k,n ,0, ..., 0, 1)] k ñèñòåì½ȳ 0 = A(x)ȳ= ȳi½ ȳ(a)0ȳ = A(x)ȳ + f¯(x)ȳ(a) = ȳ÷kPȳ(x) = ȳk+1 (x) +di ȳi (x)(∗)kPi=1PA(x)ȳk+1 + f + di Aȳi (x) = A(ȳk+1 + di ȳi ) + f¯ − äåéñòâèòåëüíî ïîäõîäèòñì.

(∗). ×åìó ðàâíû di - ?C ȳ(b) = β̄C(ȳk+1 + d1 ȳ1 (b) + ... + dk ȳk (b)) = βd1 C ȳ1 (b) + d2 C ȳ2 (b) + ... + dk C ȳk (b) = β̄ − C ȳk+1 (b) − ñèñòåìà k × k(áåç ä-âà) - ìàòðèöà ñèñòåìû íåâûðîæäåíà.y 00 (x) = f (x)Çàäà÷à. y(a) = αy(b) = βÂàðèàöèîííûå ìåòîäû.½−(k(x)y 0 (x))0 + p(x)y(x) = f (x)y(a) = 0; y(b) = 0; k(x) ≥ k0 > 0ZbI(y) =p(x) ≥ 0(k(x)(y 0 (x))2 + p(x)y 2 (x) − 2f (x)y(x))dxaZby ∈W21 [a, b]- ïð-âî ô-öèé, ïîëó÷. çàìûêàíèåì íîðìûkyk2W 12(y 2 + (y 0 )2 )dx, y 0 -=aîáîáù. ïðîèçâîäíàÿÊàêàÿ ñâÿçü çàäà÷è è I(y) ? Åñëè y - ðåøåíèå, òî äëÿ ëþáîé u ∈ W21 îêàçûâàåòñÿI(u) ≥ I(y).Z bZ b022I(u) =(ku + pu )dx − 2f udxaa80ïîäñòàâèì u =Z y + u − ybI(u) = I(y) +(k(x)(u0 − y 0 )2 + p(x)(u − y)2 )dx +|a{z}≥0Z+2|b(k(x)y 0 (u − y)0 + p(x)y(u − y) − f (u − y))dxa{z}=0 äîê-òü ñàìèì (ïðîèíò-òü ïî ÷àñòÿì)Ïðèáëèæåííîå ðåøåíèå èùåì â âèäåNXyN (x) =cj ϕj (x)j=1I(yN ) =NXΛ(ϕk , ϕl )ck cl − 2ZbΛ(ϕk , ϕl ) =abk ck ,k=1k=1l=1ãäåNX(k(x)ϕ0k ϕ0l + p(x)ϕk ϕl )dxZbbk =f ϕk dxaÇàòåì êîýôôèöèåíòû âûáèðàþòñÿ èç óñëîâèÿ ìèíèìóìà ôóíêöèîíàëà∂I(yn ), k = 1, ..., N∂ckÏîëó÷àåòñÿ çàäà÷à ëèí.

àëã.Ac̄ = b̄, Aij = Λ(ϕk , ϕl )20Ëåêöèÿ 20(7).½Zb] I(y) =−(k(x)y 0 (x))0 + p(x)y(x) = f (x)y(a) = 0; y(b) = 0; k(x) ≥ k0 > 0p(x) ≥ 0(k(y 0 )2 + py 2 − 2f y)dxaÐåøåíèå èùåòñÿ â ïðîñòðàíñòâå y ∈ W21 ñ íîðìîé kyk2W 1 =281Rba(y 2 + (y 0 )2 )dxÁåðåì â ïðîñòðàíñòâå íàáîð ôóíêöèéϕ1 , ..., ϕN ∈W21 ,NXy=cj ϕj (x)j=1ôóíêöèè ϕj - ëèí. íåçàâ.Äëÿ íàõîæäåíèÿ cj ïîëó÷èëè ñèñòåìó ËÀÓ:ZbAc̄ = b̄, Aij = Λ(ϕk , ϕl ) =aZ(k(x)ϕ0k ϕ0lb+ p(x)ϕk ϕl )dx, bk =f ϕk dxaÌàòðèöà A ñèìì.

è ïîëîæ. îïð. (ïðîâåðèòü ñàìèì) êà÷-âå ϕj ìîæåì âûáèðàòü xj−1 èëè sin πjx è ò.ä.Âñåãî N 2 èíòåãðàëîâ (â ìàòðèöå)N âåëèêî ⇒ ñëîæíîñòü â âû÷èñëåíèÿõ, ïîýòîìó èäåÿ ñîñòîèò â òîì, ÷òî ϕj ïîëàãàþòñÿRôèíèòíûìè. Ïðè ýòîì çàäà÷à ñîñòîèò â òîì, ÷òîáû ñîêðàòèòü ÷èñëî âû÷èñëÿåìûõ -â.a := 0, b := 1 - äëÿ ïðîñòîòû.N ôèêñ. (âìåñòî N áóäåò N − 1 ô-èé ϕj )x0xip p p p p p p p p p pNxi = , i = 0, .., NN01h = 1/N1#c#c ϕk+1#c #cϕk : ##cc## ccxk 7→ ϕk (x) : â ò. xk 1, èíà÷å :##ccppxk−1xk+1xkÏîëó÷èëè ϕ1 , ..., ϕN −1Âèäíî, ÷òî ìàòðèöà A áóäåò òðåõäèàã.

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