Диссертация (1149808), страница 3
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Èç äîêàçàòåëüñòâà ëåììû ñëåäóåò, ÷òîK0 ≤ 0.02, K1 ≤ 0.125,K2 ≤ 0.8.Çàìå÷àíèå 1.2. Ïåðåõîäÿ ê ïåðåìåííîétïî ïðàâèëóx = xk + th, t ∈ [0, 1]ïîëó÷àåì ñëåäóþùèå ôîðìóëû èñõîäíûõ áàçèñíûõ ñïëàéíîâ:14ωk,0 (xk + th) =−(5t + 1)(3t − 1)(t − 1)2 , t ∈ [0, 1],−(3t + 1)(5t − 1)(1 + t)2 , t ∈ [−1, 0],0, t ∈/ [−1, 1].− 12 th(5t − 2)(t − 1)2 , t ∈ [0, 1],122 th(2 + 5t)(1 + t) , t ∈ [−1, 0],0, t ∈/ [−1, 1]. 30t2 (t − 1)2 , t ∈ [0, 1],h<1>ωk (xk + th) =0, t ∈/ [0, 1].ωk,1 (xk + th) =(14)(15)(16)Íàì ïîòðåáóþòñÿ ôîðìóëû áàçèñíûõ ñïëàéíîâ:ωk,0 (xk + th) = −(5t + 1)(3t − 1)(t − 1)2 ,ωk+1,0 (xk + th) = −(5t − 6)(3t − 2)t2 ,1ωk,1 (xk + th) = − th(5t − 2)(t − 1)2 ,21ωk+1,1 (xk + th) = h(5t − 3)(t − 1)t2 ,2<1>ωk (xk + th) = 60t(2t − 1)(t − 1)/h2 ,(17)(18)(19)(20)(21)ôîðìóëû áàçèñíûõ ñïëàéíîâ äëÿ ïîñòðîåíèÿ ïðèáëèæåíèÿ ïðîèçâîäíîé èñõîäíîé ôóíêöèè:0ωk,0(xk + th) = −12t(5t − 3)(t − 1)/h,0ωk+1,0(xk + th) = −12t(5t2 − 7t + 2)/h,<1>ω0k(xk + th) = 60t(2t − 1)(t − 1)/h2 ,0ωk,1(xk + th) = −(t − 1)(10t2 − 8t + 1),0ωk+1,1(xk + th) = t(3 − 12t + 10t2 ),(22)(23)(24)(25)(26)ôîðìóëû áàçèñíûõ ñïëàéíîâ äëÿ ïîñòðîåíèÿ ïðèáëèæåíèÿ âòîðîé ïðîèçâîäíîé15èñõîäíîé ôóíêöèè:00ωk,0(xk + th) = −12(3 − 16t + 15t2 )/h2 ,(27)00ωk+1,0(xk + th) = −12(15t2 − 14t + 2)/h2 ,(28)<1>ω 00 k(xk + th) = 60(1 − 6t + 6t2 )/h3 ,(29)00ωk,1(xk + th) = −3(3 − 12t + 10t2 )/h,(30)00ωk+1,1(xk + th) = 3(−8t + 10t2 + 1)/h,ïðè(31)t ∈ [0, 1].ωk,0 (xk +th), ωk+1,0 (xk +th),xk = 0, h = 1.Íà ãðàôèêàõ 1 2 èçîáðàæåíû áàçèñíûå ñïëàéíûωk,1 (xk + th), ωk+1,1 (xk + th), ωk<1> (xk + th)Ðèñ.
1: Ãðàôèêè Ðèñ. 2: Ãðàôèêè ωk,0 (xk + th)ïðè(ñëåâà),ωk+1,0 (xk + th)(ñïðàâà).ωk,1 (xk +th) (ñëåâà), ωk+1,1 (xk +th) (ïî öåíòðó), ωk<1> (xk +th)(ñïðàâà).16Çàìå÷àíèå 1.3. Ïðèáëèæåíèÿ äëÿu(α) (x), α = 1, 2, x ∈ [xk , xk+1 ],ïîëó÷àåìïî ôîðìóëàì(α)(α)(α)uek (x) = u(xk )ωk,0 (x) + u(xk+1 )ωk+1,0 (x)+(α)(α)+ u0 (xk )ωk,1 (x) + u0 (xk+1 )ωk+1,1 (x) +ãäå(α)(α)(α)(α)Z!xk+1u(t)dt(ωk<1> )(α) (x),xkωk,0 (x), ωk+1,0 (x), ωk,1 (x), ωk+1,1 (x), (ωk<1> )(α) (x)îïðåäåëÿþòñÿ ôîðìóëàìè(14)(31).e (x), x ∈ [a, b], ñâÿçàííóþ ñ uUe(x) ñîîòíîøåíèåìe (x) = uUe(x), x ∈ [xk , xk+1 ).
Òåïåðü èç ëåììû 1 ñëåäóþò íåðàâåíñòâà:Ââåäåì ôóíêöèþe − uk[a,b] ≤ K0 h5 ku(5) k[a,b] ,kUe 0 − u0 k[a,b] ≤ K1 h4 ku(5) k[a,b] ,kUe 00 − u00 k[a,b) ≤ K2 h3 ku(5) k[a,b] ,kUkf k[a,b) = sup[a,b) |f |. òàáëèöàõ 1, 2 ïðèh = 0.1, h = 0.01 ñîîòâåòñòâåííî, ïðåäñòàâëåíû ïîãðåø-íîñòè ïðèáëèæåíèÿ íåêîòîðûõ ôóíêöèé è èõ ïåðâûõ ïðîèçâîäíûõ èíòåãðîäèôôåðåíöèàëüíûìè ñïëàéíàìè ïÿòîãî ïîðÿäêà. Äàëåå èñïîëüçóþòñÿ ñëåäóþùèå îáîçíà÷åíèÿ: äëÿ àáñîëþòíîé âåëè÷èíû ôàêòè÷åñêîé ïîãðåøíîñòè ôóíêöèè è åå ïåðâîé ïðîèçâîäíîé e = max |u − Ue |,R[−1,1]f1 = max |u0 − Ue0 |,R[−1,1]òåîðåòè÷åñêîé ïîãðåøíîñòè ôóíêöèè è åå ïåðâîé ïðîèçâîäíîé RèR1âåòñòâåííî.Òàáëèöà 1: Ïîãðåøíîñòè ïðèáëèæåíèé ïðèNou(x)1sin(3x) cos(5x)tg(x)cos(2x)1(1 + 25x2 )234eRe1Rh = 0.1RR10.12 · 10−4 0.81 · 10−3 0.33 · 10−30.200.16 · 10−5 0.11 · 10−3 0.69 · 10−4 0.43 · 10−10.24 · 10−7 0.17 · 10−5 0.64 · 10−6 0.40 · 10−30.21 · 10−3 0.14 · 10−1170.6 · 10−13.75äëÿñîîò-Òàáëèöà 2: Ïîãðåøíîñòè ïðèáëèæåíèé ïðèNo1234u(x)eRsin(3x) cos(5x) 0.12 · 10−9tg(x)0.24 · 10−10cos(2x)0.24 · 10−1210.23 · 10−82(1 + 25x )e1Rh = 0.01RR10.85 · 10−7 0.33 · 10−8 0.20 · 10−40.17 · 10−7 0.69 · 10−9 0.43 · 10−50.17 · 10−9 0.64 · 10−11 0.40 · 10−70.16 · 10−50.63 · 10−70.39 · 10−3 [55, c.84] îòìå÷åíî, ÷òî ïðè èíòåðïîëÿöèè ôóíêöèè Ðóíãå (1901 ã.)y(x) =11+25x2 ïðè ðàâíîîòñòîÿùèõ óçëàõ íà ïðîìåæóòêå [-1, 1] è áåñêîíå÷-íîì óâåëè÷åíèè ïîðÿäêàn èíòåðïîëÿöèîííîãî ïîëèíîìà pn ïîñëåäîâàòåëüíîñòüpn (x) ðàñõîäèòñÿ âáëèçè êîíöîâ ïðîìåæóòêà [-1, 1].
Ïîñëåäîâàòåëüíîñòü èíòåðïîëÿöèîííûõ êóáè÷åñêèõ B-ñïëàéíîâ, ýðìèòîâûõ ñïëàéíîâ, âñåãäà ñõîäèòñÿ êèíòåðïîëèðóåìîé íåïðåðûâíîé ôóíêöèè [25, 55].Íà ãðàôèêàõ 3, 4 ïðåäñòàâëåíû ïîãðåøíîñòè àïïðîêñèìàöèè ôóíêöèè Ðóíãåè åå ïåðâîé ïðîèçâîäíîé ïðåäëàãàåìûìè èíòåãðî-äèôôåðåíöèàëüíûìè ñïëàéíàìè ïÿòîãî ïîðÿäêà ïåðâîé âûñîòû íà ïðîìåæóòêå [-1,1] íà ðàâíîìåðíîé ñåòêåóçëîâ ïðè ïÿòè óçëàõ (-1, -0.5, 0, 0.5, 1) è îäèííàäöàòè óçëàõ èíòåðïîëÿöèè.Ðèñ. 3: Ãðàôèêè ïîãðåøíîñòåé àïïðîêñèìàöèè ôóíêöèè Ðóíãå (ñëåâà) è åå ïåðâîé ïðîèçâîäíîé (ñïðàâà) ïðè ïÿòè óçëàõ èíòåðïîëÿöèè.18Ðèñ. 4: Ãðàôèêè ïîãðåøíîñòåé àïïðîêñèìàöèè ôóíêöèè Ðóíãå (ñëåâà) è åå ïåðâîé ïðîèçâîäíîé (ñïðàâà) ïðè îäèííàäöàòè óçëàõ èíòåðïîëÿöèè.191.2Ìîäåëèðîâàíèå îöåíîê ïîãðåøíîñòåé àïïðîêñèìàöèèâL2äëÿ öåíòðàëüíûõ ïîëèíîìèàëüíûõ ñïëàéíîâÑìîäåëèðóåì îöåíêó àïïðîêñèìàöèè âL2äëÿ öåíòðàëüíûõ ïîëèíîìèàëü-íûõ ñïëàéíîâ.
Áóäåì èñïîëüçîâàòü ðàçëîæåíèå ïî ôîðìóëå Òåéëîðà ñ îñòàòêîìâ èíòåãðàëüíîé ôîðìå â îêðåñòíîñòè òî÷êèxj .Èìååì(x − xj )4 (4)1ũ(x) = u(xj ) + . . . +u (xj ) +4!4!Zx(x − t)4 u(5) (t)dt,xjïîýòîìóxj+11h4 (4)ũ(xj+1 ) = u(xj ) + . . . + u (xj ) +4!4!Z1h3 (4)ũ (xj+1 ) = u (xj ) + . . . + u (xj ) +3!3!Z00(xj+1 − t)4 u(5) (t)dt,xjxj+1(xj+1 − t)3 u(5) (t)dt.xjòåïåðü ñ ó÷åòîì àïïðîêñèìàöèîííûõ ñîîòíîøåíèé, ïîëó÷àåì äëÿ îñòàòêà ïðèáëèæåíèÿR:2Zxj+1R =(ũ(x) − u(x))2 dx =xjZxj+1=xj14!Zxj+1(xj+1 − t)4 u(5) (t)dt ωj+1,0 (x)+xjZ1 xj+1+(xj+1 − t)3 u(5) (t)dt ωj+1,1 (x)+3! xj!2Z ξZ xZ xj+111(ξ − τ )4 u(5) (τ )dτ ωj<1> (x) −(x − t)4 u(5) (t)dt dx.+dξ4!4!xjxjxjÄàëåå èñïîëüçóåì íåðàâåíñòâo Êîøè-ÁóíÿêîâñêîãîZ!2bXYabZX2≤a20! Zab!Y2 .Èìååì sZZsZ xj+1xj+1xj+14 (5)8(xj+1 − t) u (t)dt ≤r1 = |xj+1 − t| dt|u(5) (t)|2 dt, xjxjxj sZZsZ xj+1xj+1xj+13 (5)6(xj+1 − t) u (t)dt ≤r2 = |xj+1 − t| dt|u(5) (t)|2 dt, xjxjxjZ xj+1 Z ξ4 (5)dξ(ξ − τ ) u (τ )dτ .r3 = xjxjÏî òåîðåìå Êîøè-Áóíÿêîâñêîãî, èìååìZxj+1sZξZ4(5)xj+1Zξ(ξ−τ ) |u (τ )|dτ dξ ≤xjxjxj|u(5) (τ )|2 dτ dξsZxjxj+1xjZξ(ξ − τ )8 dτ dξ,xjïîýòîìórh11sZxj+1|u(5) (η)|2 dξ.90xjZ sZsZ xxx4 (5)8r4 = (x − t) u (t)dt ≤(x − t) dt|u(5) (t)|2 dt.
xjxjxjr3 ≤Îáúåäèíÿÿ ïîëó÷åííûå ðåçóëüòàòû, èìååì11r1 max |ωj+1,0 (x)| + r2 max |ωj+1,1 (x)|+4! x∈[xj ,xj+1 ]3! x∈[xj ,xj+1 ]|R| ≤+ 4!1 r3 maxx∈[xj ,xj+1 ] |ωj<1> (x)| + 4!1 r4 .Ñ ó÷åòîì íåðàâåíñòâ äëÿ1.875/hR2 =ïðèZx ∈ [xj , xj+1 ],xj+1xj14!rh99+|ωj+1,0 (x)| ≤ 1, |ωj+1,1 (x)| ≤ 0.0226h, |ωj<1> (x)| ≤ïîëó÷àåì0.0226h3!rh77r+h111.8751+4!h904!sZxj+1(th)9xj!29×!2|u(5) (t)|2 dt×21rdx, t ∈ [0, 1],îòñþäà5sZxj+1|R| ≤ h K|u(5) (t)|2 dt = h5 Kku(5) kL2 [xj ,xj+1 ] ,xjK = 0.0374.ãäå1.3Ìîäåëèðîâàíèå äâàæäû íåïðåðûâíîäèôôåðåíöèðóåìûõ ïðèáëèæåíèéÏóñòü ñåòêà óçëîâðîâàòü ïðèáëèæåíèå{xj } ðàâíîìåðíàÿ.
Íà êàæäîì [xk , xk+1 ) áóäåì ìîäåëèäëÿ u(x) â âèäåeuek (x) = u(xk )ωk,0 (x) + u(xk+1 )ωk+1,0 (x)+Zxk+1+ Ck ωk,1 (x) + Ck+1 ωk+1,1 (x) +!u(t)dt ωk<1> (x),(32)xkCk íàõîäèì ñ ïîìîùüþ ñëåäóþùèõ[xk , xk+1 ) ðàññìîòðèì ñîîòíîøåíèåãäå âåùåñòâåííûå ÷èñëàÍà ïðîìåæóòêåðàññóæäåíèé.euek (x) = uk ωk,0 (x) + uk+1 ωk+1,0 (x) + Ck ωk,1 (x) + Ck+1 ωk+1,1 (x)+Z xk+1+u(t)dt ωk<1> (x),(33)xkà íà ñîñåäíåì ïðîìåæóòêå[xk−1 , xk )ðàññìîòðèì ñîîòíîøåíèåeuek−1 (x) = uk−1 ωk−1,0 (x) + uk ωk,0 (x) + Ck−1 ωk−1,1 (x) + Ck ωk,1 (x)+Z xk<1>+u(t)dtωk−1(x).(34)xk−1Çäåñüui = u(xi ).Äâàæäû äèôôåðåíöèðóÿ ñîîòíîøåíèÿ (33)(34), èç óñëîâèÿ00euek (xk +)00e=uek−1 (xk −)ïîëó÷àåì ñèñòåìó óðàâíåíèéCk−1 − 6Ck + Ck+1 = fk ,22(35)ãäåZxkZfk = 8(uk+1 − uk−1 )/h + 20xk+1u(t)dt −xk−1u(t)dt /h2 ,xkk = 1, . . .
, n − 1.Òåïåðü ñîîòíîøåíèÿ äëÿ äâàæäû íåïðåðûâíî äèôôåðåíöèðóåìûõ ñïëàéíîâU˜˜k (x),[xk , xk+1 ), k = 1, 2, . . . , n − 1, îïðåäåëÿþòñÿ ñîîòíîøåíèåì (32) ñ êîýôôèöèåíòàìè Ck , ÿâëÿþùèìèñÿ ðåøåíèåì ñèñòåìû (35).íà êàæäîì ïðîìåæóòêåËåììà 2.e 4, Ke > 0.|Ck − u0k | ≤ q, q = KhÄîêàçàòåëüñòâî.(36) ñèñòåìå óðàâíåíèé (35) ñäåëàåì çàìåíó ïåðåìåííûõ. Äëÿóäîáñòâà îáîçíà÷èìu0 (xk ) = u0k .ÏîëîæèìSk = Ck − u0k .Ïîëó÷àåì ñèñòåìóóðàâíåíèéSk−1 − 6Sk + Sk+1 = Fk ,ãäåFk = fk − (u0k−1 − 6u0k + u0k+1 ), k = 1, .
. . , n − 1.Ïðèìåíÿÿ ôîðìóëó Òåéëîðà äëÿ ïðåäñòàâëåíèÿñòèxk ,Fk = h4u(x), u0k−1 , u0k+1â îêðåñòíî-ïîëó÷àåì ïîñëå ïðèâåäåíèÿ ïîäîáíûõ ÷ëåíîâ2018 (5)(u (τ1 ) + u(5) (τ2 )) − (u(5) (τ3 ) + u(5) (τ4 )) − (u(5) (τ5 ) + u(5) (τ6 )) ,5!6!4!τ1 , τ4 , τ5 ∈ [xk , xk+1 ], τ2 , τ3 , τ6 ∈ [xk−1 , xk ].Îòñþäàe|Fk | ≤ h4 Kãäåmax |u(5) (x)|,[xk−1 ,xk+1 ]e = 2 8 + 2 20 + 2 1 = 49/180 ≈ 0.2722.K5!6!4!Êàê èçâåñòíî, (ñì. [25])|Sk | ≤ q = max |Fi |.iÒàêèì îáðàçîì, íåðàâåíñòâî (36) äîêàçàíî ïðè23e = 49/180 ≈ 0.2722.KÂâåäåì ôóíêöèþee (x), x ∈ [a, b],Uñâÿçàííóþ ñeuek (x)ñîîòíîøåíèåìee (x) =Ueuek (x), x ∈ [xk , xk+1 ).Òåîðåìà 1.eÏóñòü u ∈ C 5 [a, b], uek äâàæäû íåïðåðûâíî äèôôåðåíöèðóå-ìîå ïðèáëèæåíèåñïëàéíîâ(9)(13),(32),ñìîäåëèðîâàííîå ñ ïîìîùüþ ïîëèíîìèàëüíûõ áàçèñíûõòîãäàαeee α h5−α ku(5) k[a,b] , α = 0, 1, 2,kU − uα k[a,b) ≤ K(37)e 0 = 0.5464, Ke 1 = 2.2692, Ke 2 = 5.6996.ãäå KÄîêàçàòåëüñòâî.Èìååìeuek (x) = u(xk )ωk,0 (x) + u(xk+1 )ωk+1,0 (x)+!Z xk+1+ Ck ωk,1 (x) + Ck+1 ωk+1,1 (x) +u(t)dt ωk<1> (x), x ∈ [xk , xk+1 ).xkÑ ó÷åòîì ëåììû (1), ëåììû (2) è ñîîòíîøåíèé (9)(11) ïîëó÷àåìe|RT | = |uek (x) − u(x)| ≤e≤ |uek − u(x)| + |(Ck − u0 )ωk,1 (x) + (Ck+1 − u0k+1 )ωk+1,1 (x)|(5)k≤≤ h4 max |u (x)|, x ∈ [xk , xk+1 ).[xk ,xk+1 ]Îòñþäà ïîëó÷àåì íåðàâåíñòâî (37) ñe 0 = 0.5464.KÀíàëîãè÷íî äëÿ ïðîèçâîäíûõ ôóíêöèè(α)(α)(α)euek (x) = u(xk )ωk,0 (x) + u(xk+1 )ωk+1,0 (x)+!Z xk+1(α)(α)(α)+ Ck ωk,1 (x) + Ck+1 ωk+1,1 (x) +u(t)dt ωk<1> (x), x ∈ [xk , xk+1 ).xkÑ ó÷åòîì ñîîòíîøåíèé (22)(24), (27)(29) èìååì24(α)(α)ee|RαT | = |uek (x) − u(α) (x)| ≤ |uek (x) − u(α) (x)|+(α)(α)+ |(Ck − u0k )ωk,1 (x) + (Ck+1 − u0k+1 )ωk+1,1 (x)| ≤≤ h5−α max |u(5) (x)|, α = 1, 2, x ∈ [xk , xk+1 ).[xk ,xk+1 ]e 1 = 2.2692, Ke 2 = 5.6996.KÎòñþäà ïîëó÷àåì íåðàâåíñòâî (37) ñ òàáëèöàõ 3, 4 ïðåäñòàâëåíû ôàêòè÷åñêèåee |, R1F = max |u0 −RF = max |u−U[−1,1][−1,1]0ee|URT , R1T ïîãðåøíîñòèh = 0.1 è h = 0.01.è òåîðåòè÷åñêèåïðîèçâîäíîé ïðèïðèáëèæåíèé ôóíêöèè è åå ïåðâîéÒàáëèöà 3RFh = 0.1Nou(x)1sin(3x) cos(5x)tg(x)cos(2x)1(1 + 25x2 )234RTh = 0.1RFh = 0.01RTh = 0.010.51 · 10−3 0.89 · 10−1 0.11 · 10−7 0.89 · 10−60.45 · 10−4 0.19 · 10−1 0.15 · 10−8 0.19 · 10−60.19 · 10−5 0.17 · 10−3 0.21 · 10−10 0.17 · 10−80.27 · 10−21.720.18 · 10−60.17 · 10−4R1Th = 0.1R1Fh = 0.01R1Th = 0.01Òàáëèöà 4u(x)1sin(3x) cos(5x)tg(x)cos(2x)1(1 + 25x2 )234 òàáëèöå 5 ïðèR1Fh = 0.1Noe2 , R2FR0.18 · 10−10.200.36 · 10−5 0.20 · 10−40.14 · 10−20.790.50 · 10−6 0.79 · 10−40.61 · 10−4 0.73 · 10−2 0.71 · 10−8 0.73 · 10−60.89 · 10−1h = 0.01è òåîðåòè÷åñêèõ 71.480.60 · 10−4 0.71 · 10−2ïðåäñòàâëåíû àáñîëþòíûå çíà÷åíèÿ ôàêòè÷åñêèõR2 , R2Tïîãðåøíîñòåé, îïðåäåëÿåìûõ ñîîòíîøåíè-ÿìè (2), (27)(31) è (32), (27)(31) âòîðûõ ïðîèçâîäíûõ ôóíêöèè25u.Òàáëèöà 5Nou(x)1sin(3x) cos(5x)tg(x)cos(2x)1(1 + 25x2 )2341.4e2RR2FR2R2T0.11 · 10−2 0.13 · 10−1 0.10 · 10−2 0.93 · 10−10.22 · 10−3 0.28 · 10−2 0.29 · 10−3 0.20 · 10−10.22 · 10−5 0.26 · 10−4 0.28 · 10−5 0.18 · 10−30.21 · 10−10.250.18 · 10−11.80Äèñêðåòíûé âàðèàíò ïîëèíîìèàëüíûõ èíòåãðîäèôôåðåíöèàëüíûõ ñïëàéíîâÐàññìîòðèì ðåøåíèå çàäà÷è ìîäåëèðîâàíèÿ íåêîòîðîé àïïðîêñèìèðóþùåéôóíêöèè, îïèñûâàþùåé ýêñïåðèìåíòàëüíûé çàêîí ðàñïðåäåëåíèÿ ñ ïîìîùüþïðåäëàãàåìûõ ïîëèíîìèàëüíûõ ñïëàéíîâ.
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