Диссертация (1149808), страница 7
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38: Ãðàôèêè ïîãðåøíîñòåé ïðèáëèæåíèé ôóíêöèè Ðóíãå, ãäå (ñëåâà)10, hj ïîëó÷àåì èç (80), j = 0, 1, 2, 7, 8, 9, xj+1è (ñïðàâà) n = 10, xj+1 = xj + hj , hj = 0.2.n== xj + hj , hj = h2 /2, j = 3, 4, 5, 6,Ðèñ. 39: Ãðàôèêè ïîãðåøíîñòåé ïðèáëèæåíèé ôóíêöèè Ðóíãå, ãäå (ñëåâà)n=9, hj ïîëó÷àåì èç (80), 0, 1, 2, 6, 7, 8, xj+1 = xj + hj , hj = 0.7h2 , j = 3, 4, 5, è(ñïðàâà) xj+1 = xj + hj , hj = 2/9, j = 0, ..., 9.67Ïðèìåð 3. Ðèñóíîê 40 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíê-x0 = −1, x1 = −0.4107, x2 =−0.123, x3 = 0.123, x4 = 0.4107, x5 = 1, Ik ≈ 0.617, j = 0, 1, 2, 3, 4, ||u − Ũ ||[−1,1] =ε, ε = 0.00361.öèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè ïðèÐèñóíîê 40 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåx0 = −1, x1 = −0.361, x2 = −0.1105, x3 =0.1106, x4 = 0.361, x5 = 1, I0 = I4 ≈ 0.683, I1 = I3 ≈ 0.593, I2 ≈ 0.532.
Çäåñüε = 0.00219 .ïîëèíîìèàëüíûìè ñïëàéíàìè, ãäåÐèñ. 40: Ãðàôèêè ïîãðåøíîñòåé ïðèáëèæåíèé ôóíêöèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè, ãäå(ñïðàâà)Ik ≈ 0.617 (ñëåâà), I0 = I4 ≈ 0.683, I1 = I3 ≈ 0.593, I2 ≈ 0.532Ïðèìåð 4. Ðèñóíîê 41 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíê-x0 = −1, x1 = −0.31, x2 = 0, x3 =0.31, x4 = 1, I0 = I3 ≈ 0.761, I1 = I2 ≈ 0.781, ||u − Ũ ||[−1,1] = ε, ε = 0.0039.öèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè ïðèÐèñóíîê 41 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóí-x0 = −1, x1 = −0.30404, x2 = 0, x3 =0.30404, x4 = 1, Ik ≈ 0.771, k = 0, 1, 2, 3. Çäåñü ε = 0.00416.Íà êàæäîé ëèíèè, ïàðàëëåëüíîé îñè y , ïðèáëèæåíèå ñòðîèòñÿ ïî ôîðìóëå:ãå ïîëèíîìèàëüíûìè ñïëàéíàìè, ãäåue(y) = u(yk )ωk,0 (y) + u(yk+1 )ωk+1,0 (y)+u0 (yj )ωk,1 (y) + u0 (yk+1 )ωk+1,1 (y)+yk+1Z+ u(t)dt ωk<0> (y), y ∈ [yk , yk+1 ].yk68(81)Ðèñ.
41: Ãðàôèêè ïîãðåøíîñòåé ïðèáëèæåíèé ôóíêöèè Ðóíãå ïîëèíîìèàëüíû-I0 = I3 ≈ 0.761, I1 = I2 ≈ 0.781 (ñëåâà), Ik ≈ 0.771, k =ìè ñïëàéíàìè, ãäå0, 1, 2, 3 (ñïðàâà)Òåïåðü ïîëó÷åíû ñëåäóþùèå ôîðìóëû äëÿÄëÿy = yk + t1 h, t1 ∈ [0, 1]:ωk,0 (yk + t1 h) = −18 t21 + 32 t31 − 15 t41 + 1,(82)ωk+1,0 (yk + t1 h) = −12 t21 + 28 t31 − 15 t41 ,(83)ωk,1 (yk +t1 h)=−(9/2) h t21 +6 h t31 −(5/2) h t41 +t1 h,(84)ωk+1,1 (yk + t1 h) = (3/2) h t21 − 4 h t31 + (5/2) h t41 ,(85)ωk<0> (yk + t1 h) = (30 t21 − 60 t31 + 30 t41 )/h.(86)(x, y) ∈ Ωj,kñëåäóþùèå ôîðìóëû ïîëó÷àåì ñ èñïîëüçîâàíèåì òåíçîð-íîãî ïðîèçâåäåíèÿ:ũ(x, y) =11 XXu(xj+i , yk+p )ωj+i,0 (x)ωk+p,0 (y)+i=0 p=0+1 X1Xu0y (xj+i , yk+p )ωj+i,0 (x)ωk+p,1 (y)+i=0 p=0y1 Zk+1Xu(xj+i , t)dtdyωj+i,0 (x)ωk<0> (y)i=0 yk69x1 Zj+1X+u(t, yk+i )dtωj<0> (x)ωk+i,0 (y)+i=0 xjx1 Zj+1X+u0y (t, yk+i )dtωj<0> (x)ωk+i,1 (y)+i=0 xjxj+1yk+1 ZZu(x, y)dxdyωk<0> (y)ωj<0> (x)yk+xj1Xu0x (xj , yk+i )dtωj,0 (x)ωk+i,0 (y)+i=0+1Xu00xy (xj , yk+i )dtωj,0 (x)ωk+i,1 (y)+i=0yk+1Z+u0x (xj , t)dtωj,1 (x)ωk<0> (y).(87)yk5.1.1Òðèãîíîìåòðè÷åñêèå ñïëàéíû îäíîé ïåðåìåííîéÎáîçíà÷èì çàeue(x)ïðèáëèæåíèå ôóíêöèèu(x)íà èíòåðâàëå[xj , xj+1 ]:eue(x) = u(xj )eωj,0 (x) + u(xj+1 )eωj+1,0 (x)+u0 (xj )eωj,1 (x) + u0 (xj+1 )eωj+1,1 (x)+xj+1Z+ u(t)dt ωej<0> (x), x ∈ [xj , xj+1 ].(88)xjÁàçèñíûå ñïëàéíûωej,0 (x), ωej+1,0 (x), ωej,1 (x), ωej+1,1 (x), ωej<0> (x),ïîëó÷àåì èçñèñòåìû:eue(x) ≡ u(x), u(x) = 1, sin(kx), cos(kx), k = 1, 2.(89)x = xj + th, t ∈ [0, 1] ñëåäóþùèå ôîðìóëû:ωej,0 (xj + th) = (6 sin(t h + h) − 8 cos(t h) h + 12 h cos(t h + h) − 4 h cos(−3 h +t h) + 15 sin(h) − 6 sin(2 h + t h) + 6 sin(−2 h + t h) + 3 sin(3 h) − 12 sin(2 h) +Ïîëó÷àåì äëÿ703 sin(−3 h + 2 t h) + 2 h cos(−3 h + 2 t h) + 6 h cos(−h + 2 t h) + 3 sin(h + 2 t h) −6 sin(−h + 2 t h) − 8 h cos(2 t h) − 6 sin(−3 h + t h))/(30 sin(h) − 24 sin(2 h) −16 h + 18 h cos(h) − 2 h cos(3 h) + 6 sin(3 h)),ωej+1,0 (xj + th) = (3 sin(h + 2 t h) − 6 sin(−h + 2 t h) + 8 h cos(t h − h) −6 h cos(−h + 2 t h) − 2 h cos(h + 2 t h) + 6 sin(t h + h) + 12 sin(2 h) − 3 sin(3 h) −15 sin(h)+8 h cos(2 t h−2 h)−6 sin(−3 h+t h)+3 sin(−3 h+2 t h)−12 h cos(−2 h+t h)+4 h cos(2 h+t h)−6 sin(2 h+t h)+6 sin(−2 h+t h))/(−30 sin(h)+24 sin(2 h)+16 h − 18 h cos(h) + 2 h cos(3 h) − 6 sin(3 h)),ωej,1 (xj + th) = (10 + 6 cos(2 h) − 2 cos(2 t h) − 4 h sin(2 t h) − 15 cos(h) −8 cos(t h)+5 cos(−3 h+2 t h)+6 h sin(−h+2 t h)−2 h sin(−3 h+2 t h)−8 h sin(t h)+6 cos(−2 h+t h)+2 cos(2 h+t h)−4 cos(−3 h+t h)+4 cos(t h−h)+12 cos(−h+2 t h)−cos(h+2 t h)−14 cos(2 t h−2 h)+6 h sin(t h+h)−cos(3 h)+2 h sin(−3 h+t h))/(30 sin(h) − 24 sin(2 h) − 16 h + 18 h cos(h) − 2 h cos(3 h) + 6 sin(3 h)),ωej+1,1 (xj + th) = (10 + 6 cos(2 h) + 2 cos(−3 h + t h) − 15 cos(h) − cos(3 h) −2 h sin(2 h + t h) − 6 h sin(−2 h + t h) + 12 cos(−h + 2 t h) + 5 cos(h + 2 t h) −cos(−3 h + 2 t h) + 2 h sin(h + 2 t h) − 6 h sin(−h + 2 t h) − 2 cos(2 t h − 2 h) −14 cos(2 t h) − 4 cos(2 h + t h) + 4 h sin(2 t h − 2 h) + 8 h sin(t h − h) + 4 cos(t h) −8 cos(t h − h) + 6 cos(t h + h))/(−30 sin(h) + 24 sin(2 h) + 16 h − 18 h cos(h) +2 h cos(3 h) − 6 sin(3 h)),ωej<0> (xj + th) = (8 + 6 cos(−h + 2 t h) + cos(h + 2 t h) − 9 cos(h) + cos(3 h) −4 cos(2 t h) − 2 cos(−3 h + t h) − 4 cos(2 t h − 2 h) + cos(−3 h + 2 t h) + 6 cos(−2 h +t h) − 2 cos(2 h + t h) − 4 cos(t h − h) + 6 cos(t h + h) − 4 cos(t h))/(−15 sin(h) +12 sin(2 h) + 8 h − 9 h cos(h) + h cos(3 h) − 3 sin(3 h)).Ìîæåò áûòü ïîêàçàíî, ÷òî ñëåäóþùèå ôîðìóëû, óñòàíàâëèâàþùèå ñîîòíîøåíèÿ ìåæäó ïîëèíîìèàëüíûìè è òðèãîíîìåòðè÷åñêèìè áàçèñíûìè ñïëàéíàìè, âåðíû:ω̃j,0 (xj + th) = (1 + 32 t3 − 18 t2 − 15t4 ) + O(h2 ),ω̃j+1,0 (xj + th) = (28 t3 − 12 t2 − 15t4 ) + O(h2 ),ω̃j,1 (xj +th)=(6t3 −(9/2)t2 −(5/2)t4 + t)h+O(h3 ),ω̃j+1,1 (xj +th)=(−4t3 +(3/2)t2 +(5/2)t4 )h+O(h3 ),ω̃j<0> (xj + th) = (−60t3 + 30t2 + 30t4 )h−1 + O(h).71Òåîðåìà 4.Ïîãðåøíîñòü ïðèáëèæåíèÿ ñïëàéíàìè(88)òàêîâà, ÷òî âûïîë-íÿåòñÿ ñîîòíîøåíèå:e|ue(x) − u(x)| ≤ Kh5 k4u0 + 5u000 + uV k[xj ,xj+1 ] ,(90)ãäå x ∈ [xj , xj+1 ], K > 0, K íå çàâèñèò îò h è u.Äîêàçàòåëüñòâî.u(x) íà ïðîìåæóòêå [xj , xj+1 ] ìîæåò áûòü çàïèñàíàRx420000Vâ âèäå (ñì [74]): u(x) =3 xj (4u (τ )+ 5u (τ ) + u (τ )) sin (x/2 − τ /2)dτ + c1 +c2 sin(x)+c3 cos(x)+c4 sin(2x)+c5 cos(2x), ãäå ci , i = 1, 2, 3, 4, 5 ïðîèçâîëüíûåÔóíêöèÿêîíñòàíòû.
Èñïîëüçóÿ ìåòîä èç [74], ïîëó÷àåì (90).Ãðàôèê 42 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòè ïðèáëèæåíèé ôóíêöèè7 cos(2x) + 5 sin(x)xj+1 = xj + hj , hj = 2/9, n = 9 ïîëèíîìèàëüíûìè ñïëàéíàìè, çäåñüε = 0.00000896. Ãðàôèê 42 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòè ïðèáëèæåíèéïðèôóíêöèè7 cos(2x) + 5 sin(x)ïðèxj+1 = xj +hj , hj = 2/9, n = 9 òðèãîíîìåòðè÷åñêèìè ñïëàéíàìè, Digits=25.Ãðàôèê 43 ïîêàçûâàåò ïîãðåøíîñòè ïðèáëèæåíèé ôóíêöèè15 cos(2x) sin(exp(x))xj+1 = xj + hj , hj = 2/9, n = 9 ïîëèíîìèàëüíûìè ñïëàéíàìè, çäåñü ε =0.000677 (ñëåâà) è òðèãîíîìåòðè÷åñêèìè ñïëàéíàìè, ε = 0.000579 (ñïðàâà),ïðèDigits=25.727 cos(2x) + 5 sin(x) ïîëèíîìèàëü(ñëåâà), òðèãîíîìåòðè÷åñêèìè ñïëàéíàìè (ñïðàâà)Ðèñ. 42: Ãðàôèêè ïîãðåøíîñòè ïðèáëèæåíèéíûìè ñïëàéíàìè15 cos(2x) sin(exp(x)) ïîëèíîìè(ñëåâà), òðèãîíîìåòðè÷åñêèìè ñïëàéíàìè (ñïðàâà)Ðèñ.
43: Ãðàôèêè ïîãðåøíîñòè ïðèáëèæåíèéàëüíûìè ñïëàéíàìè73eue(y) ïðèáëèæåíèå ôóíêöèè u(y) íà ïðîìåæóòêå [yk , yk+1 ]. Áàçèñíûå ñïëàéíû ωek,0 (y), ωek+1,0 (y), ωek,1 (y), ωek+1,1 (y), ωek<0> (y), ïîëó÷àåì èç ñèñòåÎáîçíà÷èì çàìû óðàâíåíèé:eue(y) ≡ u(y), u(y) = 1, sin(ky), cos(ky), k = 1, 2.(91)(x, y) ∈ Ωj,k òî èñïîëüçóåì âûðàæåíèå (87), ãäå ïðèìåíÿåì òðèãîíîìåòðè÷åñêèå ωes,i (x), ωes,i (y), ωej<0> (x), ωek<0> (y) âìåñòî ïîëèíîìèàëüíûõ ôóíêöèéωs,i (x), ωs,i (y), ωj<0> (x), ωj<0> (y).˜˜˜Âîçüì¼ì Ũ (x), x ∈ [a, b], òàê ÷òî Ũ (x) = ũ(x),x ∈ [xj , xj+1 ].˜ y) − u(x, y) òåíçîðÒàáëèöà 14 ïîêàçûâàåò ïîãðåøíîñòè ïðèáëèæåíèé ũ(x,Åñëèíûì ïðîèçâåäåíèåì òðèãîíîìåòðè÷åñêèõ ñïëàéíîâ, ïîëó÷åííûõ èç (89), (91) èïîãðåøíîñòè ïðèáëèæåíèéũ(x, y) − u(x, y)òåíçîðíûì ïðîèçâåäåíèåì ïîëèíî-ìèàëüíûõ ñïëàéíîâ ôóíêöèé:u1 (x, y) =cos(x) cos(y),(1 + 25 sin2 (x))(1 + 25 sin2 (y))u2 (x, y) =ïðèxy,(1 + 25x2 )(1 + 25y 2 )[a, b] = [−1, 1], [c, d] = [−1, 1], h = 0.1, y = 0.05.
Âû÷èñëåíèÿ ïðîèçâîäèëèñüâ Maple, Digits=25.ÇäåñüRI = max |Ũ − u|, RII = max |Ũ˜ − u|.x∈[−1,1]x∈[−1,1]Òàáëèöà 14u(x, y)RIRIIu1 (x, y) 0.70998 · 10−5 0.70395 · 10−5u2 (x, y) 0.48035 · 10−6 0.47316 · 10−6745.2Ìåòîä 25.2.1Ëåâîñòîðîííèå ïîëèíîìèàëüíûå ñïëàéíû îäíîé ïåðåìåííîéÎáîçíà÷èì çàue(x)ïðèáëèæåíèå ôóíêöèèu(x)íà ïðîìåæóòêå[xj , xj+1 ]:ue(x) = u(xj )ωj,0 (x) + u(xj+1 )ωj+1,0 (x)+u0 (xj )ωj,1 (x) + u0 (xj+1 )ωj+1,1 (x)+Zxj+ u(t)dt ωj<−1> (x).(92)xj−1ωj,0 (x), ωj+1,0 (x), ωj,1 (x), ωj+1,1 (x), ωj<−1> (x),Áàçèñíûå ñïëàéíûïîëó÷èì èçñèñòåìû óðàâíåíèé:ue(x) ≡ u(x), u(x) = xi−1 , i = 1, 2, 3, 4, 5.Äëÿx = xj + th, t ∈ [0, 1]èìååì ñëåäóþùèå ôîðìóëû:ωj,0 (xj +th)=(32/31)t3 − (78/31)t2 +(15/31)t4 +1,(94)ωj+1,0 (xj + th)=(48/31)t2 +(28/31)t3 −(45/31)t4 ,(95)ωj,1 (xj + th) = (h/62)(85t4 − 39t2 − 108t3 + 62t),(96)ωj+1,1 (xj + th) = (h/62)(35t4 − 27t2 − 8t3 ),(97)ωj<−1> (xj + th) = (1/31) (30 t2 − 60 t3 + 30 t4 )/h,(98)Ãðàôèêè 44, 45, 46 (ñëåâà) ïîêàçûâàþò áàçèñíûå ôóíöèèωj,1 (x), ωj+1,1 (x), ωj<−1> (x)ïðèh = 1.ωj,0 (x), ωj+1,0 (x),Ðèñóíîê 46 (ñïðàâà) ïîêàçûâàåò ïî-u(x) = 1/(1 + 25x2 ) ïîëèíîìèàëüíûìèh = 0.1, x ∈ [−1, 1], ku − Ũ k = ε, ε = 0.00141.ãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåñïëàéíàìè,(93)75Ðèñ.
44: Ãðàôèêè áàçèñíûõ ôóíêöèé:Ðèñ. 45: Ãðàôèêè áàçèñíûõ ôóíêöèé:h = 1 (ñïðàâà)ωj,0 (x) (ñëåâà), ωj+1,0 (x) (ñïðàâà)ωj,1 (x)ïðèh = 1 (ñëåâà), ωj+1,1 (x)ïðèh = 1 (ñëåâà),è ïîãðåøíî-ñòè ïðèáëèæåíèÿ ôóíêöèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéñíàìè,h = 0.1, x ∈Ðèñ. 46: Ãðàôèêè áàçèñíîé ôóíêöèèωj<−1> (x)[−1.1] (ñïðàâà)76ïðè×òîáû óëó÷øèòü êà÷åñòâî àïïðîêñèìàöèè, ìîæíî âûáèðàòü÷òîh−1 = x0 − x−1 , xj+1 = xj + hj ,xZj +hjãäåhjïîëó÷àåì èç:Zx0 pp1 + (u0 (x))2 dx =1 + (u0 (x))2 dx.xjxj ∈ [a, b] òàêèå,(99)x0 −hjÏðèìåð 1. Ðèñóíîê 47 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíê-h−1 = 0.25, hj ïîëó÷àåì èç (99),j = 5, 6, 7, 8, 9, 10, 11, çäåñü ku − Ũ k = ε,öèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè, ïðèj = 0, 1, 3, 4, 12, 13, 14, è hj = 0.8h4 ,ε = 0.00019.
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