Численные методы. Ионкин (2012) (v1.1) (косяки есть), страница 10

÷ðåçâû÷àéíî óäîáíà äëÿ ðåàëèçàöèè2. ïåðâûé ïîðÿäîê ïîτ,âòîðîé ïîh3. óñëîâíî ñõîäÿùàÿñÿ è óñëîâíî óñòîé÷èâàÿ82×èñòî íåÿâíàÿ ðàçíîñòíàÿ ñõåìà (ñõåìà ñ îïåðåæåíèåì).Ñòàâèì â ñîîòâåòñòâèå èñõîäíîé çàäà÷å 4.14.3 ñëåäóþùóþ ðàçíîñòíóþ ñõåìó:n+1n+1yi−1− 2yin+1 + yi+1yin+1 − yin=+ f (xi , tn+1 ), (xi , tn+1 ) ∈ ωτ h ,τh2(y0n+1 = µ1 (tn+1 ), tn+1 ∈ ω τ ,n+1yN= µ2 (tn+1 ), tn+1 ∈ ω τ ,yi0 = u0 (xi ),xi ∈ ω h .(4.21)(4.22)(4.23)Ðàçíîñòíàÿ ñõåìà çàïèñàíà íà ÷åòûðåõòî÷å÷íîì øàáëîíåÐèñ. 4.5:Ýòà ðàçíîñòíàÿ ñõåìà áóäåò àáñîëþòíî ñõîäÿùàÿñÿ è àáñîëþòíî óñòîé÷èâàÿ,â òîé æå íîðìå, ñ òåì æå ïîðÿäêîì ÷òî è ÿâíàÿ.Íî ïðè ýòîì íåèçâåñòíî, èìååò ëè ðåøåíèå ÑËÀÓ (ñõåìà íåÿâíàÿ, íóæíî îáðàùàòü ìàòðèöó).

Ýòó ðàçíîñòíóþ ñõåìó ïðèäåòñÿ ðåøàòü ìåòîäîì ïðîãîíêè, ýòîñëîæíåå ÷åì íàõîäèòü ðåøåíèå íà ÿâíîé ñõåìå.Ìû çàïèñàëè òàê:n+1n+1γyi−1− (1 + 2γ)yin+1 + γyi+1= −Fi (y n ),ãäåi = 1, . . . , N − 1,Fi = (yin + fin+1 )Âûïèøåì ìàòðèöó äàííîé ñèñòåìû:1 + 2γ −γ00...0 −γ 1 + 2γ −γ 0...0  .............S = −..... 0...0 −γ 1 + 2γ −γ 00... 0−γ 1 + 2γÎáðàçóåòñÿ ìàòðèöà ñ äèàãîíàëüíûì ïðåîáëàäàíèåì. Cëåäîâàòåëüíî, ðåøåíèåñóùåñòâóåò, åäèíñòâåííî, àëãîðèòì ðåøåíèÿ - ïðîãîíêà.Ðàññìîòðèì íåÿâíóþ ðàçíîñòíóþ ñõåìó, ñõåìó ñ îïåðåæåíèåì:n+1n+1yi−1− 2yi + yi+1yin+1 − yin=+ f (x, tn+1 )τn283(4.24)y0n+1 = µ1 (tn+1 ); ynn+1 = µ2 (tn+1 ); tn+1 ∈ ω¯τ(4.25)yi0 = u0 (xi ); xi ∈ ω¯τ(4.26)Îáñóäèì âîïðîñ ñõîäèìîñòè è óñòîé÷èâîñòè ðàçíîñòíîé ñõåìû.

Íåîáõîäèìîîïðåäåëèòüñÿ ñ òåì, â êàêîé îáëàñòè îñóùåñòâëÿåòñÿ ñõîäèìîñòü. Ïîêàæåì, ÷òîýòè ðàçíîñòíûå ñõåìû ñõîäÿòñÿ àáñîëþòíî.Ââåäåì ñåò÷àòóþ ôóíêöèþ:zin = yin − u(xi , tn ) = yin − uniÇàïèøåì óðàâíåíèån+1n+1zi−1− 2zin+1 + zi+1zin+1 − zin=+ ψin2τh(4.27)Íóëåâûå íà÷àëüíûå è êðàåâûå óñëîâèÿ:n+1z0n+1 = zN=0(4.28)zi0 = 0(4.29)Ïðèìåíÿåì ïðèíöèï ìàêñèìóìà (ïðèìåíÿåì äëÿ ÿâíîé ðàçíîñòíîé ñõåìû).Ïîãðåøíîñòü àïïðîêñèìàöèè íà ðåøåíèè:ψinn+1un+1+ un+1un+1− unii−1 − 2uii+1i+− fin+1=−2τh(4.30)Çàäà÷à:Ïîêàçàòü, ÷òîÐåøåíèå:Ðàçëîæèìψin = O(τ + h2 )u(xi , tn+1 )â òî÷êå(xn , tn )ïî ôîðìóëå Òåéëîðà:u(xi , tn+1 ) = un+1= u(xi , tn ) + ut (xi , tn )τ + O(τ 2 )iÐàçëîæèìu(xi+1 , tn )â òî÷êå(xi , tn )ïî ôîðìóëå Òåéëîðà:11u(xi+1 , tn ) = uni+1 = u(xi , tn ) + ux (xi , tn )h + uxx (xi , tn )h2 + uxxx (xi , tn )h3 + O(h4 )26Ðàçëîæèìu(xi−1 , tn )â òî÷êå(xi , tn )ïî ôîðìóëå Òåéëîðà:11u(xi−1 , tn ) = uni+1 = u(xi , tn ) − ux (xi , tn )h + uxx (xi , tn )h2 − uxxx (xi , tn )h3 + O(h4 )26Ïîäñòàâèì äàííûå ðàçëîæåíèÿ â 4.30, ïðèâåäåì ïîäîáíûå ñëàãàåìûå è, ïðèíèìàÿ âî âíèìàíèå âèä ðàññìàòðèâàåìîé çàäà÷è, ïîëó÷èìψin = O(τ + n2 )84Ïîëó÷àåì îöåíêó â ñèëüíîé íîðìå C.

Ìû çíàåì, ÷òî ñóùåñòâóþò íåòðèâèàëü-⇒ ∃xi0 : max0≤i≤N | zin+1 |=| zin+1|= kz n+1 kC0íûå ðåøåíèÿ ýòîì óçëå çàïèøåì óðàâíåíèå 4.27:(1 +Ïóñòüγ=τh2τ n+1τ n+1)z=(zi0 −1 + zin+1) + zin0 + τ ψin0i00 +122hh⇒(1 + 2γ) | zin+1|≤ γ(| zin+1| + | zin+1|)+ | zin0 | +τ | ψin0 |00 −10 +1Óñèëèì íîðìó:(1 + 2γ) | zin+1|≤ γ(kz n+1 kC + kz n+1 kC ) + kz n kC + τ kψ n kC0(1 + 2γ)kz n+1 kC ≤ 2γkz n+1 kC + kz n kC + τ kψ n kCkz n+1 kC ≤ kz n kC + τ kψ n kCÌû ïîëó÷èëè òó æå ñàìóþ îöåíêó, ÷òî è äëÿ ÿâíîé ðàçíîñòíîé ñõåìû, íî áåçêàêèõ-ëèáî îãðàíè÷åíèé íà øàãè ñåòêèτèh.kz n+1 kC ≤ kz 0 kC + Σnk=0 τ kψ k kC ,ãäåkz 0 kC = 0kz n+1 kC ≤ Σnk=0 τ kψ n kCkψ n k ≤ M (τ + h2 ),ãäå Ì íå çàâèñèò îòτè h.

Ýòî âàæíî ïðè ñòðåìëåíèè Ì ê áåñêîíå÷íîñòè.kz n+1 kC ≤ M1 (τ + h2 ), M1 = T Mkz n+1 kC → 0, τ, h → 0 ⇒Çíà÷èò èìååò ìåñòî ñõîäèìîñòü ðàçíîñòíîé ñõåìû ê ðåøåíèþ òî÷íîé çàäà÷è.Äàííàÿ ðàçíîñòíàÿ ñõåìà èìååò 1-é ïîðÿäîê òî÷íîñòè ïîn+1Åñëè â 4.244.26 ïðè y0íóëåâàÿ}⇒=n+1yN=0⇒τè âòîðîé ïî h.{ Åñëè äàæå íà÷àëüíàÿ ôóíêöèÿ íåïîëó÷èì òó æå ñàìóþ îöåíêó:ky n+1 kC ≤ ku0 kC + τ Σnk=0 kf k kC ,ãäåu0- íà÷àëüíîå óñëîâèå, àfk -ïðàâàÿ÷àñòü. ýòîì ñëó÷àå ãîâîðÿò, ÷òî ðàçíîñòíàÿ ñõåìà óñòîé÷èâà ïî íà÷àëüíîìó óñëîâèþ è ïðàâîé ÷àñòè. Åñëèky n+1 k ≤ M1 ku0 kC + M2 kf kC , ãäå M1 , M2íå çàâèñÿò îòøàãîâ. Îòñþäà ñëåäóåò àïðèîðíàÿ îöåíêà, îçíà÷àþùàÿ óñòîé÷èâîñòü ïî íà÷àëüíîìó óñëîâèþ è ïðàâîé ÷àñòè.85Ðèñ.

4.6: Øàáëîí òèïà "ÿùèê"Ÿ 3. Ñèììåòðè÷íàÿÊðàíêà-Íèêîëüñîíà)ðàçíîñòíàÿñõåìà(ÑõåìàÍà ïðàêòèêå äàííàÿ ñõåìà ïðèìåíÿåòñÿ ÷àùå, ïîñêîëüêó â îöåíêå âòîðîé ïîðÿäîê òî÷íîñòè ïîτè h.∂u ∂ 2 u= 2 + f (x, t), 0 ≤ x ≤ 1, 0 ≤ t ≤ T∂t∂x(4.31)y(0, t) = µ1 (t), y(1, t) = µ2 (t), 0 ≤ t ≤ T(4.32)y(x, 0) = u0 (x), 0 ≤ x ≤ 1(4.33)nyx̄x,in− 2yin + yi−1yi+1=n2Âèä ñõåìû:yin+1 − yinn+1n= 0.5(yx̄x,i+ yx̄x,i) + f (xi , tn+ 21 ), (xi,tn+ 1 ) ∈ ωτ n ,2τ(4.34)(xi , tn+ 12 ) = (xi , tn + 0.5τ )n+1y0n+1 = µ1 (tn+1 ), yN= µ2 (tn+1 ), tn+1 ∈ ω̄τ(4.35)yi0 = u0 (xi ), xi ∈ ω̄h(4.36)Ìû ñîïîñòàâëÿåì çàäà÷å 4.314.33 ðàçíîñòíóþ ñõåìó 4.344.36.Ñèììåòðè÷íûé øàáëîí.

Ðåøåíèå ðàçíîñòíîé ñõåìû ñóùåñòâóåò è åäèíñòâåííî.Ñïîñîá ðåøåíèÿ - ïðîãîíêà. Åñëè íå ãîâîðèòü ïðî ñõîäèìîñòü ïî êàêîé èìåííîíîðìå, òî òàêàÿ ôðàçà íå èìååò ñìûñëà.86Ïîêàæåì àáñîëþòíóþ ñõîäèìîñòü. Ââåäåì ïîãðåøíîñòüzin = yin − uni :zin+1 − zinn+1n= 0.5(zx̄x,i+ zx̄x,i) + ψinτψin = −1un+1− unini+ 0.5(un+1x̄x,i + ux̄x,i ) + f (xi , tn + τ )τ2(4.37)Çàäà÷à:Ïîêàçàòü, ÷òîÐåøåíèå:Ðàçëîæèìψin = O(τ 2 + n2 )un+1i±1èuniâ ðÿä Òåéëîðà â îêðåñòíîñòè òî÷êè(xi , tn+0.5 ) :1 n+0.5 τ 2n+0.5n+0.5 τ+utt,i ( ) + O(τ 3 )un+1=u+uiit,i2 22τ 1 n+0.5 τ 2(uni = uin+0.5 − ut,i n + 0.5) − un+0.5+ utt,i ( ) + O(τ 3 )t,i2 22Ïîäñòàâèì ïîëó÷åííûå ðàçëîæåíèÿ â ôîðìóëó 4.37:n+0.5nn+0.5ψin = −ut,i+ O(τ 2 ) + 0.5(un+1x̄x,i + ux̄x,i ) + fi ïðåäñòàâëåíèè ôòîðîé ðàçíîñòíîé ïðîèçâîäíîé ðàçëîæèì âñå âõîæäåíèÿôóíêöèè â ðÿä Òåéëîðà.

Ïîëó÷èì:unx̄x,i=unxx,i+h2nuxxxx,i12Ïðèìåíèì ïîëó÷åííîå ðàçëîæåíèå êíèå â ðÿä Òåéëîðà â òî÷êån+1n+1un+1x̄x,i = uxx,i +uxxxx,iÀíàëîãè÷íî äëÿun+1x̄x,i ,+ O(h4 )çàòåì ïðîâåäåì åù¼ îäíî ðàçëîæå-(xi , tn+0.5 ):22h2n+0.5 τn+0.5 hn+0.5 h τ+O(h4 ) = un+0.5+u+u+u+O(τ 2 +h4 )xx,ixxt,ixxxx,ixxxxt,i1221212 2unx̄x,i :Ïîäñòàâèì ïîëó÷åííûå ðàçëîæåíèÿ â âûðàæåíèå äëÿn+0.5n+0.5ψin = (−ut,i+ uxx,i+ fin+0.5 ) + un+0.5xxxx,iψin :h2+ O(τ 2 + h4 ) = O(τ 2 + h2 )12Ÿ 4. Çàäà÷à Øòóðìà-Ëóèâèëëÿd2 u+ λu(x) = 0,dx287(4.38)u(x) 6= 0, 0 ≤ x ≤ 1, u(0) = u(1) = 0çíà÷åíèå, u(x)- ñîáñòâåííàÿ ôóíêöèÿ.ãäåÐåøåíèå çàäà÷èλ=0√- íà÷àëüíûå óñëîâèÿ.λ- ñîáñòâåííîåλk = π 2 k 2 , k = 1, 2, .

. .íå äàåò ðåøåíèå, ïîýòîìó â ñïåêòð íå âõîäèò.Îòâå÷àþùèå èì ñîáñòâåííûå çíà÷åíèÿ ôóíêöèè√2 → uk (x) = 2sin(πkx)L2 - ïðîñòðàíñòâî ôóíêöèé,ïðîèçâåäåíèå: (uk , ul ) = σkl .{uk }∞ -îðòîíîðìèðîâàííûé0 < λ1 < . . . < λn < . . .uk (x) = c sin(πkx), c 6= 0, c =èíòåãðèðóåìûõ ñ êâàäðàòîì.  íåì ñêàëÿðíîåáàçèñ⇒∀f ∈ L2 , f (x) =kf k2L2 =∞Xk=1∞Xck uk (x) ⇒c2kk=1- ðàâåíñòâî Ïàðñåâàëÿ.yx̄xi + λy(xi ) = 0, xi ∈ ωn , y(xi ) 6= 0(4.39)y0 = y N = 0(4.40)Ïðîáëåìà ñîáñòâåííûõ çíà÷åíèé àíàëèòè÷åñêè ðåøàåòñÿ ðåäêî.yi+1 − 2yi + yi+1 + h2 λyi = 0y(xi ) = sin(αxi ), α ∈ R, i = 1, . .

. , N − 1yi+1 + yi−1 = (2 − h2 λ)yiyi+1 + yi+1 = y(xi + h) + y(xi − h) = sinα(xi + h) + sinα(xi − h) = 2sin(αxi )cos(αh)2cos(αx)sin(αxi ) = (2 − h2 λ)sin(αxi )2cos(αh) = 2 − h2 λ4αhαh⇒ λ = 2 sin22h24πkhyN = 0 = sinα ⇒ α = πk ⇒ λk = 2 sin2, k = 1, . . . , N − 1h2√yk (xi ) = csin(πkxi ), xi ∈ ω̄n , k = 1, . . . , N − 1, c = 2k 2 λ = 2(1 − cos(αh)) = 4sin2HN −1 : ∀f, g ∈ Hn−1 , dimHn−1 = N − 1, (f, g) =N−1Xi=188fi gi h ⇒(4.41)(4.42)vuN −1uX= kf k = tf 2hkf kL2 (ωn )ii=1N −1{yk }k=1HN −1 ; HN −1 Îðòîíîðìèðîâàííûé áàçèñïðîñòðàíñòâî ñåò÷àòûõôóíêöèé.).√ yk (xi ) = µk (xi√c = 2 ⇒ yk (xi ) = 2sinπkxi , i, k = 1, .

. . , N − 1X−1⇒ ∀f ∈ HN −1 f = ΣN(yk , yl ) =k=1 ck µkÎáîçíà÷èìÅñëèk,lÈñïîëüçóåì ðàâåíñòâî Ïàðñåâàëÿ:kf k2L2 (ωn )=N−1Xc2kk=1zik=NXck (tn )µk (xi ), xi ∈ ω¯hk=1ψin=NXψ (k) (tn )µk (xi )k=1- ïîãðåøíîñòü àïïðîêñèìàöèè.NXck (tn+1 ) − ck (tn )τk=1µk (xi ) =N−1X0.5(ck (tn+1 ) +(k)ck (tn ))µx̄x,ik=1+N−1Xψ (k) µk (xi )k=1(k)µx̄x,i = −λk µk (xi ) ⇒ðàññìàòðèâàåì òîëüêî ïðè k=1:ck (tn+1 ) − ck (tn ))+ 0.5λk (ck (tn+1 ) + ck (tn )) = ψ (k) (tn )τçàäà÷ó íàõîæäåíèÿ ck .⇒Ïîëó÷èëè(1 + 0.5τ λk )ck (tn+1 ) = (1 − 0.5τ λk )ck (tn ) + τ ψ (k) (tn )ck (tn+1 ) =τ1 − 0.5τ λkck (tn ) +ψ k (tn ) = qk1 + 0.5τ λk1 + 0.5τ λk89ck (tn+1 ) − ck (tn )+ 0.5λk (ck (tn+1 ) + ck (tn )) = ψ (k) (tn );τk = 1, 2, ...., n − 1(1 + 0, 5λk τ )ck (tn+1 ) = (1 − 0.5τ λk )ck (tn ) + τ ψ (k) (tn )ck (tn+1 ) =(1 − 0.5λk )τck (tn ) +ψ (k) (tn )1 + 0.5τ λk1 + 0.5λk|qk | ≤ 1Zin+1=N−1Xck (tn+1 )µk (xi ) =ïîñëåäíÿÿ ñóììà =Òîãäà ïîëó÷èìkVk =qk ck (tn )µk (xi ) +N−1Xk=1k=1k=1n+1 2N−1Xτψ (k) (tn )µk (xi );1 + 0.5τ λkWin+1kZ n+1 kL+2(wn ) = kZ n+1 k ≤ kV n+1 k + kW n+1 k{ñîãëàñíî ðàâåíñòâó Ïàðñåâàëÿ}=N−1Xqk2 c2k (tn )k=1kWn+1 2k ={àíàëîãè÷íî}=XkZ n+1 k ≤ kZ n k +nPn+1ñëîå ÷åðåç íóëåâîé ñëîé:τ kψ(tj )kj=0Òîãäàc2k (tn ) = kZ n k2k=1τ2(ψ (k) (tn ))2 = τ 2 kψ n k2(1 + 0.5τ λk )kW n+1 k ≤ τ kψ n kkZ n+1 k ≤ kZ n k + τ kψ n kÇàïèøåì ðåøåíèå íà≤N−1Xkz 0 k = 0ψ n = 0(τ n + h2 )M íå çàâèñèò îò τ è h22Ýòî çíà÷èò, ÷òî kψ(tj )k ≤ M (τ + h )90kZ n+1 kL2 (wn )n+1y0n+1 = yN=0ky n+1 kL2 (wn ) ≤ kUo k +nPτ kf kj=0Ââåäåì âåùåñòâåííûé ïàðàìåòð, ïðè îïðåäåëåííûõ çíà÷åíèÿõ êîòîðîãî áóäóòïîëó÷àòüñÿ ñõåìû, êîòîðûå ìû óæå ðàññìàòðèâàëè.

Íî òàêæå ìîæåò ïîëó÷àòüñÿáåñêîíå÷íîå ÷èñëî äðóãèõ ñõåì. Ýòî ñåìåéñòâî ñõåì íàçûâàåòñÿ ñõåìû ñ âåñàìè.Âûâåäåì è ïîêàæåì, êàê âûâîäèòñÿ ïîãðøåíîñòü àïïðîêñèìàöèè ïðè ðåøåíèèäëÿ ëþáûõ ðàçíîñòíûõ ñõåì, êîòîðûå ìû ðàññìàòðèâàåì.Ðàçíîñòíàÿ ñõåìà ñ âåñàìè:Ïîãðåøíîñòü àïïðîêñèìàöèè íà ðåøåíèè.+ êðàåâûåyin+1 − yinn+1n= σyx̄x,i+ (1 − σ)yx̄x,i+ φniτóñëîâèÿ (5) + íà÷àëüíûå óñëîâèÿ (6). σ ∈ R1. Åñëè ìû âîçüìåì2.σ = 1; φni = fin+13.σ = 0.5;σ = 0; φni = fin(4.43)òî ìû ïîëó÷èì ÿâíóþ ðàçíîñòíóþ ñõåìó.- ÷èñòî íåÿâíàÿ ñõåìàn+ 12φni = fi- ñèììåòðè÷íàÿ ñõåìà Êðàíêà-ÍèêîëüñîíàÄëÿ âñåõ ñõåì ñ âåñàìè ëåãêî ïîëó÷èòü âñå îöåíêè, êîòîðûå ìû ïîëó÷àëè äëÿñðåäíåêâàäðàòè÷íûõ íîðì - ïîëó÷èòü ìåòîäîì ðàçäåëåíèÿ ïåðåìåííûõ.ÂâîäèìZin = yin − UinZin+1 − Zinn+1n= σZx̄x,i+ (1 − σ)yx̄,i+ ψinτψin óçëå=n+1σUx̄x,i+ (1 −nσ)Ux̄x,iUin+1 − Uin+ φni−τ(xi , tn+ 21 )hUi+1 = Ui + U2Äîãîâîðèìñÿ îá îáîçíà÷åíèÿõ äëÿ ÷àñòè÷íûõ ïðîèçâîäíûõ:∂U1(x, tn + ) = U̇∂t291(4.44)∂U= U0δxh2 h2 00 h3 000Ui+1 = U i +++ Ui + Ui + ....226h2 00 h3 0000Ui−1 = Ui − hU i + Ui − Ui + ...26hUi0À íà ïîëóöåëîì ñëîå:ττ2 0= Ui (tn+ 12 ) − U̇i (tn+ 21 ) + Üi (tn+ 12 )28ττUIn = Ui (tn+ 21 ) − U̇i (tn+ 12 ) + U̇i (tn+ 21 ) − ....28Uin+1Ux̄x,ih2 0000Ui+1 − 2Ui + Ui−100= Ui + ui + O(h4 )=2h12Ïîýòîìó êîãäà ìû ãîâîðèìUx̄x,i − Ui00 =ïåðâàÿ ïðîèçâîäíàÿ ïî âðåìåíè:Òåïåðü ìîæåì ðàññìàòðèâàòü= O(n2 )= Ui (.)(tn+ 21 ) + O(τ 2 )ψin :τ 00h2 0000= σ(U + U (.) + U + O(τ h2 )) + O(h4 ))+2122τh+(1 − σ)(U 00 − U 00 (.) + U 0000 ) − U (.) + φn + O(τ 2 + h4 )212000000íîâûõ îáîçíà÷åíèÿõ: U = U̇ − f U= u̇0 − f 0000ψinÂUIn+1 −uniτh2 000012 UiÒîãäà ìû ïîëó÷àåìh2U − U̇ + + (σ − 0.5)τ U +̇ + O(τ 2 + h4 ) =12U 00 − U̇ + f (xi , tn+ 12 ) = φ00i − f (xi , tn+ 21 ) + τ [(σ − 0.5)+0000φnih2 0000 h2+ ]U̇ − f (íåïîíÿòíî) + O(τ 2 + h4 )12122hσ∗ = σ = 21 − 12τφni = f (xi , tn+ 12 ) +h2 00112 f (xi , tn+ 2 )92ψ = O(τ 2 + h4 ) - ñõåìà ïîâûøåííîãî ïîðÿäêà àïïðîêñèìàöèèσ = 0; φni = fin ; O(τ + h2 )σ = 1; φni = fin+1 ; O(τ + h2 )σ = 21 ; φni = fi (tn + 12 )2Êîãäà σ 6= ðàññìîòðåííûì σ - áóäåì ïîëó÷àòü O(τ + h )Ïåðåõîäèì ê ïîñòðîåíèþ è èçó÷åíèþ íîâîãî êëàññà ðàçíîñòíûõ ñõåì - äëÿóðàâíåíèÿ ÏóàññîíàŸ 5.