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Çàäà÷à î êîëüöå82Ýòèì çàêàí÷èâàåòñÿ ïåðâûé ýòàï ìåòîäà Ôóðüå.Äàëåå íà âòîðîì ýòàïå ìåòîäà ïîñòðîèì ðåøåíèå, óäîâëåòâîðÿþùååíà÷àëüíîìó óñëîâèþ (5.15).Ïî àíàëîãèè ñ îáûêíîâåííûìè äèôôåðåíöèàëüíûìè óðàâíåíèÿìè îáùåå ðåøåíèå áóäåì ðàçûñêèâàòü â âèäå ëèíåéíîé êîìáèíàöèè ÷àñòíûõ ðåøåíèé.  îòëè÷èå îò îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé äëÿóðàâíåíèé ñ ÷àñòíûìè ïðîèçâîäíûìè áóäåì ðàññìàòðèâàòü ëèíåéíóþ êîìáèíàöèþ íå êîíå÷íîãî, à áåñêîíå÷íîãî íàáîðà ÷àñòíûõ ðåøåíèé, ò.å. ðÿäñëåäóþùåãî âèäà∞A0 Xu(x, t) =+2µk=1πkπkBk sinx + Ak cosx``¶22 πke−a ( ` ) t . (5.30)Íåèçâåñòíûå ïîñòîÿííûå A0 , Ak è Bk îïðåäåëèì èç íà÷àëüíîãî óñëîâèÿ(5.15).Ïðè t = 0 èìååì∞A0 Xu(x, 0) =+2k=1µ¶πkπkBk sinx + Ak cosx .``(5.31)Ïîäñòàâëÿÿ (5.31) â (5.15), ïîëó÷èì∞A0 Xϕ(x) =+2µk=1¶πkπkBk sinx + Ak cosx .``(5.32)Òàêèì îáðàçîì, Bk , Ak , A0 êîýôôèöèåíòû ðàçëîæåíèÿ â ðÿä Ôóðüåôóíêöèè ϕ(x) íà îòðåçêå [−`, `].Âñïîìíèì, ÷òî ñèñòåìà ñîáñòâåííûõ ôóíêöèé îðòîãîíàëüíà íà [−`, `] :Z`µπksinx`¶³π n ´sinx dx =`(−`µZ`cos−`πkx`¶³π n ´cosx dx =`(0,`,k 6= nk = n.(5.33)0,`,k 6= nk = n.(5.34)5.2.
Çàäà÷à î êîëüöå83Äåéñòâèòåëüíî, ïðè k = n èìååìµZ`sin2πkx`¶Z`1 − cos2dx =−`¡2 π k¢Z``dx =−`µZ`cos2πkx`¶−`Z`1 + cos2dx =−`1dx = `2¡2 π k¢`Z`dx =−`1dx = `2−`Äëÿ íàõîæäåíèÿ A0 ïðîèíòåãðèðóåì îáå ÷àñòè ðàâåíñòâà (5.32)îò −` äî `Z`ϕ(x) dx =A0(2 `)2−`Âûðàçèì A01A0 =`Z`(5.35)ϕ(x) dx−`Äëÿ íàõîæäåíèÿ îñòàëüíûõ êîýôôèöèåíòîâ Ak , k = 1, 2, . . . óìíîæèìîáå ÷àñòè ðàâåíñòâà (5.32) íà cos¡π n x¢`è ïðîèíòåãðèðóåì îò −` äî ` èïîëó÷èìZ`ϕ(x) cos³π n x´`dx =∞Xk=1−`Z`Akcos³π n x´`µπkxcos`¶(5.36)dx.−` ïðàâîé ÷àñòè ðàâåíñòâà (5.36) ñ ó÷åòîì (5.34) îñòàåòñÿ ñëàãàåìîå ëèøüïðè k = n. Îòñþäà ïîëó÷àåòñÿ âûðàæåíèå äëÿ AkR`Ak =ϕ(x) cos¡π k x¢`−`R`−`dx(5.37)¡¢cos2 π`k x dxèëè1Ak =`Z`−`µπkxϕ(x) cos`¶dx.(5.38)5.2.
Çàäà÷à î êîëüöå84Òåïåðü îïðåäåëèì Bk , Äëÿ ýòîãî óìíîæèì îáå ÷àñòè ðàâåíñòâà (5.32)íà sin¡π n x¢`è ïðîèíòåãðèðóåì îò −` äî `.  ðåçóëüòàòå èìååìZ`ϕ(x) sin³π n x´`dx =∞Xk=1−`Z`Bksin³π n x´`µ¶πkxsindx.(5.39)`−`Ñ ó÷åòîì (5.33) â ñóììå îñòàåòñÿ ëèøü ñëàãàåìîå ïðè k = n. Ïðèõîäèì êâûðàæåíèþR`ϕ(x) sin¡π k x¢`−`Bk =R`−`dx(5.40)¡¢sin2 π`k x dxèëè1Bk =`Z`µπkxϕ(x) sin`¶(5.41)dx.−`Îïðåäåëèâ êîýôôèöèåíòû â ðåøåíèè (5.30) ïîëó÷àåì ðåøåíèå êðàåâîéçàäà÷è î êîëüöå (5.14)-(5.17).Îòâåò:∞A0 Xu(x, t) =+2k=11ãäå A0 =`µπkπkBk sinx + Ak cosx``¶22 πke−a ( ` )t(5.42)Z`(5.43)ϕ(x) dx−`1Ak =`Z`µπkxϕ(x) cos`¶dx.(5.44)dx.(5.45)−`1Bk =`Z`µπkxϕ(x) sin`¶−`Ïðèâåäåì ïðèìåðû ïðèìåíåíèÿ ìåòîäà Ôóðüå ê ðåøåíèþ êîíêðåòíûõêðàåâûõ çàäà÷.5.2.
Çàäà÷à î êîëüöå85Ïðèìåð 5.1. Íàéäèòå ðàñïðåäåëåíèå òåìïåðàòóðû â òîíêîì êîëüöå åäèíè÷íîãî ðàäèóñà ñ òåïëîèçîëèðîâàííîé áîêîâîé ïîâåðõíîñòüþ, åñëè â íà÷àëüíûé ìîìåíò âðåìåíè êîëüöî áûëî íàãðåòî ïî ëèíåéíîìó çàêîíóϕ(x) = u0 + u1 x,ãäå u0 , u1 êîíñòàíòû.Ðåøåíèå. Çàïèøåì ìàòåìàòè÷åñêóþ ïîñòàíîâêó çàäà÷è:ut = a2 uxx ,(5.46)−π 6 x 6 πu(x, 0) = u0 + u1 x(5.47)u(−π, t) = u(π, t)(5.48)ux (−π, t) = ux (π, t)(5.49)Íà÷íåì ñ ïåðâîãî ýòàïà ìåòîäà Ôóðüå. Áóäåì ðàçûñêèâàòü ÷àñòíûåðåøåíèÿ â âèäå ïðîèçâåäåíèÿ(5.50)ũ(x, t) = X(x) T (t).Ïîñëå ïîäñòàíîâêè (5.50) â óðàâíåíèå (5.46) è êðàåâûå óñëîâèÿ (5.48)(5.49) è ðàçäåëåíèÿ ïåðåìåííûõ ïîëó÷àåì äèôôåðåíöèàëüíîå óðàâíåíèåäëÿ T (t):T 0 (t) + λ a2 T (t) = 0,(5.51)ðåøåíèå êîòîðîãî îïðåäåëÿåòñÿ ïî ôîðìóëå (5.28) è êðàåâóþ çàäà÷ó äëÿX(x):(X 00 (x) + λ X(x) = 0,X(−π) = X(π),x ∈ (−π, π)(5.52)X 0 (−π) = X 0 (π).Âûïèøåì ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè ïîëó÷åííîéêðàåâîé çàäà÷è (5.52)λ0 = 0,λk = k 2 ,X0 (x) = 1,k∈NXk (x) = {sin( k x); cos( k x)} ,(5.53)k ∈ N.(5.54)5.2.
Çàäà÷à î êîëüöå86 èòîãå íà ïåðâîì ýòàïå ìåòîäà Ôóðüå íàéäåíû ÷àñòíûå ðåøåíèÿũ0 (x, t) = 1no−a2 k 2 t−a2 k 2 tũk (x, t) = sin(k x) e; cos(k x) e, k ∈ N.(5.55)Åùå ðàç ïîä÷åðêíåì, ÷òî ïîñòðîåííûå ÷àñòíûå ðåøåíèÿ óäîâëåòâîðÿþò îäíîðîäíîìó óðàâíåíèþ òåïëîïðîâîäíîñòè (5.46) è êðàåâûì óñëîâèÿì (5.48)(5.49).Ïðèñòóïèì êî âòîðîìó ýòàïó ìåòîäà Ôóðüå. Áóäåì ñòðîèòü ðåøåíèå,óäîâëåòâîðÿþùåå íà÷àëüíîìó óñëîâèþ (5.47).Áóäåì ðàçûñêèâàòü ðåøåíèå â âèäå ëèíåéíîé êîìáèíàöèè áåñêîíå÷íîãîíàáîðà ÷àñòíûõ ðåøåíèé, ò.å. â âèäå ñëåäóþùåãî ðÿäà∞A0 X2 2u(x, t) =+(Bk sin(k x) + Ak cos(k x)) e−a k t .2(5.56)k=1Ïðè t = 0 ïîòðåáóåì âûïîëíåíèÿ íà÷àëüíîãî óñëîâèÿ (5.47):∞A0 Xu0 + u1 x =+(Bk sin(k x) + Ak cos(k x)) .2(5.57)k=1Íåèçâåñòíûå ïîñòîÿííûå A0 , Ak è Bk îïðåäåëèì ïî ôîðìóëàì (5.43)(5.45).1A0 =π1Ak =πBk =1πZπ(u0 + u1 x) dx = 2 u0(5.58)(u0 + u1 x) cos(k x) dx(5.59)(u0 + u1 x) sin(k x) dx.(5.60)−πZπ−πZπ−πÍàéäåì, ïîëüçóÿñü ôîðìóëàìè èíòåãðèðîâàíèÿ ïî ÷àñòÿì, èíòåãðàë â5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à87âûðàæåíèè äëÿ AkZπ(u0 + u1 x) cos(k x) dx =−π¯π u Zπ1¯1= (u0 + u1 x) sin(k x)¯ −sin(k x) dx = 0.−πkk(5.61)−πÇíà÷èò, êîýôôèöèåíòû Ak = 0, k ∈ N.Òåïåðü, èíòåãðèðóÿ ïî ÷àñòÿì, íàéäåì èíòåãðàë äëÿ êîýôôèöèåíòîâBkZπ−π¯π u Zπ1¯1cos(k x) dx =(u0 + u1 x) sin(k x) dx = − (u0 + u1 x) cos(k x)¯ +−πkk=−−πu12π u12π cos(kπ) =(−1)k+1 .kk(5.62)Òàêèì îáðàçîì, çíà÷åíèå êîýôôèöèåíòîâ BkBk =2 u1(−1)k+1 ,kk ∈ N.(5.63)Çíàÿ âñå êîýôôèöèåíòû â ðàçëîæåíèè (5.56), ïîëó÷àåì îêîí÷àòåëüíîåðåøåíèå çàäà÷è (5.46)-(5.49)Îòâåò:∞X(−1)k+12 2u(x, t) = u0 + 2 u1sin(k x) e−a k t .kk=15.3 Ïåðâàÿ êðàåâàÿ çàäà÷àÏîñòàíîâêà çàäà÷è. Íàéäèòå ðàñïðåäåëåíèå òåìïåðàòóðû â òîíêîì îäíîðîäíîì ñòåðæíå (0 6 x 6 `) ñ òåïëîèçîëèðîâàííîé áîêîâîé ïîâåðõíîñòüþ, åñëè íà÷àëüíàÿ òåìïåðàòóðà ñòåðæíÿ ÿâëÿåòñÿ ïðîèçâîëüíîé ôóíêöèåé ϕ(x), è íà êîíöàõ ñòåðæíÿ x = 0, x = ` ïîääåðæèâàåòñÿ íóëåâàÿ5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à88òåìïåðàòóðà.ut = a2 uxx ,06x6`(5.64)u(x, 0) = ϕ(x)(5.65)u(0, t) = 0,(5.66)u(`, t) = 0Ðåøåíèå. Íà÷íåì ñ ïåðâîãî ýòàïà ìåòîäà Ôóðüå. Áóäåì ðàçûñêèâàòü÷àñòíîå ðåøåíèå äèôôåðåíöèàëüíîãî óðàâíåíèÿ (5.64), óäîâëåòâîðÿþùååêðàåâûì óñëîâèÿì (5.66). Ïðåäñòàâèì ðåøåíèå â âèäå ïðîèçâåäåíèÿũ(x, t) = X(x) T (t).(5.67)Ïîñòàâèì ũ(x, t) â âèäå (5.67) â óðàâíåíèå (5.64). ÈìååìX(x) T 0 (t) = a2 X 00 (x) T (t).(5.68)Ðàçäåëÿÿ ïåðåìåííûå, ïîëó÷èìT 0 (t)X 00 (x)=.a2 T (t)X(x)(5.69)Ñëåâà â ýòîì ðàâåíñòâå ñòîèò ôóíêöèÿ îò t, à ñïðàâà ôóíêöèÿ îò x.Òàêîå òîæäåñòâî âîçìîæíî ëèøü òîãäà, êîãäà êàæäàÿ ôóíêöèÿ ÿâëÿåòñÿíåêîòîðîé ïîñòîÿííîé, îáîçíà÷èì åå −λ.T 0 (t)X 00 (x)== −λ.a2 T (t)X(x)(5.70)Äëÿ ôóíêöèé T (t) è X(x) ïîëó÷àåì îáûêíîâåííûå äèôôåðåíöèàëüíûåóðàâíåíèÿ ïåðâîãî è âòîðîãî ïîðÿäêîâ:T 0 (t) + λ a2 T (t) = 0,(5.71)X 00 (x) + λ X(x) = 0.(5.72)Åñëè ïîäñòàâèòü ðåøåíèå (5.67) â êðàåâûå óñëîâèÿ (5.66), òî èìååìX(0) T (t) = 0;X(`) T (t) = 0.(5.73)Äëÿ ëþáîé ôóíêöèè T (t) ìîæíî óäîâëåòâîðèòü ýòèì ðàâåíñòâàì, åñëèX(0) = 0;X(`) = 0.(5.74)5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à89Òàêèì îáðàçîì, ôóíêöèÿ X(x) äîëæíà áûòü ðåøåíèåì êðàåâîé çàäà÷èíà ñîáñòâåííûå çíà÷åíèÿ ïåðâîãî ðîäà (5.4):(X 00 (x) = −λ X(x), x ∈ (0, `)X(0) = 0; X(`) = 0.(5.75)Ýòà êðàåâàÿ çàäà÷à èìååò ñëåäóþùèé íàáîð ñîáñòâåííûõ çíà÷åíèé è ñîáñòâåííûõ ôóíêöèé (5.5)µλk =πk`¶2k ∈ N,,¶πkXk (x) = sinx ,`(5.76)µk ∈ N.(5.77)Òåïåðü íàéäåì ôóíêöèþ T (t), êîòîðàÿ äîëæíà óäîâëåòâîðÿòü äèôôåðåíöèàëüíîìó óðàâíåíèþ (5.71). Ðàçäåëÿÿ ïåðåìåííûå è èíòåãðèðóÿ, èìååìT (t) = e−a2λt(5.78),ãäå λ = λk ñîáñòâåííîå çíà÷åíèå (5.76).Òàêèì îáðàçîì, íà ïåðâîì ýòàïå ìåòîäà Ôóðüå ïîëó÷àåì áåñêîíå÷íûéíàáîð ÷àñòíûõ ðåøåíèé èñõîäíîé êðàåâîé çàäà÷è ñëåäóþùåãî âèäàµπkũk (x, t) = sinx`¶e2−a2 ( π`k ) t,k ∈ N,(5.79)Ýòèì çàêàí÷èâàåòñÿ ïåðâûé ýòàï ìåòîäà Ôóðüå.Âòîðîé ýòàï ìåòîäà çàêëþ÷àåòñÿ â íàõîæäåíèè òàêîãî ðåøåíèÿ, êîòîðîå óäîâëåòâîðÿåò íà÷àëüíîìó óñëîâèþ.
Áóäåì ðàçûñêèâàòü ðåøåíèå ââèäå áåñêîíå÷íîé ëèíåéíîé êîìáèíàöèè ÷àñòíûõ ðåøåíèéu(x, t) =∞Xk=1µπkxBk sin`¶e2−a2 ( π`k ) t,(5.80)ãäå íåèçâåñòíûå ïîñòîÿííûå Bk îïðåäåëèì èç íà÷àëüíîãî óñëîâèÿ.Ïîëîæèì t = 0 â ðåøåíèè (5.80) è ïîëó÷èìu(x, 0) =∞Xk=1¶πkx .Bk sin`µ(5.81)5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à90Ïîäñòàâëÿÿ (5.81) â (5.65), èìååìϕ(x) =∞Xk=1µ¶πkBk sinx .`(5.82)Òàêèì îáðàçîì, Bk êîýôôèöèåíòû ðàçëîæåíèÿ â ðÿä Ôóðüå ôóíêöèèϕ(x) íà îòðåçêå [0, `].Óìíîæèì îáå ÷àñòè ðàâåíñòâà (5.82) íà sin¡π n x¢`è ïðîèíòåãðèðóåì îò0 äî `, ïîëó÷èìZ`ϕ(x) sin³π n x´`0=∞XZ`sinBkk=1dx =³π n x´`0µπkxsin`¶(5.83)dx.Äëÿ îïðåäåëåíèÿ êîýôôèöèåíòîâ Ôóðüå ôóíêöèè ϕ(x) âîñïîëüçóåìñÿòåì, ÷òî ñèñòåìà ñîáñòâåííûõ ôóíêöèé îðòîãîíàëüíà íà [0, `]. 0, k 6= n³´πkπnsinx sinx dx =(5.84) ` , k = n.``02 ïðàâîé ÷àñòè ðàâåíñòâà (5.83) ñ ó÷åòîì îðòîãîíàëüíîñòè ñîáñòâåííûõôóíêöèé (5.84) îñòàåòñÿ ñëàãàåìîå ëèøü ïðè k = n. Îòñþäà ïîëó÷àåòñÿµZ`¶âûðàæåíèå äëÿ BkR`Bk =ϕ(x) sin¡π k x¢`0R`sin2¡π k x¢0`dxdx2=`Z`0µπkxϕ(x) sin`¶dx.(5.85)Îïðåäåëèâ Bk , çàïèøåì ðåøåíèå êðàåâîé çàäà÷è (5.64)-(5.66)Îòâåò:u(x, t) =∞Xk=12ãäå Bk =`µπkxBk sin`Z`0µ¶22 πke−a ( ` ) t ,πkxϕ(x) sin`(5.86)¶dx.(5.87)5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à91Ïðèìåð 5.2. Íàéäèòå ðàñïðåäåëåíèå òåìïåðàòóðû â òîíêîì îäíîðîäíîìñòåðæíå (0 6 x 6 π ) ñ òåïëîèçîëèðîâàííîé áîêîâîé ïîâåðõíîñòüþ, åñëèíà÷àëüíàÿ òåìïåðàòóðà ñòåðæíÿ çàäàíà è ðàâíàϕ(x) = u0 + u1 x,ãäå u0 , u1 êîíñòàíòû. Íà êîíöàõ ñòåðæíÿ x = 0, x = ` ïîääåðæèâàåòñÿíóëåâàÿ òåìïåðàòóðà.Ðåøåíèå. Çàïèøåì ìàòåìàòè÷åñêóþ ïîñòàíîâêó çàäà÷è:ut = a2 uxx ,06x6π(5.88)u(x, 0) = u0 + u1 x(5.89)u(0, t) = 0,(5.90)u(π, t) = 0Ïðèìåíèì ìåòîä Ôóðüå äëÿ ðåøåíèÿ ýòîé çàäà÷è.
Íà ïåðâîì ýòàïåáóäåì ðàçûñêèâàòü ÷àñòíûå ðåøåíèÿ â âèäåũ(x, t) = X(x) T (t).(5.91)Ïîäñòàâèì ðåøåíèå (5.91) â óðàâíåíèå (5.88) è êðàåâûå óñëîâèÿ (5.90)è ðàçäåëèì ïåðåìåííûå. Ïîëó÷èì äèôôåðåíöèàëüíîå óðàâíåíèå äëÿ T (t):T 0 (t) + λ a2 T (t) = 0,(5.92)è êðàåâóþ çàäà÷ó äëÿ X(x):(X 00 (x) + λ X(x) = 0,X(0) = 0, X(π) = 0.x ∈ (0, `)(5.93)Âûïèøåì ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè ïîëó÷åííîéêðàåâîé çàäà÷èλk = k 2 ,k∈NXk (x) = sin( k x),(5.94)k ∈ N.Âîñïîëüçóåìñÿ ðåøåíèåì óðàâíåíèÿ äëÿ T (t) (5.78).(5.95)5.3.
Ïåðâàÿ êðàåâàÿ çàäà÷à92Òàêèì îáðàçîì, â èòîãå íà ïåðâîì ýòàïå ìåòîäà Ôóðüå íàéäåíû ÷àñòíûåðåøåíèÿũk (x, t) = sin(k x) e−a2k2 t, k ∈ N.(5.96)Åùå ðàç ïîä÷åðêíåì, ÷òî ïîñòðîåííûå ðåøåíèÿ óäîâëåòâîðÿþò çàäàííîìóóðàâíåíèþ òåïëîïðîâîäíîñòè (5.88) è êðàåâûì óñëîâèÿì (5.90).Ïðèñòóïèì ê âòîðîìó ýòàïó ìåòîäà Ôóðüå. Áóäåì ñòðîèòü ðåøåíèå,óäîâëåòâîðÿþùåå íà÷àëüíîìó óñëîâèþ (5.89).Ðàçûñêèâàåì ýòî ðåøåíèå â âèäå ëèíåéíîé êîìáèíàöèè áåñêîíå÷íîãîíàáîðà ÷àñòíûõ ðåøåíèé, ò.å.
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