А.Д. Алексеев, С.Н. Кудряшов - Уравнения с частными производными в примерах и задачах (1120422), страница 8
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B ïîëóïîëîñå 0 < x < l,t > 0 äëÿ óðàâíåíèÿ∂2u∂t22= a2 ∂∂xu2ðåøèòü ñìåøàííûå çàäà÷è ñî ñëåäóþùèìè óñëîâèÿìè:1) u(0, t) = 0,u(x, 0) = f (x),∂2u∂t2 (l, t)= −h ∂u∂x (l, t),∂u∂t (x, 0)= F (x);2) u(0, t) = 0,∂2u∂t2 (l, t)= −h ∂u∂x (l, t),u(x, 0) = Ax,∂u∂t (x, 0)= 0;= 0,∂2u∂t2 (l, t)= −h ∂u∂x (l, t),u(x, 0) = f (x),∂u∂t (x, 0)= F (x);3)4)∂u∂x (0, t)∂2u∂t2 (0, t)= h ∂u∂x (0, t),u(x, 0) = f (x),∂u∂t (x, 0)∂2u∂t2 (l, t)= −h ∂u∂x (l, t),= F (x).2.4.3. Çàäà÷è î êîëåáàíèè â ñðåäå ñ ñîïðîòèâëåíèåì. ýòîì ðàçäåëå ðàññìàòðèâàþòñÿ çàäà÷è î êîëåáàíèè ñòðóíû â ñðåäå ñ ñîïðîòèâëåíèåì, êîòîðîå ëèáî ïðîïîðöèîíàëüíî ïåðâîé ñòåïåíè ñêîðîñòè,∂u∂t ,èîïèñûâàþòñÿ îäíîðîäíûì óðàâíåíèåì (1.4) ïðè g(x, t) = 0.
Ýòî çàäà÷è 2.7, 2.8,2.9. Ëèáî ñîïðîòèâëåíèå ïðîïîðöèîíàëüíî îòêëîíåíèþ u(x, t), çàäà÷è 2.10 2.14. Ïðèâåäåì íåñëîæíûé ïðèìåð.Ïðèìåð 4. Íàéòè ðåøåíèå óðàâíåíèÿ2∂u∂ 2u2∂ u+ 2ν=a 2,∂t2∂t∂t0<ν<πa,`(4.16)óäîâëåòâîðÿþùåå óñëîâèÿìu(0, t) = 0,u(`, t) = 0;74(4.17)2πx4πx+ B sin;``(4.18)A, B ïîñòîÿííûå.(4.19)u(x, 0) = A sin∂u(x, 0) = 0,∂tÐåøåíèå. Ïðèìåíèì ìåòîä Ôóðüå.
Ïîñêîëüêó óðàâíåíèå (4.16) âñòðå÷àåòñÿâïåðâûå ïðèõîäèòüñÿ âñþ ïðîöåäóðó ïðîäåëàòü ñ ñàìîãî íà÷àëà. Èùåì ÷àñòíîåðåøåíèå (4.16), óäîâëåòâîðÿþùåå (4.17), â âèäå ïðîèçâåäåíèÿu(x, t) = X(x)T (t).Ïîäñòàâëÿÿ u(x, t) â (4.16) è ðàçäåëÿÿ ïåðåìåííûå, ïîëó÷àåìT (00) (t) + 2νT 0 (t) X 00 (x)== −λ2 .2a T (t)X(x)Äëÿ X(x) ïîëó÷àåì èçâåñòíóþ ïî ïðèìåðó 1 êðàåâóþ çàäà÷ó:X 00 (x) + λ2 X(x) = 0,Îòñþäà λk =πk`X(0) = X(`) = 0.è Xk (x) = sin πkx` , k ∈ N . Äëÿ T (t) çàïèøåòñÿ òàêæå ëèíåéíîåóðàâíåíèå ñ ïîñòîÿííûìè êîýôôèöèåíòàìèT (00) (t) + 2vνT 0 (t) +πka`2T (t) = 0.(4.20)Ñîîòâåòñòâóþùåå åìó õàðàêòåðèñòè÷åñêîå óðàâíåíèå èìååò âèäp2 + 2νp + (πka 2) = 0.`2Äèñêðèìèíàíò ýòîãî êâàäðàòíîãî óðàâíåíèÿ, ∆ = ν 2 − ( πka` ) ïî óñëîâèþ çà22äà÷è îòðèöàòåëåí. Äëÿ ïðîñòîòû ïîëàãàåì v 2 − ( πk` ) = −ωk .
Îáùåå ðåøåíèåóðàâíåíèÿ (4.20) ðàâíîTk (t) = ak e−νt cos ωk t + bk e−νt sin ωk t, k ∈ N.75Ôîðìàëüíîå ðåøåíèå ñòðîèòñÿ ïî ôîðìóëå−νtu(x, t) = e∞X(ak cos ωk t + bk sin ωk t) sink=1Ïðîèçâîäíóþ∂u∂tπkx`(4.21).âû÷èñëèì çàðàíåå è îíà ðàâíà∞Xπkx∂u(−ak sin ωk t + bk cos ωk t)ωk sin(x, t) = −νu(x, t) + e−νt.∂t`k=1 ðàâåíñòâå (4.21) ïîëàãàåì t = 0. Ïîëó÷àåì∞Xk=1Îòñþäà a2 = A,ak sin2πkx4πkxπkx= A sin+ B sin.```a4 = B , âñå îñòàëüíûå ak = 0. Âû÷èñëÿåì∞∞k=1k=1X∂uπkx Xπkx(x, 0) = −νak sin+ωk bk sin.∂t`` ñèëó (4.19)∂u∂t (x, 0)= 0, îòêóäà−νak + ωk k = 0,b2 =νa2Aν=,ω2ω2b4 =Bν.ω4Âñå îñòàëüíûå bk = 0. Îòâåò çàïèñûâàåòñÿ ðàâåíñòâîì−νtu(x, t) = e2πxA(ω2 cos ω2 t + ν sin ω2 t) sin+ω2`B4πx+ (ω4 cos ω4 t + ν sin ω4 t) sin.ω4`Çàìå÷àíèå.  ðàíåå ðàññìîòðåííûõ ïðèìåðàõ êîýôôèöèåíòû ïðè êîñèíóñàõ ñ t (ak )çàâèñÿò òîëüêî îò íà÷àëüíûõ îòêëîíåíèé (f (x), à ïðè ñèíóñàõ ñ t(bk ) îò íà÷àëüíûõ ñêîðîñòåé (ôóíêöèè F (x)) è ðàâåíñòâî íóëþ îäíîé èç ýòèõôóíêöèé îçíà÷àëî îáðàùåíèå â íóëü âñåé ñåðèè êîýôôèöèåíòîâ.
Äëÿ ïðèâåäåííîé çàäà÷è ðîëü êîýôôèöèåíòîâ ïðè êîñèíóñàõ ñîõðàíèëàñü, íî óæå bk76çàâèñÿò è îò íà÷àëüíîãî îòêëîíåíèÿ è îò íà÷àëüíîé ñêîðîñòè è íå âñå ðàâíûíóëþ, åñëè F (x) ≡ 0. Áóäüòå âíèìàòåëüíåé.2.7. B ïîëóïîëîñå 0 < x < `,t > 0 äëÿ óðàâíåíèÿ2∂u∂ 2u2∂ u+2ν=a,∂t2∂t∂x2ν>0(êîëåáàíèÿ â ñðåäå ñ ñîïðîòèâëåíèåì)ðåøèòü íà÷àëüíî-êðàåâûå çàäà÷è ñî ñëåäóþùèìè óñëîâèÿìè:1) ν <πa` ,u(0, t) = 0, u(`, t) = 0,u(x, 0) = f (x),2) u(0, t) = 0,u(x, 0) =∂u∂t (x, 0)u(`, t) = 0,hc x,0 ≤ x ≤ c,h(`−x)(`−c) ,c < x ≤ `,∂u∂t (x, 0)= 0;3) u(0, t) = 0,u(x, 0) = kx,4) ∂u∂x (0, t) = 0,u(x, 0) = f (x),5) ∂u∂x (0, t) = 0,u(x, 0) = f (x),∂u∂x (`, t)= 0,∂u∂t (x, 0)= 0;∂u∂x (`, t)= 0,∂u∂t (x, 0)∂u∂x (`, t)∂2u∂t2+ 2 ∂u∂t == F (x);+ hu(`, t) = 0, h>0∂u∂t (x, 0)Äëÿ 0 < x < π2 ,2.8.= F (x);= F (x).t > 0 ðåøèòü ñìåøàííûå çàäà÷è:∂2u∂x2 ,77∂u∂x (0, t)= 0, u( π2 , t) = 0,u(x, 0) = f (x),2.9.∂2u∂t2∂u∂x (0, t)+ 2 ∂u∂t =∂u∂t (x, 0)∂2u∂x2 ,= 0, u( π2 , t) = 0,u(x, 0) = cos x,2.10.∂2u∂t2− 2u =u(0, t) = 0,∂2u∂t2− 5u =u(0, t) = 0,2.12.∂2u∂t2∂2u∂t2= 0,∂u∂t (x, 0)= F (x).∂2u∂x2 ,= 0,∂u∂t (x, 0)− 10u =u(0, t) = 0,= F (x).∂2u∂x2 ,∂u π∂x ( 2 , t)u(x, 0) = f (x),= 0,∂u∂t (x, 0)− 10u =u(0, t) = 0,2.13.∂2u∂x2 ,∂u π∂x ( 2 , t)u(x, 0) = f (x),= 0.∂u∂t (x, 0)∂u π∂x ( 2 , t)u(x, 0) = f (x),2.11.= F (x).= 0.∂2u∂x2 ,∂u π∂x ( 2 , t)= 0,u(x, 0) = 91 sin x + sin 3x,2.14.∂2u∂t2− 17u =u(0, t) = 0,= F (x).∂2u∂x2 ,∂u π∂x ( 2 , t)u(x, 0) = f (x),∂u∂t (x, 0)= 0,∂u∂t (x, 0)= F (x).782.4.4.
Íåñêîëüêî òåêñòîâûõ çàäà÷.2.15. Îäíîðîäíàÿ ñòðóíà äëèíîé l, çàêðåïëåííàÿ íà îáîèõ êîíöàõ, íàõîäèòñÿ â ïðÿìîëèíåéíîì ïîëîæåíèè ðàâíîâåñèÿ.  íåêîòîðûé ìîìåíò âðåìåíè,ïðèíèìàåìûé çà íà÷àëüíûé, îíà ïîëó÷àåò â òî÷êå x = c óäàð îò ìîëîòî÷êà, êîòîðûé ñîîáùàåò ýòîé òî÷êå ïîñòîÿííóþ ñêîðîñòü v0 . Íàéòè îòêëîíåíèåu(x, t) äëÿ ëþáîãî ìîìåíòà âðåìåíè.Ðàññìîòðåòü äâà ñëó÷àÿ.à) Ñòðóíà âîçáóæäàåòñÿ íà÷àëüíîé ñêîðîñòüþ∂u∂t (x, 0) v0 , |x − c| <= 0, |x − c| >π2h ,π2h ,Ýòîò ñëó÷àé ñîîòâåòñòâóåò ïëîñêîìó æåñòêîìó ìîëîòî÷êó, èìåþùåìó øèðèíóπhè óäàðÿþùåìó â òî÷êå x = c.á) Ñòðóíà âîçáóæäàåòñÿ íà÷àëüíîé ñêîðîñòüþ v0 cos h(x − c), |x − c| < π ,2h∂u∂t (x, 0)=0,|x − c| >π2h ,Ýòîò ñëó÷àé ñîîòâåòñòâóåò æåñòêîìó âûïóêëîìó ìîëîòî÷êó øèðèíîé πh . Òàêîé ìîëîòî÷åê â öåíòðå èíòåðâàëà âîçáóæäàåò íàèáîëüøóþ ñêîðîñòü.2.16.
Ðåøèòü çàäà÷ó î ìàëûõ ïîïåðå÷íûõ êîëåáàíèÿõ ñòðóíû äëèíîé 2l ñçàêðåïëåííûìè êîíöàìè x = −l, x = l, êîòîðàÿ îòòÿãèâàåòñÿ â äâóõ òî÷êàõx = −c è x = c íà íåáîëüøîå ðàññòîÿíèå h îò ïîëîæåíèÿ ðàâíîâåñèÿ è â ìîìåíò t = 0 îòïóñêàåòñÿ áåç íà÷àëüíîé ñêîðîñòè.2.17. Îäíîðîäíûé ñòåðæåíü äëèíîé 2l ñæàò ñèëàìè, ïðèëîæåííûìè ê åãîêîíöàì, òàê, ÷òî îí óêîðîòèëñÿ äî äëèíû 2l(1−).Ïðè t = 0 íàãðóçêà ñíèìàåò79ñÿ. Ïîêàçàòü, ÷òî ñìåùåíèå u(x, t) ñå÷åíèÿ ñ àáñöèññîé õ ñòåðæíÿ îïðåäåëÿåòñÿôîðìóëîé∞(2n + 1)πx(2n + 1)πat8l X (−1)( n + 1)sincos,u(x, t) = 2π(2n + 1)22`2`k=0åñëè òî÷êà x = 0 íàõîäèòñÿ ïîñåðåäèíå ñòåðæíÿ è a-ñêîðîñòü ïðîäîëüíûõâîëí â ñòåðæíå.2.18.
Èçó÷èòü ñâîáîäíûå ïðîäîëüíûå êîëåáàíèÿ îäíîðîäíîãî öèëèíäðè÷åñêîãî ñòåðæíÿ äëèíîé l, ó êîòîðîãî îáà êîíöà ñâîáîäíû è u(x, 0) = a,∂u(x,0)∂t= b, a, b - ïîñòîÿííûå.2.19. Îäíîðîäíûé ñòåðæåíü äëèíîé l íàõîäèòñÿ â ïðÿìîëèíåéíîì ïîëîæåíèè ðàâíîâåñèÿ. Îäèí êîíåö ñòåðæíÿ çàêðåïëåí óïðóãî, à äðóãîé ñâîáîäåí.Íàéòè ïðîäîëüíûå êîëåáàíèÿ ñòåðæíÿ, åñëè â íà÷àëüíûé ìîìåíò âðåìåíè åãîòî÷êàì ñîîáùàåòñÿ ñêîðîñòü f (x).2.20. Êîíöû îäíîðîäíîé ñòðóíû äëèíîé l óäåðæèâàþòñÿ ñ ïîìîùüþ óïðóãèõ ñèë.
Èçó÷èòü ñâîáîäíûå ïîïåðå÷íûå êîëåáàíèÿ ñòðóíû, åñëè èçâåñòíî âíà÷àëüíûé ìîìåíò âðåìåíè ñìåùåíèå, à íà÷àëüíûå ñêîðîñòè îòñóòñòâóþò.5.1 Íåîäíîðîäíûå çàäà÷è. ýòîì ïóíêòå ìû ðàññìàòðèâàåì òàê íàçûâàåìûå íåîäíîðîäíûå çàäà÷è, òîåñòü â çàäà÷àõ ëèáî ïðèñóòñòâóþò âíåøíèå ñèëû, ëèáî íåîäíîðîäíûå êðàåâûåóñëîâèÿ (ïîäâèæíûå êîíöû), ëèáî òî è äðóãîå âìåñòå.80ÎÒÂÅÒÛ Ê ÏÐÈÂÅÄÅÍÍÛÌ ÇÀÄÀ×ÀÌ.Òèïû óðàâíåíèé1.1. Ýëëèïòè÷åñêèé. 1.2. Ãèïåðáîëè÷åñêèé. 1.3. Ïàðàáîëè÷åñêèé.1.4. Ýëëèïòè÷åñêèé. 1.5.
Ãèïåðáîëè÷åñêèé. 1.6. Ãèïåðáîëè÷åñêèé.1.7. Ïàðàáîëè÷åñêèé.1.8. Ýëëèïòè÷åñêèé. 1.9. Ãèïåðáîëè÷åñêèé.1.10. Ýëëèïòè÷åñêèé. 1.11. Ýëëèïòè÷åñêèé. 1.12. Ãèïåðáîëè÷åñêèé.1.13. ξ = 2y + x,η = x,1.14. ξ = 3x − 2y ,1.15. ξ = 5x + y ,1.16. ξ = ex ,η = 2x + y,η = x,η = y,1.17. ξ = x2 − 2ey ,1.18. ξ = y − x2 ,1.21. ξ = 2ex − y 2 ,1.22. ξ = ey cosx,η = x,∂2u∂ξ 2+,∂2u∂ξ 281∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).+∂2u∂ξ∂η∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).ãèïåðáîëè÷åñêèé,ãèïåðáîëè÷åñêèé,∂2u∂η 2∂2u∂ξ∂ηãèïåðáîëè÷åñêèé,ýëëèïòè÷åñêèé,eyx,,ïàðàáîëè÷åñêèé,η = x + y,η=∂2u∂η 2ïàðàáîëè÷åñêèé,η = x2 + y 2 ,+ãèïåðáîëè÷åñêèé,ýëëèïòè÷åñêèé,η = 2x,∂2u∂ξ 2,ïàðàáîëè÷åñêèé,η = x,1.19.
ξ = cosx + y 3 ,1.20. ξ = xy ,ýëëèïòè÷åñêèé,∂2u∂ξ∂η∂2u∂ξ∂η22= F (ξ, η, u, ∂∂ξu2 , ∂∂ 2uη ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).1.23. ξ = cosx − siny ,1.24. ξ = 2x − y ,η=1.25. ξ = tgy − x,1.26. ξ = cosy ,η = x,1x+ y1 ,η = x;1.27. ξ = lny − x1 ,η = x,η = x,1.29. ξ = 2y + e−2x ,η = e−2x ,1.31. ξ = ysinx,η = x,ïàðàáîëè÷åñêèé,1.32. ξ = x − ey ,η = 2x − ey ,η = x1 ,1.34. ξ = 2x − siny ,1.35.
ξ = y − lnsinx,∂2u∂η 2η = x,∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂2u∂ξ 2 + ∂η 2∂2u∂ξ 2∂2u∂η 2∂2u∂ξ 2ïàðàáîëè÷åñêèé,ïàðàáîëè÷åñêèé,82∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).+∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂ξ∂ηãèïåðáîëè÷åñêèé,ýëëèïòè÷åñêèé,η = y,∂2u∂η 2+ýëëèïòè÷åñêèé,ýëëèïòè÷åñêèé,∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂η 2ïàðàáîëè÷åñêèé,η = tgx,1.33. ξ = y + x2 ,∂2u∂ξ 2ïàðàáîëè÷åñêèé,1.28. ξ = y + ctgx,1.30. ξ = ctgy ,∂2u∂η 2ýëëèïòè÷åñêèé,∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂2u∂ξ∂ηãèïåðáîëè÷åñêèé,ïàðàáîëè÷åñêèé,η = sinx;∂2u∂η 2ïàðàáîëè÷åñêèé,+∂2u∂η 2∂2u∂ξ 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).+∂2u∂η 2∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).1.36. ξ = lncosy ,y1.37. ξ = earctg x∂2u∂ξ∂ηη = lnsinx,ýëëèïòè÷åñêèé,px2 + y 2 , η = x − y ,∂2u∂2u+2∂ξ∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).ãèïåðáîëè÷åñêèé,∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).1.38.
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