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Êàæäàÿ èç ïîëó÷åííûõ ôîðìóë èìååò ñâîè ïðåèìóùåñòâà. Ôîðìóëà Ïóàññîíà ïîçâîëÿåò ðàñøèðèòü êëàññ íà÷àëüíûõ äàííûõ.Ôîðìóëó 7.4 ìîæíî èñïîëüçîâàòü äëÿ èññëåäîâàíèÿ àñèìïòîòè÷åñêîãîïîâåäåíèÿ ðåøåíèÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè ïðè t → +∞.Çàäà÷à 7.20. Óñòàíîâèòü íàñêîëüêî ñèëüíî ìîæåò ðàñòè ôóíêöèÿ ϕ(x) íàáåñêîíå÷íîñòè, ÷òîáû ôîðìóëà Ïóàññîíà îïðåäåëÿëà ðåøåíèå óðàâíåíèÿòåïëîïðîâîäíîñòè ïðè t ≤ T .Ïðèìåð 7.11. Èñïîëüçóÿ ôîðìóëó 7.4 , èññëåäîâàòü àñèìïòîòè÷åñêîåïîâåäåíèå ðåøåíèÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè ïðè t → +∞.7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè148Ðåøåíèå.
Áóäåì ñ÷èòàòü, ÷òî ϕ(x), à, ñëåäîâàòåëüíî, è ϕ(y)bïðèíàäëåæàò S(R). Ñíà÷àëà îòìåòèì î÷åâèäíûé ôàêò, ÷òî èíòåãðàë1Iδ (t, x) = √2πZ2 2−aϕ(y)eby t iyxedy|y|>δóäîâëåòâîðÿåò îöåíêå2 2Iδ (t, x) = O(e−aδ t).Ïîýòîìó îñíîâíîé âêëàä â ðåøåíèå u(t, x) ïðè t → +∞ äàåò èíòåãðàë1Jδ (t, x) = √2πZ−aϕ(y)eb2 2y t iyxedy.|y|<δÑ÷èòàÿ δ äîñòàòî÷íî ìàëûì, ìîæíî çàìåíèòü ϕ(x)bíà åå çíà÷åíèå âíóëå1ϕ(0)b =√2πÎñòàâøèéñÿ èíòåãðàë1√2πZ+∞ϕ(x) dx.−∞Ze−a2 2y t iyxedy|y|<δìîæíî çàìåíèòü èíòåãðàëîì1√2πZ+∞2 2e−a y t eiyx dy,(7.8)−∞2 2òàê êàê ðàçíîñòü ýòèõ èíòåãðàëîâ èìååò ïîðÿäîê O(e−aδ t).  ñâîþ î÷åðåäü,èíòåãðàë 7.8, êàê áûëî óñòàíîâëåíî â ïðèìåðå 7.10 ñîâïàäàåò ñ ôóíêöèåéx2e− 4a2 tG0 (x, t) = √ .2a πtÑëåäîâàòåëüíî, èìååò ìåñòî àñèìïòîòè÷åñêàÿ ôîðìóëàx2e− 4a2 t√ .u(x, t) ∼ ϕ(0)b2a πt7.6.
Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè149Êîíå÷íî, ïðèâåäåííûå ðàññóæäåíèÿ ìîæíî ïðèçíàòü ëèøü íàâîäÿùèìèñîîáðàæåíèÿìè. Îäíàêî, â ýòîì íàïðàâëåíèè ìîæíî äâèãàòüñÿ äàëüøå.Ïðåäâàðèòåëüíî çàìåòèì, ÷òî ñïðàâåäëèâî ðàâåíñòâî1√2π(2 )Z+∞− 4ax 2 tnde2 2√y n e−a y t eiyx dy = (−i)n n.dx2a πt−∞Ïîýòîìó, çàìåíèâ ôóíêöèþ ϕ(y)bâ èíòåãðàëå 7.4 åå òåéëîðîâñêèì ðàçëîæåíèåì:∞Xynϕ(y)b∼ϕb (0) ,n!n=0(n)ïîëó÷àåì äëÿ u(x, t) ñëåäóþùèé àñèìïòîòè÷åñêèé ðÿä∞X(−i)n dn(n)u(x, t) ∼ϕb (0)n! dxnn=0(x2e− 4a2 t√2a πt).Ïðèâåäåííûé ñïîñîá àñèìïòîòè÷åñêîãî ðàçëîæåíèÿ èíòåãðàëîâ íîñèòíàçâàíèå ìåòîäà Ëàïëàñà.
Åãî îáîñíîâàíèå ìîæíî íàéòè âî ìíîãèõ êíèãàõ.Ñìîòðèòå, íàïðèìåð, Ì.Â. Ôåäîðþê. Ìåòîä ïåðåâàëà. Ì.: Íàóêà. 1977.Çàäà÷à 7.21. Íàéòè ðåøåíèå óðàâíåíèÿ òåïëîïðîâîäíîñòèut = uxx ,óäîâëåòâîðÿþùåå íà÷àëüíîìó óñëîâèþ u|t=0 = ϕ(x), äëÿ ñëåäóþùèõ ôóíêöèé ϕ(x):a).
e−(x−x0 )24ττ > 0, b). xn , c). sin αx, d). eβx cos αx.Ïðåäëîæèòü äâà ñïîñîáà ðåøåíèÿ: à) ñ ïîìîùüþ ôîðìóëû Ïóàññîíà, b)èñïîëüçîâàòü äðóãèå ñîîáðàæåíèÿ.Ïðèìåð 7.12. Ïðèìåíÿÿ ïðåîáðàçîâàíèå Ôóðüå, íàéòè ðåøåíèå íåîäíîðîäíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè2∂u2∂ u= a 2 + f (x, t)∂t∂ xñ íóëåâûì íà÷àëüíûì óñëîâèåìu|t=0 = 0.7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè150Ðåøåíèå. Ïðèìåíÿÿ ïðåîáðàçîâàíèå Ôóðüå ê óðàâíåíèþ òåïëîïðîâîäíîñòè, ïîëó÷àåì ñëåäóþùóþ çàäà÷ó Êîøè(∂bu∂t= −a2 y 2 ub + fb(y, t)ub|t=0 = 0.Åå ðåøåíèå èìååò âèäZt2 2e−aub(y, t) =y (t−τ )fb(y, t) dy.0Àíàëîãè÷íî ïðèìåðó 7.10 ïîëó÷àåì, ÷òîZ+∞−1Fy→x(e−a2 2y (t−τ )fb(y, t)) =−∞2− 4a(x−s)2 (t−τ )epf (s, τ ) ds.2a π(t − τ )Ïîýòîìó ôîðìóëà Ïóàññîíà äëÿ ðåøåíèÿ íåîäíîðîäíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè ñ íóëåâûì íà÷àëüíûì óñëîâèåì èìååò âèäZ t Z+∞u(x, t) =0 −∞2− 4a(x−s)2 (t−τ )epf (s, τ ) ds dτ.2a π(t − τ )Çàäà÷à 7.22.
Äîêàçàòü, ÷òî ôóíêöèÿ2− 4a(x−s)2 (t−τ )ep(7.9)2a π(t − τ )ïðè ôèêñèðîâàííûõ s, τ è ïðè t > τ óäîâëåòâîðÿåò óðàâíåíèþ òåïëîïðîG(x, t|s, τ ) =âîäíîñòè∂G∂ 2G= a2 2 .∂t∂ xÍå ïðîòèâîðå÷èò ëè ýòîò ôàêò ðåçóëüòàòó ïðåäûäóùåãî ïðèìåðà?Ôóíêöèÿ G(x, t|s, τ ), ïðîäîëæåííàÿ íóëåì ïðè τ ≤ t, íàçûâàåòñÿ ôóíäàìåíòàëüíûì ðåøåíèåì óðàâíåíèÿ òåïëîïðîâîäíîñòè.Ïðèìåð 7.13.
Ïðèìåíÿÿ ïðåîáðàçîâàíèå Ôóðüå, äîêàçàòü, ÷òî ôóíäàìåíòàëüíîå ðåøåíèå G(x, t|s, τ ) ÿâëÿåòñÿ åäèíñòâåííûì ðåøåíèåì óðàâíåíèÿ òåïëîïðîâîäíîñòè2∂G2∂ G= a 2 + δ(x − s)δ(t − τ ),∂t∂ xðàâíûì íóëþ ïðè t < τ.7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè151Ðåøåíèå. Åñòåñòâåííî, çäåñü ïðîèçâîäíûå ïîíèìàþòñÿ â ñìûñëå îáîáùåííûõ ôóíêöèé.
Äîñòàòî÷íî ðàññìîòðåòü ñëó÷àé s = 0. Ïðèìåíÿÿ ïðåîáðàçîâàíèå Ôóðüå ïî ïåðåìåííîé x ïðèõîäèì ê ñëåäóþùåìó äèôôåðåíöè-b:àëüíîìó óðàâíåíèþ äëÿ Gb∂Gb + √1 δ(t − τ ).= −a2 y 2 G∂t2π2 2Èñïîëüçóÿ èíòåãðèðóþùèé ìíîæèòåëü eaäóy tïðèâåäåì óðàâíåíèå 7.10 ê âè-b ayt∂ Geea y τ= √ δ(t − τ ).∂t2π(2 2Ó÷èòûâàÿ, ÷òî δ(t − τ ) =ddt η(t÷òî(7.10)2 2− τ ), ãäå η(t) =1, ïðè t > 0,0, ïðè t < 0,ïîëó÷àåì,2 2a y (τ −t)b=e√Gη(t − τ ).2πÏîñëå ýòîãî ïðèìåíåíèå ôîðìóëû îáðàùåíèÿ ïðèâîäèò ê íóæíîìó ðåçóëüòàòó.Çàìå÷àíèå 7.2.
Òåïåðü ìîæíî îòâåòèòü íà âîïðîñ â ïðåäûäóùåì ïðèìåðå ñëåäóþùèì îáðàçîì. Íåò, íå ïðîòèâîðå÷èò. Íî íåëüçÿ äèôôåðåíöèðîâàòü ïîä çíàêîì èíòåãðàëà â îáû÷íîì ñìûñëå. Äèôôåðåíöèðîâàíèåâ ñìûñëå îáîáùåííûõ ôóíêöèé íå ïðèâîäèò ê ïðîòèâîðå÷èþ.Çàìå÷àíèå 7.3. Îáîáùåííàÿ ôóíêöèÿ δ(x−s)δ(t−τ ) èíòåðïðåòèðóåòñÿêàê ìãíîâåííûé òî÷å÷íûé èñòî÷íèê òåïëà â òî÷êå x = s, äåéñòâóþùèéâ ìîìåíò âðåìåíè t = τ .
Ïîýòîìó ôóíäàìåíòàëüíîå ðåøåíèå íàçûâàþò ôóíêöèåé âëèÿíèÿ ìãíîâåííîãî òî÷å÷íîãî èñòî÷íèêà, ò. å. ôóíäàìåíòàëüíîå ðåøåíèå G(x, t|s, τ ) îïðåäåëÿåò ðàñïðåäåëåíèå òåìïåðàòóðûïðè òàêîì èñòî÷íèêå. Ïðåäñòàâëÿÿ èñòî÷íèêè òåïëà êàê ñóïåðïîçèöèþòî÷å÷íûõ èñòî÷íèêîâZ+∞ Z+∞f (x, t) =δ(x − s)δ(t − τ )f (s, τ ) ds dτ−∞ −∞7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè152ïîëó÷àåì ðåøåíèå êàê ñóïåðïîçèöèþ ôóíêöèé âëèÿíèÿ ñ òîé æå ïëîòíîñòüþZ+∞ Z+∞u(x, t) =G(x, t|s, τ )f (s, τ ) ds dτ.−∞ −∞Ñ÷èòàÿ, ÷òî èñòî÷íèêè òåïëà ðàâíû íóëþ ïðè t < 0 è ó÷èòûâàÿ, ÷òîôóíäàìåíòàëüíîå ðåøåíèå ðàâíî íóëþ ïðè τ > t, âûâîäèìZ+∞Z tu(x, t) =G(x, t|s, τ )f (s, τ ) ds dτ.−∞ 0Ïîëó÷åííàÿ ôîðìóëà ñîâïàäàåò ñ ôîðìóëîé Ïóàññîíà äëÿ ðåøåíèÿ íåîäíîðîäíîãî óðàâíåíèÿ ñ íóëåâûì íà÷àëüíûì óñëîâèåì.
Êîíå÷íî, ïðèâåäåííûå ðàññóæäåíèÿ íîñÿò ýâðèñòè÷åñêèé õàðàêòåð, íî ìîãóò áûòü ñòðîãîîáîñíîâàíû â ðàìêàõ òåîðèè îáîáùåííûõ ôóíêöèé.Ïðèìåð 7.14. Äîêàçàòü, ÷òî ÿäðî Ïóàññîíà G0 (x − s, t) ïðè t > 0 ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ òåïëîïðîâîäíîñòèut = a2 uxxñ íà÷àëüíûì óñëîâèåìu|t=0 = δ(x − s).Ðåøåíèå. Îãðàíè÷èìñÿ äîêàçàòåëüñòâîì ðàâåíñòâàG0 (x − s, 0) = δ(x − s),êîòîðîå ñëåäóåò ïîíèìàòü â ñìûñëå ñõîäèìîñòè îáîáùåííûõ ôóíêöèélim G0 (x − s, t) = δ(x − s).t→+0Íàïîìíèì, ÷òî ïîñëåäíåå îçíà÷àåò, ÷òî äëÿ ëþáîé ïðîáíîé ôóíêöèè f (x)äîëæíî âûïîëíÿòüñÿ óñëîâèåZ+∞G0 (x − s, t)f (s) ds = f (x).limt→+0−∞7.6.
Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè153Íî äàííàÿ ôîðìóëà åñòü ñëåäñòâèå òîãî ôàêòà, ÷òî èíòåãðàëZ+∞G0 (x − s, t)f (s) ds−∞ÿâëÿåòñÿ ðåøåíèåì îäíîðîäíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè ñ íà÷àëüíîéôóíêöèåé f (x).Çàäà÷à 7.23. Íàéòè ôîðìóëó Ïóàññîíà äëÿ îáùåãî ñëó÷àÿ íåîäíîðîäíîãîóðàâíåíèÿ òåïëîïðîâîäíîñòè ñ íåíóëåâûì íà÷àëüíûì óñëîâèåì.Çàäà÷à 7.24. Ïðèìåíèòü ïðåîáðàçîâàíèå Ôóðüå ê ðåøåíèþ óðàâíåíèÿ2∂u∂u2∂ u=a 2 +b+ cu + f (x, t).∂t∂ x∂xÓðàâíåíèå òåïëîïðîâîäíîñòè íà ïîëóïðÿìîéÏðèìåð 7.15. Ðåøèòü íà÷àëüíî-êðàåâóþ çàäà÷ó(∂u2 ∂2u=a∂t∂ 2 x + cu∂u∂x |x=0 = α(t),+ f (x, t), 0 < x < +∞,u|t=0 = ϕ(x).ìåòîäîì èíòåãðàëüíûõ ïðåîáðàçîâàíèé.Ðåøåíèå.
Ñíà÷àëà âûáåðåì òèï èíòåãðàëüíîãî ïðåîáðàçîâàíèÿ. Èç ðåçóëüòàòà ïðèìåðà 7.7 âûòåêàåò, ÷òî ôîðìóëà cos-ïðåîáðàçîâàíèÿ Ôóðüåâòîðîé ïðîèçâîäíîé ñîäåðæèò ãðàíè÷íîå çíà÷åíèå â íóëå ëèøü ïåðâîé ïðîèçâîäíîé. Ïîýòîìó â äàííîì ñëó÷àå ñëåäóåò ïðèìåíÿòü cos-ïðåîáðàçîâàíèåÔóðüå.Äàëåå, çàäà÷ó óäîáíî ðàçáèòü íà òðè áîëåå ïðîñòûå çàäà÷è.1. Îäíîðîäíîå óðàâíåíèå ñ îäíîðîäíûì ãðàíè÷íûì óñëîâèåì è çàäàííûìíà÷àëüíûì óñëîâèåì(∂u2 ∂2u=a∂t∂2x +∂u∂x |x=0 = 0,cu, 0 < x < +∞,u|t=0 = ϕ(x);2. íåîäíîðîäíîå óðàâíåíèå ñ îäíîðîäíûìè ãðàíè÷íûì è íà÷àëüíûì óñëîâèÿìè(∂u2 ∂2u=a∂t∂2x +∂u∂x |x=0 = 0,cu + f (x, t), 0 < x < +∞,u|t=0 = 0;7.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè1543.
îäíîðîäíîå óðàâíåíèå ñ íóëåâûì íà÷àëüíûì óñëîâèåì è çàäàííûì ãðàíè÷íûì óñëîâèåì(∂u2 ∂2u=a∂t∂ 2 x + cu,∂u∂x |x=0 = α(t),0 < x < +∞,u|t=0 = 0.Ïîñëå ïðèìåíåíèÿ cos-ïðåîáðàçîâàíèÿ Ôóðüå ê êàæäîé èç ýòèõ çàäà÷ ïðèõîäèì ê ñëåäóþùèì çàäà÷àì äëÿ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé:1.(∂bu∂t= −a2 y 2 ub + cbu, 0 < y < +∞,ub|t=0 = ϕ(y).b2.(∂bu∂t= −a2 y 2 ub + cbu + fb(y, t), 0 < y < +∞,ub|t=0 = 0.3.∂bu∂tq2 2= −a y ub + cbu−2π α(t),0 < y < +∞,ub|t=0 = 0.Êàæäóþ èç ýòèõ çàäà÷ ìîæíî ðàññìàòðèâàòü êàê çàäà÷ó Êîøè äëÿ ëèíåéíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ïåðâîãî ïîðÿäêà. Îáîçíà÷èâ ðåøåíèÿñîîòâåòñòâåííî ÷åðåç ub1 , ub2 , ub3 , èìååì1.ub1 (y, t) = ϕ(y)b exp{−a2 y 2 t + ct};2.Ztexp{(−a2 y 2 + c)(t − τ )}fb(y, τ ) dτ ;ub2 (y, t) =03.r Zt2exp{(−a2 y 2 + c)(t − τ )}α(τ ) dτ.ub3 (y, t) = −π07.6.
Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè155Òåïåðü äëÿ îïðåäåëåíèÿ ðåøåíèé u1 (x, t) u2 (x, t), u3 (x, t) ñëåäóåò âîñïîëüçîâàòüñÿ ôîðìóëîé îáðàùåíèÿ äëÿ cos-ïðåîáðàçîâàíèÿ Ôóðüå.  ðåçóëüòàòåïîëó÷èì ñëåäóþùèå ôîðìóëû.1.r Z∞2u1 (x, t) =ϕ(y)b exp{−a2 y 2 t + ct} cos yx dy;π(7.11)02.r Z∞ Z t2u2 (x, t) =exp{(−a2 y 2 + c)(t − τ )}fb(y, τ ) dτ cos yx dy; (7.12)π003.u3 (x, t) = −2πZ∞ Z texp{(−a2 y 2 + c)(t − τ )}α(τ ) dτ cos yx dy;0(7.13)0Ðåøåíèå èñõîäíîé çàäà÷è ÿâëÿåòñÿ ñóììîé uj :u(x, t) = u1 (x, t) + u2 (x, t) + u3 (x, t).Ïðèìåð 7.16. Èñïîëüçóÿ ôîðìóëû (7.11) - (7.13), âûâåñòè ôîðìóëûÏóàññîíà.Ðåøåíèå.
Èñïîëüçóåì âû÷èñëåííûé â ïðèìåðå 7.8 èíòåãðàë√Z+∞x2π2 2I(x, t) =e−a y t cos yx dy = √ e− 4a2 t .2a t0Òîãäà äëÿ u3 (x, t) ñïðàâåäëèâà ôîðìóëà1u3 (x, t) = − √a πZt2− 4a2x(t−τ )e√0t−τec(t−τ ) α(τ ) dτ.(7.14)Äëÿ âû÷èñëåíèÿ u1 (x, t) çàìåíÿåì ϕ(y)b èíòåãðàëîì cos-ïðåîáðàçîâàíèÿ Ôóðüå. Âîçíèêàþùèé ïðè ýòîì èíòåãðàëZ+∞2 2J(x, s, t) =e−a y t cos yx cos ys dy07.6. Ïðèìåíåíèå èíòåãðàëüíûõ ïðåîáðàçîâàíèé ê îäíîìåðíîìó óðàâíåíèþòåïëîïðîâîäíîñòè156âûðàæàåòñÿ ÷åðåç I(x, t) ñëåäóþùèì îáðàçîì1J(x, s, t) = (I(x + s, t) + I(x − s, t)).2Ïîýòîìó√2(x−s)2π − (x+s)−2J(x, s, t) = √ (e 4a t + e 4a2 t ).4a tÂñëåäñòâèå ýòîãî ñïðàâåäëèâà ôîðìóëàectu1 (x, t) = √2a πt¶Z∞ µ2(x−s)2− (x+s)−e 4a2 t + e 4a2 t ϕ(s) ds.(7.15)0Àíàëîãè÷íî ïîëó÷àåìZtu2 (x, t) =0ec(t−τ )p2a π(t − τ )Z∞ µ2e− 4a(x+s)2 (t−τ )2+e− 4a(x−s)2 (t−τ )¶f (s, τ )(s) ds dτ.
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