задача штурма-лиувилля - Институт математики и

Ìèíèñòåðñòâî îáðàçîâàíèÿ è íàóêè Ðîññèéñêîé Ôåäåðàöèè
Ôåäåðàëüíîå àãåíòñòâî ïî îáðàçîâàíèþ
Äàëüíåâîñòî÷íûé ãîñóäàðñòâåííûé óíèâåðñèòåò
Ë.À. ÌÎË×ÀÍÎÂÀ
ÇÀÄÀ×À ØÒÓÐÌÀ-ËÈÓÂÈËËß
Ìåòîäè÷åñêèå óêàçàíèÿ äëÿ còóäåíòîâ
ìàòåìàòè÷åñêèõ cïåöèàëüíîcòåé
Âëàäèâîñòîê
Èçäàòåëüñòâî Äàëüíåâîñòî÷íîãî óíèâåðñèòåòà
2006
ÁÁÊ 22.161
Ì75
Ðåöåíçåíòû:
À.Ã. Êîëîáîâ, ê.ô.-ì.í. (ÈÌÊÍ ÄÂÃÓ);
Ì.À Êíÿçåâà, ê.ò.í. (ÈÀÏÓ ÄÂÎ ÐÀÍ).
M75
Ìîë÷àíîâà Ë.À.
Çàäà÷à Øòóðìà-Ëèóâèëëÿ. Ó÷åáíî-ìåòîäè÷åñêîå
ïîñîáèå. - Âëàäèâîñòîê: Èçä-âî Äàëüíåâîñò. óí-òà, 2006. - 16ñ.
Ìåòîäè÷åñêèå óêàçàíèÿ ðàçðàáîòàíû äëÿ ñòóäåíòîâ Èíñòèòóòà ìàòåìàòèêè è êîìïüþòåðíûõ íàóê ÄÂÃÓ. Â
íèõ äàåòñÿ òåîðåòè÷åñêèé ìàòåðèàë, ïîçâîëÿþùèé ñòóäåíòàì
èñïîëüçîâàòü ñðåäñòâà ìàòåìàòè÷åñêîãî ïàêåòà Maple â
ñâîåé ïðàêòè÷åñêîé äåÿòåëüíîñòè ïðè âûïîëíåíèè çàäàíèé
ïî ñïåöêóðñó Ïàêåòû ïðèêëàäíûõ ïðîãðàìì è ìàòåìàòè÷åñêèõ äèñöèïëèí, ñâÿçàííûõ ñ ðåøåíèåì çàäà÷è Øòóðìà Ëèóâèëëÿ.
Äëÿ ñòóäåíòîâ ìàòåìàòè÷åñêèõ ñïåöèàëüíîñòåé.
1702050000
Ì 180(03)−2006
ÁÁÊ 22.161
c Ìîë÷àíîâà Ë.À., 2006
°
c ÈÌÊÍ ÄÂÃÓ, 2006
°
Çàäà÷à Øòóðìà-Ëèóâèëëÿ
Ðàññìîòðèì îáûêíîâåííîå îäíîðîäíîå ëèíåéíîå äèôôåðåíöèàëüíîå
óðàâíåíèå âòîðîãî ïîðÿäêà
[p(x)X 0 (x)]0 + [λr(x) − q(x)]X(x) = 0,
a < x < b,
(1)
ãäå p(x), q(x), r(x) - âåùåñòâåííûå ôóíêöèè îò x. Ïðåäïîëàãàåòñÿ, ÷òî p(x),
p0 (x), q(x), r(x) íåïðåðûâíû â (a,b); p(x) è r(x) ïîëîæèòåëüíû â (a,b); λ ïàðàìåòð, ïðèíèìàþùèé ëþáûå çíà÷åíèÿ.
Êîíöû èíòåðâàëà (a,b) ìîãóò áûòü êàê îáûêíîâåííûìè òî÷êàìè, òàê
è îñîáûìè (ñèíãóëÿðíûìè). Íàïîìíèì, ÷òî åñëè ïðè íåêîòîðîì x õîòÿ
áû îäèí èç êîýôôèöèåíòîâ óðàâíåíèÿ (2) èìååò áåñêîíå÷íûé ðàçðûâ èëè
p(x) = 0, òî ãîâîðÿò, ÷òî êîýôôèöèåíòû óðàâíåíèÿ èìåþò îñîáåííîñòü â
òî÷êå x.
Ãðàíè÷íàÿ çàäà÷à, â êîòîðîé ðåøåíèÿ óðàâíåíèÿ (2) óäîâëåòâîðÿþò îäíîðîäíûì ëèíåéíûì ãðàíè÷íûì óñëîâèÿì ñ âåùåñòâåííûìè êîýôôèöèåíòàìè íàçûâàþò çàäà÷åé Øòóðìà-Ëèóâèëëÿ. Òàêèì îáðàçîì, ïîä çàäà÷åé
Øòóðìà-Ëèóâèëëÿ ïîíèìàåòñÿ ñëåäóþùàÿ çàäà÷à: íàéòè ðåøåíèå óðàâíåíèÿ (2), ïðèíàäëåæàùåå êëàññó C (2) (a, b) è óäîâëåòâîðÿþùèå íåêîòîðûì
îäíîðîäíûì ãðàíè÷íûì óñëîâèÿì, çàäàííûì íà êîíöàõ èíòåðâàëà (a,b).
Ïðèìåðîì òàêèõ óñëîâèé ìîãóò áûòü óñëîâèÿ
X|x→a+0 = 0,
X|x→b−0 = 0.
Ðàçëè÷àþò çàäà÷è äâóõ òèïîâ - ðåãóëÿðíóþ çàäà÷ó è ñèíãóëÿðíóþ çàäà÷ó. Çàäà÷à Øòóðìà-Ëèóâèëëÿ íàçûâàåòñÿ ðåãóëÿðíîé, åñëè èíòåðâàë (a,b)
êîíå÷åí, êîíöû èíòåðâàë (a,b) - îáûêíîâåííûå òî÷êè ðàññìàòðèâàåìîãî
óðàâíåíèÿ. Çàäà÷à íàçûâàåòñÿ ñèíãóëÿðíîé, åñëè õîòÿ áû îäíî èç ýòèõ
óñëîâèé íå âûïîëíåíî. Ñèíãóëÿðíàÿ çàäà÷à ìîæåò áûòü ñ îäíèì èëè äâóìÿ ñèíãóëÿðíûìè êîíöàìè. Õàðàêòåð îäíîðîäíûõ ãðàíè÷íûõ óñëîâèé ðåãóëÿðíîé è ñèíãóëÿðíîé çàäà÷ ðàçíûé.
Ñôîðìóëèðóåì òèïîâûå ãðàíè÷íûå óñëîâèÿ.  ðåãóëÿðíîé çàäà÷å ðàçëè÷àþò ãðàíè÷íûå óñëîâèÿ ïåðâîãî ðîäà
X(a) = 0, X(b) = 0;
ãðàíè÷íûå óñëîâèÿ âòîðîãî ðîäà
X 0 (a) = 0, X 0 (b) = 0;
ãðàíè÷íûå óñëîâèÿ òðåòüåãî ðîäà
X 0 (a) − ha X(a) = 0, X 0 (b) + hb X(b) = 0, ha , hb ≥ 0;
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ãðàíè÷íûå óñëîâèÿ ÷åòâåðòîãî ðîäà
X(a) = X(b), X 0 (a) = X 0 (b), p(a) = p(b).
Ãðàíè÷íûå óñëîâèÿ ÷åòâåðòîãî ðîäà íàçûâàþò óñëîâèÿìè ïåðèîäè÷íîñòè.
 ñëó÷àå ñèíãóëÿðíîé çàäà÷è ðàçëè÷àþò äâà âàðèàíòà çàäà÷ â çàâèñèìîñòè îò òîãî, îäèí èëè äâà êîíöà ñèíãóëÿðíû. Ïóñòü x = a - ñèíãóëÿðíûé
êîíåö, x = b - ðåãóëÿðíûé êîíåö. Òîãäà íà ñèíãóëÿðíîì êîíöå ñòàâèòñÿ
óñëîâèå îãðàíè÷åííîñòè ôóíêöèè
X|x→a+0 = 0(1),
à íà ðåãóëÿðíîì êîíöå ìîãóò áûòü óñëîâèÿ ïåðâîãî, âòîðîãî èëè òðåòüåãî
ðîäà.
Åñëè îáà êîíöà ñèíãóëÿðíû, òî ñòàâÿòñÿ óñëîâèÿ îãðàíè÷åííîñòè ôóíêöèè
X|x→a+0 = 0(1), X|x→b−0 = 0(1).
Íàñ áóäóò èíòåðåñîâàòü íåòðèâèàëüíûå X 6= 0 ðåøåíèÿ çàäà÷è. Îäíàêî íåòðèâèàëüíûõ ðåøåíèé ïðè äàííîì ïðîèçâîëüíîì λ ìîæåò è íå áûòü.
Ïîýòîìó ñîäåðæàíèåì çàäà÷è Øòóðìà-Ëèóâèëëÿ ÿâëÿåòñÿ íå òîëüêî îòûñêàíèå ðåøåíèé (ñîáñòâåííûõ ôóíêöèé ) ïðè äàííîì λ, íî è îïðåäåëåíèå
ñîâîêóïíîñòè (ñïåêòðà) ñîáñòâåííûõ çíà÷åíèé λ, ïðè êîòîðûõ ñóùåñòâóåò
íåòðèâèàëüíûå ðåøåíèÿ. Ñîáñòâåííûå ôóíêöèè, ïî îïðåäåëåíèþ, íàõîäÿòñÿ ñ òî÷íîñòüþ äî ïðîèçâîëüíîé êîíñòàíòû. Åñëè íàêëàäûâàåòñÿ óñëîâèå
Z b
r(x)|X(x)|2 dx = 1,
a
òî èìåþò äåëî ñ íîðìèðîâàííûìè ñîáñòâåííûìè ôóíêöèÿìè.
Äëÿ ðåãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ ñ ãðàíè÷íûìè óñëîâèÿìè
ïåðâîãî, âòîðîãî, òðåòüåãî èëè ÷åòâåðòîãî ðîäà èìåþò ìåñòî ñëåäóþùèå
òåîðåìû.
Òåîðåìà 1. Âñå ñîáñòâåííûå çíà÷åíèÿ ðåãóëÿðíîé çàäà÷è Øòóðìà Ëèóâèëëÿ âåùåñòâåííû.
Òåîðåìà 2. Âñå ñîáñòâåííûå çíà÷åíèÿ ðåãóëÿðíîé çàäà÷è Øòóðìà Ëèóâèëëÿ îãðàíè÷åíû ñíèçó.
Íåêîòîðûå ñâîéñòâà ñîáñòâåííûõ çíà÷åíèé ðåãóëÿðíîé
çàäà÷è Øòóðìà-Ëèóâèëëÿ
Äëÿ ðåãóëÿðíîé çàäà÷è â ñîîòâåòñòâèè ñ îáùåé òåîðèåé Êîøè, ìîæíî
óòâåðæäàòü, ÷òî ñóùåñòâóåò èíòåãðàë X(x) ∈ C (2) óðàâíåíèÿ (1), óäîâëåòâîðÿþùèé óñëîâèÿì X(c) = α, X 0 (c) = β , α, β - ëþáûå ÷èñëà, c ∈ (a, b).
4
Åñëè ÷èñëà α,β íå çàâèñÿò îò ïàðàìåòðà λ, òî ðåøåíèå X(x) ïðè ôèêñèðîâàííîì x ÿâëÿåòñÿ öåëîé ôóíêöèåé îò λ.
Ïîä öåëîé ôóíêöèåé êîìïëåêñíîãî ïåðåìåííîãî λ ïîäðàçóìåâàåòñÿ òàêàÿ ôóíêöèÿ, êîòîðàÿ ðàçëàãàåòñÿ â ðÿä Òåéëîðà âî âñåé îáëàñòè (ðàäèóñ
ñõîäèìîñòè ðàâåí ∞), òî åñòü f (λ) - öåëàÿ ôóíêöèÿ, åñëè
f (λ) =
∞
X
C n λn =
n=0
∞
X
f (n) (0) n
λ , |λ| < ∞.
n!
n=0
 êà÷åñòâå òî÷êè x = c âîçüìåì ãðàíè÷íóþ òî÷êó x = a. Òîãäà ñóùåñòâóþò äâà èíòåãðàëà ϕ(x, λ) è ψ(x, λ), óäîâëåòâîðÿþùèå óñëîâèÿì
ϕ(x, λ) : X(a) = 1, X 0 (a) = 0,
ψ(x, λ) : X(a) = 0, X 0 (a) = 1.
Ìîæíî çàïèñàòü
ϕ(a, λ) = 1, ϕ0 (a, λ) = 0,
ψ(a, λ) = 0, ψ 0 (a, λ) = 1.
(2)
(2)
Ïðè ýòîì èíòåãðàëû ϕ(x, λ), ψ(x, λ) ∈ C (a, b) è ÿâëÿþòñÿ, ïðè ôèêñèðîâàííîì x, öåëûìè ôóíêöèÿìè îò λ. Ýòè ôóíêöèè ëèíåéíî íåçàâèñèìû è
íàçûâàþòñÿ ôóíäàìåíòàëüíîé ñèñòåìîé ðåøåíèé Øòóðìà-Ëèóâèëëÿ.
Îáùèé èíòåãðàë óðàâíåíèÿ Øòóðìà-Ëèóâèëëÿ (1) ìîæíî ïðåäñòàâèòü
â âèäå
X(x, λ) = Aϕ(x, λ) + Bψ(x, λ), A, B = const.
(3)
×èñëà A, B è λ âûáèðàþòñÿ òàê, ÷òîáû ïîëó÷åííîå ðåøåíèå îòâå÷àëî
ãðàíè÷íûì óñëîâèÿì.  ñëó÷àå óñëîâèé ïåðâîãî ðîäà
Aϕ(a, λ) + Bψ(a, λ) = 0,
Aϕ(b, λ) + Bψ(b, λ) = 0,
èëè
A = 0, ψ(b, λ) = 0.
 ñëó÷àå óñëîâèé âòîðîãî ðîäà
Aϕ0 (a, λ) + Bψ 0 (a, λ) = 0,
Aϕ0 (b, λ) + Bψ 0 (b, λ) = 0,
èëè
B = 0, ϕ0 (b, λ) = 0.
5
 ñëó÷àå óñëîâèé òðåòüåãî ðîäà óðàâíåíèå äëÿ îïðåäåëåíèÿ λ èìååò âèä
ϕ0 (b, λ) + hb ϕ(b, λ) + ha ψ 0 (b, λ) + ha hb ψ(b, λ) = 0.
Îêàçûâàåòñÿ, ÷òî âñå ïîëó÷åííûå óðàâíåíèÿ äëÿ îïðåäåëåíèÿ λ èìååò
áåñ÷èñëåííîå ìíîæåñòâî ðåøåíèé.
Ñïåêòð ðåãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ - ñ÷åòíîå ìíîæåñòâî âåùåñòâåííûõ ÷èñåë áåç òî÷åê ñãóùåíèÿ. Ïóñòü λ1 < λ2 < · · · < λn - ïîñëåäîâàòåëüíîñòü ñîáñòâåííûõ çíà÷åíèé. Òîãäà â ñëó÷àå óñëîâèé ïåðâîãî ðîäà
áóäåì èìåòü ñîáñòâåííûå ôóíêöèè
Xn (x) = Bn ψ(x, λn ), n = 1, 2, 3, · · ·
 ñëó÷àå óñëîâèé âòîðîãî ðîäà áóäåì èìåòü
Xn (x) = An ϕ(x, λn ), n = 1, 2, 3, · · ·
è â ñëó÷àå óñëîâèé òðåòüåãî ðîäà
Xn (x) = An [ϕn (x, λn ) + ha ψ(x, λn )], n = 1, 2, 3, · · ·
Èìååò ìåñòî ñëåäóþùàÿ òåîðåìà.
Òåîðåìà. Ñèñòåìà ñîáñòâåííûõ ôóíêöèé ðåãóëÿðíîé çàäà÷è Øòóðìà
- Ëèóâèëëÿ îðòîãîíàëüíà íà [a,b] ñ âåñîì r(x), òî åñòü
Z
½
b
r(x)Xm (x)Xn (x)dx =
a
0,
m 6= n,
||Xn ||2 , m = n.
Ëèòåðàòóðà
1. Ãîâîðóõèí Â.Í., Öèáóëèí Â.Ã. Êîìïüþòåð â ìàòåìàòè÷åñêîì èññëåäîâàíèè. Ó÷åáíûé êóðñ. - ÑÏá.: Ïèòåð, 2001.
2. Ãîëîñêîêîâ Ä. Ï. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ðåøåíèå çàäà÷
â ñèñòåìå Maple. Ó÷åáíèê äëÿ âóçîâ. -ÑÏá.: Ïèòåð, 2004.
3 Áóäàê Á.Ì., Ñàìàðñêèé À.À., Òèõîíîâ À.Í. Ñáîðíèê çàäà÷ ïî ìàòåìàòè÷åñêîé ôèçèêå. Ì.: Íàóêà, 1972.
4. Ñáîðíèê çàäà÷ ïî ìàòåìàòèêå äëÿ âòóçîâ. Ñïåöèàëüíûå êóðñû. Ì.:
Íàóêà, 1984.
5. Ñìèðíîâ Ì.Ì. Çàäà÷è ïî óðàâíåíèÿì ìàòåìàòè÷åñêîé ôèçèêè. Ì.:
Íàóêà, 1975.
6. Òèõîíîâ À.Í., Ñàìàðñêèé À.À. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè.
Ì.: Íàóêà, 1984.
6
Çàäàíèÿ
1 Íàéòè ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè êðàåâîé çàäà÷è
1. y 00 + λy = 0, y(0) = 0, y(l) = 0, x ∈ [0, l];
2. y 00 + λy = 0, y(0) = 0, y 0 (l) + hy(l) = 0, x ∈ [0, l], h > 0;
3. y 00 + λy = 0, y(l) = 0, −y 0 (0) + hy(0) = 0, x ∈ [0, l], h > 0;
4. y 00 + λy = 0, y(0) = 0, y 0 (l) = 0, x ∈ [0, l];
5. y 00 + λy = 0, y 0 (0) = 0, y(l) = 0, x ∈ [0, l];
6. y 00 + λy = 0, y 0 (0) = 0, y 0 (l) = 0, x ∈ [0, l];
7. y 00 + λy = 0, y 0 (0) = 0, y 0 (l) + hy(l) = 0, x ∈ [0, l], h > 0;
8. y 00 + λy = 0, y 0 (l) = 0, −y 0 (0) + hy(0) = 0, x ∈ [0, l], h > 0;
9. y 00 + λy = 0, −y 0 (0) + hy(0) = 0, y 0 (l) + hy(l) = 0, x ∈ [0, l], h > 0;
10. (xy 0 )0 + λ/xy = 0, y(a) = 0, y(b) = 0, x ∈ [a, b], a 6= 0, b 6= 0;
11. (x2 y 0 )0 + λx2 y = 0, y(a) = 0, y(b) = 0, x ∈ [a, b], a 6= 0, b 6= 0;
12. y 00 + λy = 0, y(−l) = 0, y(l) = 0, x ∈ [−l, l];
13. y 00 + λy = 0, y 0 (−l) = 0, y 0 (l) = 0, x ∈ [−l, l];
14. y 00 + λy = 0, −y 0 (−l) + hy(−l)=0, y 0 (l) + hy(l)=0, x ∈ [−l, l], h > 0;
15. (xy 0 )0 + λ/xy = 0, y 0 (a) = 0, y 0 (b) = 0, x ∈ [a, b], a 6= 0, b 6= 0;
16. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y(0) − y 0 (0) = 0, y(1) = 0, x ∈ [0, 1];
17. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y(0) = 0, y(1) + y 0 (1) = 0, x ∈ [0, 1];
18. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y(0) − y 0 (0) = 0 y 0 (1) = 0, x ∈ [0, 1];
19. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y 0 (0) = 0 y(1) + y 0 (1) = 0, x ∈ [0, 1];
20. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y(0) = 0 y(1) + y 0 (1) = 0, x ∈ [0, 1];
21. ((x + 0.2 cos πx)y 0 )0 + λy = 0, 2y(0) + y 0 (0) = 0 y(1) = 0, x ∈ [0, 1];
22. ((x + 0.2 cos πx)y 0 )0 + λy = 0, y(0) − y 0 (0) = 0, y(1) = 0, x ∈ [0, 1];
23. ((4x + cos πx)y 0 )0 + λy = 0, y(0) = 0, y(1) + y 0 (1) = 0, x ∈ [0, 1];
24. ((4x + cos πx)y 0 )0 + λy = 0, y(0) − y 0 (0) = 0 y 0 (1) = 0, x ∈ [0, 1];
25. ((4x + cos πx)y 0 )0 + λy = 0, y 0 (0) = 0 y(1) + y 0 (1) = 0, x ∈ [0, 1].
7
Ïðèìåðû ðåøåíèÿ òèïîâûõ çàäà÷
Ïðèìåð 1. Íàéòè ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè çàäà÷è
Øòóðìà-Ëèóâèëëÿ:
(x2 y 0 )0 + λx2 y = 0, y(0) = O(1), y 0 (a) = 0, x ∈ [0, a].
Ðåøåíèå. Çàäàäèì óðàâíåíèÿ ïðè λ 6= 0 è λ = 0:
>restart:eq:=diff(x^2*diff(y(x),x),x)+lambda*x^2*y(x)=0;
>lambda:=0;eq0:=subs(y(x)=y0(x),eq);lambda:='lambda';
eq := 2 x (
d
d2
y(x)) + x2 ( 2 y(x)) + λ x2 y(x) = 0
dx
dx
λ := 0
eq0 := 2 x (
d
d2
y0(x)) + x2 ( 2 y0(x)) = 0
dx
dx
λ := λ
Íàõîäèì îáùåå ðåøåíèå óðàâíåíèÿ ïðè λ 6= 0:
>dsol:=dsolve(eq,y(x));assign(dsol):
√
√
_C1 sin( λ x) _C2 cos( λ x)
dsol := y(x) =
+
x
x
Ïðîâåðèì ïðàâèëüíîñòü ýòîãî ðåøåíèÿ:
>simplify(value(eq));
0=0
Âèäèì, ÷òî ðåøåíèå áóäåò îãðàíè÷åíî â íóëå, åñëè ïðèíÿòü êîíñòàíòó ïðè
êîñèíóñå ðàâíîé íóëþ:
y:=subs(_C2=0,y(x));
√
_C1 sin( λ x)
y :=
x
Íàõîäèì ïðîèçâîäíóþ:
>y1:=simplify(diff(y,x));
√
√
√
_C1 (−sin( λ x) + cos( λ x) λ x)
y1 :=
x2
8
Ïîëó÷àåì óðàâíåíèå äëÿ îïðåäåëåíèÿ ñîáñòâåííûõ çíà÷åíèé:
>eq1:=expand(subs(x=a,y1)/_C1*a^2)=0;
>eq1:=expand(lhs(eq1)/cos(lambda^(1/2)*a))=0;
√
√
√
eq1 := −sin( λ a) + cos( λ a) λ a = 0
√
sin( λ a) √
√
eq1 := −
+ λa = 0
cos( λ a)
Ïðåîáðàçîâûâàåì ýòî óðàâíåíèå è îïðåäåëÿåì ôóíêöèþ, íóëÿìè êîòîðîé
çàäàþòñÿ ñîáñòâåííûå çíà÷åíèÿ:
>eq1:=convert(eq1,tan);
>f:=subs(lambda^(1/2)*a=mu,lhs(eq1));
√
√
eq1 := −tan( λ a) + λ a = 0
f := −tan(µ) + µ
√
Çäåñü ìû ââåëè îáîçíà÷åíèå f (µ) = −tg(µ) + µ, ïðè÷åì µ = λa.
Ïóñòü µk , k = 1, 2, 3, · · · - ïîëîæèòåëüíûå êîðíè óðàâíåíèÿ f (µ) = 0. Òîãäà ñîáñòâåííûå çíà÷åíèÿ çàäà÷è îïðåäåëÿþòñÿ ïî ôîðìóëå λk = (µk /a)2 .
Ñîáñòâåííûå ôóíêöèè áóäóò sin(µk x/a)/x. Îïðåäåëÿåì ýòè ôóíêöèè:
>y:=(x,n)->sin((mu[n]/a)*x)/x;
µn x
)
a
y := (x, n) →
x
Ñîãëàñíî îáùåé òåîðèè, ýòè ôóíêöèè îðòîãîíàëüíû íà [0,a] ñ âåñîì x2 .
Ïðîâåðèì ýòî:
>Int(x^2*y(x,n)*y(x,m),x=0..a);res:=value(%);
>simplify(res,{sin(mu[m])=mu[m]*cos(mu[m]),
>sin(mu[n])=mu[n]*cos(mu[n])});
Z a
µn x
µm x
sin(
) sin(
) dx
a
a
0
0
Âû÷èñëèì êâàäðàò íîðìû:
>Int(x^2*y(x,n)^2,x=0..a);Norma:=simplify(value(%));
Z a
µn x 2
sin(
) dx
a
0
sin(
N orma :=
1 a (cos(µn ) sin(µn ) − µn )
2
µn
9
Óïðîñòèì ðåçóëüòàò:
>Norma:=subs(sin(mu[n])=mu[n]*cos(mu[n]),Norma);
>Norma:=simplify(Norma,{cos(mu[n])^2=-sin(mu[n])^2+1});
Norma :=
1 a (−cos(µn )2 µn + µn )
2
µn
1
N orma := a sin(µn )2
2
Ïðîâåðèì, áóäåò ëè λ = 0 ñîáñòâåííûì çíà÷åíèåì çàäà÷è:
>sol0:=dsolve(eq0,y0(x));assign(sol0):simplify(value(eq0));
sol0 := y0(x) = _C1 +
_C2
x
0=0
Âèäèì, ÷òî ýòî ðåøåíèå áóäåò îãðàíè÷åííûì â íóëå, åñëè _C2 = 0, à
òîãäà îíî óäîâëåòâîðÿåò âòîðîìó êðàåâîìó óñëîâèþ ïðè ëþáîì _C1. Òàêèì îáðàçîì, ïîëó÷àåì, ÷òî λ = 0 ÿâëÿåòñÿ ñîáñòâåííûì çíà÷åíèåì çàäà÷è,
êîòîðîìó îòâå÷àåò ñîáñòâåííàÿ ôóíêöèÿ, ðàâíàÿ åäèíèöå.
Ïîñìîòðèì íà ðàñïðåäåëåíèå êîðíåé õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ
f (µ) = −tg(µ) + µ = 0;
ïîñòðîèì ãðàôèê ëåâîé ÷àñòè ýòîãî óðàâíåíèÿ:
>plot(f,mu=0..Pi,-0.5..0.5,discont=true,color=black);
0.4
0.2
0
0.5
1
1.5
mu
2
2.5
–0.2
–0.4
Ðèñ. 1. Ãðàôèê ôóíêöèè f (µ)
10
3
Íàéäåì ïåðâûå òðè êîðíÿ õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ:
>K:=3:mu:=array(1..K):
>for i from 1 to K do
>mu[i]:=fsolve(f=0,mu,mu=i*Pi..(i+1)*Pi):
>print(mu[i]);
>end do:
4.493409458
7.725251837
10.90412166
Ïðèìåð 2. Íàéòè ñîáñòâåííûå çíà÷åíèÿ è ñîáñòâåííûå ôóíêöèè çàäà÷è
Øòóðìà-Ëèóâèëëÿ:
y 00 + λy = 0, y(a) = 0, y 0 (b) = 0, x ∈ [a, b].
Ðåøåíèå. Çàäàäèì óðàâíåíèÿ ïðè λ 6= 0
>restart:eq:=diff(y(x),x,x)+lambda*y(x)=0;
³ d2
´
y(x)
+ λy(x) = 0
dx2
Íàõîäèì îáùåå ðåøåíèå óðàâíåíèÿ:
>dsolve(eq,y(x));y:=unapply(rhs(%),x);
eq :=
√
√
y(x) = _C1 sin( λx) + _C2 cos( λx)
√
√
y := x → _C1 sin( λx) + _C2 cos( λx)
Çàäàåì ãðàíè÷íûå óñëîâèÿ:
>assume(b>a):
>eq1:=y(a)=0;eq2:=D[1](y)(b)=0;
√
√
eq1 := _C1 sin( λã) + _C2 cos( λã) = 0
√ √
√ √
eq2 := _C1 cos( λb̃) λ − _C2 sin( λb̃) λ = 0
Ôîðìèðóåì ìàòðèöó êîýôôèöèåíòîâ è âû÷èñëÿåì åå îïðåäåëèòåëü:
>with(linalg):genmatrix({eq1,eq2},{_C1,_C2});A:=det(%);
>Delta:=combine(%));
√
√
·
¸
sin(
λã)
cos(
√ √
√ λã)√
A :=
cos( λb̃) λ − sin( λb̃) λ
√
√
√
∆ := − λ cos( λã − λb̃)
11
Ïðèðàâíèâàåì íóëþ ýòîò îïðåäåëèòåëü è ðåøàåì ïîëó÷åííîå õàðàêòåðèñòè÷åñêîå óðàâíåíèå:
>Delta:=select(has,Delta,[cos]);
√
√
∆ := cos( λã − λb̃)
>_EnvAllSolutions:=true:lambda:=solve(Delta,lambda);
λ :=
π 2 (1 + 2 _Z1 ˜)2
4 (−b̃ + ã)2
>lambda:=subs(_Z1='k',lambda);
λ :=
Íàõîäèì ñîáñòâåííûå ôóíêöèè:
>assume(k,posint):y(x);
 s
√
π 2 (1 + 2 k̃)2
 4

(−b̃ + ã)2
_C1 sin 

4

π 2 (1 + 2 k)2
4 (−b̃ + ã)2
 s
√
π 2 (1 + 2 k̃)2
x
 4


(−b̃ + ã)2
 + _C2 cos 


4



>C1:=solve(eq1,_C1);
C1 :=
k̃) ã
_C2 cos( π2(1+2
)
(−b̃+ã)
k̃) ã
sin( π2(1+2
)
(−b̃+ã)
>simplify(subs(_C1=C1,y(x))):combine(%);
π ã k̃−2 π x k̃
)
_C2 sin( π ã−π x+2
−2 b̃+2 ã
ã+2 π ã k̃
sin( π−2
)
b̃+2 ã
>Yn:=unapply(select(has,%,[x]),x,k);
Yn := (x, k̃) → sin(
π ã − π x + 2 π ã k̃ − 2 π x k̃
)
−2 b̃ + 2 ã
Ïðîâåðèì äèôôåðåíöèàëüíîå óðàâíåíèå:
12

x




>y:='y':Yn(x,k):simplify(subs(y(x)=%,eq));
0=0
Ïðîâåðèì ãðàíè÷íûå óñëîâèÿ:
>Yn(a,k)=0;simplify(D[1](Yn)(b,k))=0;
0=0
0=0
Ïðîâåðèì îðòîãîíàëüíîñòü ñîáñòâåííûõ ôóíêöèé íà îòðåçêå [a,b]:
>assume(n,posint):assume(m,posint):
>Int(Yn(x,n)*Yn(x,m),x=a..b);simplify(value(%));
Z
b̃
sin(
ã
π ã − π x + 2 π ã m̃ − 2 π x m̃
π ã − π x + 2 π ã ñ − 2 π x ñ
) sin(
) dx
−2 b̃ + 2 ã
−2 b̃ + 2 ã
0
Âû÷èñëèì íîðìó ñîáñòâåííûõ ôóíêöèé:
>Norma:=Int(Yn(x,n)^2,x=a..b):simplify(value(%));
b̃ ã
−
2 2
Ìîæíî ïðåîáðàçîâàòü àðãóìåíò ó ñîáñòâåííûõ ôóíêöèé ê áîëåå óäîáíîìó âèäó:
>simplify(collect(
>(Pi*a-Pi*x+2*Pi*a*k-2*Pi*x*k)/(-2*b+2*a),x));
−
(1 + 2 k̃) π (ã − x)
2 (−b̃ + ã)
Òàêèì îáðàçîì, ñîáñòâåííûå çíà÷åíèÿ çàäà÷è áóäóò
λk =
(2k + 1)2 π 2
, k = 1, 2, 3, . . . ,
4(b − a)2
à ñîáñòâåííûå ôóíêöèè yk (x) = sin(
Ðàññìîòðèì ñëó÷àé λ = 0 :
> lambda:=0;eq;
(2k + 1)π(a − x)
), k = 1, 2, 3, . . . .
2(b − a)
λ := 0
13
d2
y(x) = 0
dx2
>dsolve(eq,y(x));assign(%):y0:=unapply(y(x),x);
y(x) = _C1 x + _C2
y0 := x → _C1 x + _C2
>eq0_1:=y0(a)=0;eq0_2:=D(y0)(b)=0;
eq0 _1 := _C1 a˜ + _C2 = 0
eq0 _2 := _C1 = 0
>genmatrix({eq0_1,eq0_2},{_C1,_C2});
·
¸
1 0
a˜ 1
>det(%);
1
Òàê êàê îïðåäåëèòåëü îòëè÷åí îò íóëÿ, çíà÷èò, ñóùåñòâóåò òîëüêî òðèâèàëüíîå ðåøåíèå. Ñëåäîâàòåëüíî, λ = 0 íå åñòü ñîáñòâåííîå çíà÷åíèå
çàäà÷è.
14
Ó÷åáíîå èçäàíèå
Ëèëèÿ Àëåêñàíäðîâíà Ìîë÷àíîâà
ÇÀÄÀ×À ØÒÓÐÌÀ-ËÈÓÂÈËËß
Ìåòîäè÷åñêèå óêàçàíèÿ äëÿ còóäåíòîâ
ìàòåìàòè÷åñêèõ cïåöèàëüíîcòåé
 àâòîðñêîé ðåäàêöèè
Òåõíè÷åñêèé ðåäàêòîð Ë.Ì. Ãóðîâà
Êîìïüþòåðíûé íàáîð è âåðñòêà àâòîðà
Ïîäïèñàíî â ïå÷àòü 31.01.06
Ôîðìàò 60 × 84 1/16. Óñë. ïå÷. ë. 0.9. Ó÷.-èçä. ë. 0,8.
Òèðàæ 25 ýêç.
Èçäàòåëüñòâî Äàëüíåâîñòî÷íîãî óíèâåðñèòåòà
690950, Âëàäèâîñòîê, óë. Îêòÿáðüñêàÿ, 27.
Îòïå÷àòàíî â ëàáîðàòîðèè
êàôåäðû êîìïüþòåðíûõ íàóê ÈÌÊÍ ÄÂÃÓ
690950, Âëàäèâîñòîê, óë. Îêòÿáðüñêàÿ, 27, ê. 132.