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.  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; 3 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.