задача (Задача Штурма-Лиувилля и функция Грина)

ÔÓÍÊÖÈß ÃÐÈÍÀ È ÇÀÄÀ×À ØÒÓÐÌÀ-ËÈÓÂÈËËßÑ. Â. ÏÈÊÓËÈÍ1.Ïîñòàíîâêè çàäà÷ è îïðåäåëåíèÿ∩Ïóñòü x ∈ R1 , Ω = (a, b) èíòåðâàë ÷èñëîâîé ïðÿìîé, , u(x) ∈ C 2 (Ω) C 1 (Ω) ôóíêöèÿ íà îòðåçêå Ω = [a, b], òàêàÿ ÷òî u′′ (x) ∈ L2 (Ω). Ðàññìîòðèì ëèíåéíûéîïåðàòîð äèâåðãåíòíîãî âèäà:Lu(x) ≡ −(p(x)u′ (x))′ + q(x)u(x),(1)ãäå p(x) ∈ C 1 (Ω) äèôôåðåíöèðóåìàÿ ôóíêöèÿ, íå îáðàùàþùàÿñÿ â íîëü íèãäå â Ω,q(x) ∈ C(Ω) íåïðåðûâíàÿ ôóíêöèÿ. Íà u(x) íàêëàäûâàþòñÿ ãðàíè÷íûå óñëîâèÿ âòî÷êàõ a è b:α1 u(a) + α2 u′ (a) = 0, α12 + α22 ̸= 0,(2)β1 u(b) + β2 u′ (b) = 0, β12 + β22 ̸= 0.(3)Áóäåì ðàññìàòðèâàòü çàäà÷ó(4)Lu(x) = f (x)Ñîãëàñíî ôîðìóëå èíòåãðèðîâàíèÿ ïî ÷àñòÿì, ôóíêöèÿ u(x) ÿâëÿåòñÿ ðåøåíèåìóðàâíåíèÿ (4) â òîì è òîëüêî â òîì ñëó÷àå, êîãäà äëÿ ëþáîé áåñêîíå÷íî ãëàäêîéôóíêöèè φ(x) ñ êîìïàêòíûì íîñèòåëåì â Ω âûïîëíåíî ñîîòíîøåíèå:∫∫∫′′p(x)u (x)φ (x)dx + q(x)u(x)φ(x)dx =φ(x)f (x).(5)ΩΩΩ(Êîìïàêòíîñòü íîñèòåëÿ ôóíêöèè φ îçíà÷àåò, ÷òî φ(x) = 0, äëÿ âñåõ x èç íåêîòîðîéîêðåñòíîñòè òî÷åê a è b).Åñëè äëÿ ôóíêöèè u(x), íå ðàâíîé òîæäåñòâåííî íóëþ, è ÷èñëà λ ∈ R âûïîëíÿåòñÿðàâåíñòâîLu(x) = λu(x),(6)òî u(x) íàçûâàþò ñîáñòâåííîé ôóíêöèåé, à λ ñîáñòâåííûì çíà÷åíèåì îïåðàòîðà L.Çàäà÷à ïî îòûñêàíèþ ñîáñòâåííûõ ÷èñåë è ñîáñòâåííûõ ôóíêöèé íàçûâàåòñÿ çàäà÷åéØòóðìà-Ëèóâèëëÿ.

Ñîáñòâåííûå ôóíêöèè è çíà÷åíèÿ èãðàþò âàæíóþ ðîëü â èíæåíåðíûõè ôèçè÷åñêèõ çàäà÷àõ.Íåïðåðûâíàÿ ôóíêöèÿ äâóõ ïåðåìåííûõ G(x, y) : Ω × Ω → R, òàêàÿ ÷òîLG(x, y) = δ(x − y),(7)íàçûâàåòñÿ ôóíêöèåé Ãðèíà îïåðàòîðà L; çäåñü δ(x) äåëüòà-ôóíêöèÿ Äèðàêà.Óñëîâèå (7) îçíà÷àåò (ïî îïðåäåëåíèþ), ÷òî äëÿ ëþáîé ãëàäêîé ôóíêöèè φ(x) ñêîìïàêòíûì íîñèòåëåì âûïîëíåíî ñîîòíîøåíèå:∫∫′′p(x)Gx (x, y)φ (x)dx + q(x)G(x, y)φ(x)dx = φ(y).(8)ΩΩ12Ñ. Â. ÏÈÊÓËÈÍÑ ïîìîùüþ ôóíêöèè Ãðèíà ðåøåíèå çàäà÷è (4) âûðàæàåòñÿ ôîðìóëîé∫u(x) =G(x, y)f (y)dx,(9)Ωòàêèì îáðàçîì, èíòåãðàëüíûé îïåðàòîð â ïðàâîé ÷àñòè (9) ÿâëÿåòñÿ îáðàòíûì ê L.Çàäà÷à Øòóðìà-Ëèóâèëëÿ (6), (2), (3) ñâîäèòñÿ ñ ïîìîùüþ ôóíêöèè Ãðèíà G(x, y)ê èíòåãðàëüíîìó óðàâíåíèþ∫u(x) = λ G(x, y)u(y)dx.(10)Ω îòëè÷èå îò äèôôåðåíöèàëüíîãî óðàâíåíèÿ (6), èíòåãðàëüíîå óðàâíåíèå (10) óæåñîäåðæèò â ñåáå èíôîðìàöèþ î ãðàíè÷íûõ óñëîâèÿõ (2) è (3).2.Ïîñòðîåíèå ôóíêöèè ÃðèíàÏîñòðîèì ôóíêöèþ Ãðèíà äëÿ îïåðàòîðà L (1).

Ïëàí ïîñòðîåíèÿ òàêîâ:1) ðåøèòü (4) êàê îáûêíîâåííîå äèôôåðåíöèàëüíîå óðàâíåíèå ïî îòäåëüíîñòèäëÿ óñëîâèÿ (2) è (3), ïîëó÷èòü ñîîòâåòñòâåííî ðåøåíèÿ ua (x) è ub (x);2) äëÿ äàííîé òî÷êè y ∈ Ω ñêîíñòðóèðîâàòü èç ua (x) è ub (x) íåïðåðûâíóþôóíêöèþ gy (x), óäîâëåòâîðÿþùóþ óñëîâèÿì (2), (3) è óðàâíåíèþ (4) âåçäåâ Ω, êðîìå òî÷êè y .  òî÷êå x = y ôóíêöèÿ gy (x) äîëæíà áûòü íåïðåðûâíà,à åå ïðîèçâîäíàÿ ðàçðûâíà; òîãäà ïðè ïîâòîðíîì äèôôåðåíöèðîâàíèè â (1)âîçíèêíåò äåëüòà-ôóíêöèÿ ñ íåêîòîðûì (íå ðàâíûì íóëþ) êîýôôèöèåíòîì cy ;3) íîðìèðîâàòü gy (x) íà cy ïîëîæèòü G(x, y) = gy (x)/cy .Ïóñòü ua (x) è ub (x) ðåøåíèÿ, îïèñàííûå â ï.

1). Çàôèêñèðóåì òî÷êó y ∈ Ω.Ïîëîæèì{ka ua (x), åñëè a < x <= ygy (x) =(11)kb ub (x), åñëè y <= x < bãäå ka = ub (y), kb = ua (y). Ôóíêöèÿ gy (x) óäîâëåòâîðÿåò óñëîâèÿì ïóíêòà 2).Îáîçíà÷èâ u(x) = ka ua (x)−kb ub (x), ïîäñòàâèì gy (x) â îïðåäåëåíèå (8) âìåñòî G(x, y):∫∫′′p(x)gy (x)φ (x)dx + q(x)gy (x)φ(x)dx =(12)∫ Ωy∫Ωyp(x)u′ (x)φ′ (x)dx +q(x)u(x)φ(x)dx =(13)aa∫ yy′[p(x)u (x)φ(x)]a +Lu(x)φ(x)dx = p(y)u′ (y)φ(y).(14)a ïåðâîì ðàâåíñòâå ó÷òåíî îïðåäåëåíèå (4) äëÿ ôóíêöèè ub (x) è ñîîòíîøåíèå u(x) =gy (x) − ub (x), âûïîëíåííîå ïðè x ∈ (a, y).  ïîñëåäíåì ðàâåíñòâå èñïîëüçóåòñÿ òî,÷òî Lu(x) = 0. Êàê è ñëåäîâàëî îæèäàòü, ôóíêöèÿ gy (x) îòëè÷àåòñÿ îò G(x, y)òîëüêî ïîñòîÿííûì ìíîæèòåëåì cy = −k(y) = p(y)u′ (y), ãäå k(y) ðàâíî ñêà÷êóôóíêöèè p(x)gy′ (x) â òî÷êå x = y :k(y) = p(y)[ua (y)u′b (y) − ub (y)u′a (y)] = p(y)w(y).(15) u (y) ub (y) îïðåäåëèòåëü Âðîíñêîãî (âðîíñêèàí) ðåøåíèé ua (y)Çäåñü w(y) ≡ a′ua (y) u′b (y) è ub (y).

Ñîãëàñíî èçâåñòíîìó ñâîéñòâó âðîíñêèàíà (ñì. ñëåäóþùèé ïóíêò), ñïðàâåäëèâîÔÓÍÊÖÈß ÃÐÈÍÀ È ÇÀÄÀ×À ØÒÓÐÌÀ-ËÈÓÂÈËËß3ñîîòíîøåíèå: p(y)w(y) = const, òî åñòü k = p(a)w(a) íå çàâèñèò îò y .  ÷àñòíîñòè,ôóíêöèÿ Ãðèíà îêàçûâàåòñÿ ñèììåòðè÷íîé îòíîñèòåëüíî x è y :{1 ub (y)ua (x), åñëè a < x <= yG(x, y) = −(16)ua (y)ub (x), åñëè y <= x < bk3.Îïðåäåëèòåëü ÂðîíñêîãîÏóñòü f1 (x), f2 (x), . . . , fn (x), n−1 ðàç äèôôåðåíöèðóåìûå ôóíêöèè.

ÎïðåäåëèòåëåìÂðîíñêîãî (âðîíñêèàíîì) íàáîðà ôóíêöèé fi (x) íàçûâàåòñÿ äåòåðìèíàíò f1 (x)f2 (x)···fn (x) f1′ (x)···fn′ (x) f2′ (x)′′′′f2 (x)···fn′′ (x) (17)w(f1 , . . . , fn )(x) = f1 (x)............ (n−1)(n−1)(n−1) f1(x) f2(x) · · · fn(x) Ðàññìîòðèì îäíîðîäíîå ëèíåéíîå (îáûêíîâåííîå) äèôôåðåíöèàëüíîå óðàâíåíèå ïîðÿäêà n:an (x)f (n) (x) + an−1 (x)f (n−1) (x) + · · · + a1 (x)f ′ (x) + a0 (x)f (x) = 0Óòâåðæäåíèå.(18)Ïóñòü f1 (x), f2 (x), . . .

, fn (x) ðåøåíèÿ óðàâíåíèÿ (18), èw(x) = w(f1 , . . . , fn )(x) èõ âðîíñêèàí. Òîãäà ñïðàâåäëèâî óðàâíåíèåan (x)w′ (x) + an−1 (x)w(x) = 0.(19)Åñëè u1 (x) è u2 (x) ðåøåíèÿ óðàâíåíèÿ Lu(x) = 0, ãäå L îïåðàòîðâèäà (1), è w(x) = w(u1 , u2 )(x) èõ âðîíñêèàí, òî (p(x)w(x))′ = 0.Ñëåäñòâèå.Äîêàæåì óòâåðæäåíèå äëÿ n = 2; â îáùåì ñëó÷àå äîêàçàòåëüñòâî àíàëîãè÷íî.Íåïîñðåäñòâåííî èç îïðåäåëåíèÿ äåòåðìèíàíòà ñëåäóåò ïðàâèëî Ëåéáíèöà: a(x) b(x) ′ a′ (x) b′ (x) a(x) b(x) (20) c(x) d(x) = c(x) d(x) + c′ (x) d′ (x) ,ãäå a(x), b(x), c(x), d(x) ïðîèçâîëüíûå äèôôåðåíöèðóåìûå ôóíêöèè. Ïðîèçâåäåìâû÷èñëåíèå: ′ u1 (x) u′2 (x) u1 (x) u2 (x) ′ + a2 (x) ′′a2 (x)w (x) = a2 (x) ′ u1 (x) u′′2 (x) =u1 (x) u′2 (x) u1 (x) u2 (x) u1 (x) u2 (x) − a0 (x) − a1 (x) ′ u1 (x) u2 (x) =u1 (x) u′2 (x) − a1 (x)w(x).Ýòî äîêàçûâàåò óòâåðæäåíèå.4.ÏðèìåðÍàéäåì ôóíêöèþ Ãðèíà äëÿ çàäà÷è1Lu(x) ≡ xu′′ (x) + u′ (x) − u(x) = 0,xu(1) = 0u(2) − u′ (2) = 0x ∈ (1, 2)(21)(22)(23)4Ñ.

Â. ÏÈÊÓËÈÍÇàïèøåì îïåðàòîð çàäà÷è (21) â ôîðìå (1):1Lu(x) = (xu′ (x))′ − u(x),(24)xòî åñòü, p(x) = −x, q(x) = − x1 . ×àñòíûå ðåøåíèÿ óðàâíåíèÿ (24) áóäåì èñêàòü ââèäå u(x) = xm . Èìååì: m2 xm−1 − xm−1 = 0, îòêóäà m = ±1:u1 (x) = x1u2 (x) = .xÍàéäåì ðåøåíèå çàäà÷è (24), (22) â âèäå ëèíåéíîé êîìáèíàöèè u1 (x) è u2 (x): ua (x) =c1 u1 (x) + c2 u2 (x), 0 = u(1) = c1 u1 (1) + c2 u2 (1) = c1 + c2 , ñëåäîâàòåëüíî1ua (x) = u1 (x) − u2 (x) = x − .xÀíàëîãè÷íî, íàõîäèì ub (x) = d(1 u1 (x) + d2)u2 (x),1130 = ub (2) − u′b (2) = (2 − 1)d1 +− (− ) d2 = d1 + d2 , îòêóäà2444ub (x) = 3x − .xÂû÷èñëèì îïðåäåëèòåëü Âðîíñêîãî:4 13x−x− ua (x) ub (x) xx 2= ,w(x) = ′=ua (x) u′b (x) 4 x1 1+3+ 2 x2xîòêóäà k = p(x)w(x) = −2.

Äëÿ íàõîæäåíèÿ k äîñòàòî÷íî âû÷èñëèòü p(x)w(x)|x=1 : 0 −1 = −2.k = p(1)w(1) = −1 × 2 7 Òàêèì îáðàçîì, () ()14 x − x × 3y − y ,1G(x, y) =() ()214× 3x −, y−yxåñëè 1 < x ≤ y(25)åñëè y ≤ x < 2..