Теория обратных спектральных задач для оператора Лапласа с потенциалом получила развитиев работах В.А. Садовничего, В.В. Дубровского и их учеников [2]. 24. Романов, В.Г. Обратные задачи математической физики / В.Г. Романов.


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ÌèíèñòåðñòâîîáðàçîâàíèÿèíàóêèÐîññèéñêîéÔåäåðàöèè Ôåäåðàëüíîåãîñóäàðñòâåííîåàâòîíîìíîå îáðàçîâàòåëüíîåó÷ðåæäåíèåâûñøåãîîáðàçîâàíèÿ ¾ÞÆÍÎ-ÓÐÀËÜÑÊÈÉÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉÓÍÈÂÅÐÑÈÒÅÒ (ÍÀÖÈÎÍÀËÜÍÛÉÈÑÑËÅÄÎÂÀÒÅËÜÑÊÈÉÓÍÈÂÅÐÑÈÒÅÒ)¿ Èíñòèòóòåñòåñòâåííûõèòî÷íûõíàóê Ôàêóëüòåòìàòåìàòèêè,ìåõàíèêèèêîìïüþòåðíûõòåõíîëîãèé Êàôåäðàóðàâíåíèéìàòåìàòè÷åñêîéôèçèêè ÐÀÁÎÒÀÏÐÎÂÅÐÅÍÀÄÎÏÓÑÒÈÒÜÊÇÀÙÈÒÅ Ðåöåíçåíò,äîöåíòêàôåäðûÇàâåäóþùèéêàôåäðîéóðàâíåíèé ìàòåìàòè÷åñêîãîèêîìïüþòåðíîãîìàòåìàòè÷åñêîéôèçèêèÞÓðÃÓ, ìîäåëèðîâàíèÿ,êàíä.ôèç.-ìàò.íàóê,ä-ðôèç.-ìàò.íàóê, äîöåíòÑàãàäååâàÌ.À.ïðîôåññîðÃ.À.Ñâèðèäþê ¾ ¿ 2017ã.¾ ¿ 2017ã. ×èñëåííîåðåøåíèåîáðàòíîéñïåêòðàëüíîéçàäà÷è äëÿîäíîéìîäåëèãèäðîäèíàìèêè ÂÛÏÓÑÊÍÀßÊÂÀËÈÔÈÊÀÖÈÎÍÍÀßÐÀÁÎÒÀ 01.04.01.2017.145.00.ÌÄ Ðóêîâîäèòåëü ,êàíä.ôèç.-ìàò. íàóê,äîöåíò /Ã.À.Çàêèðîâà/ ¾ ¿ 2017ã. Àâòîð ,ìàãèñòðàíòãðóïïûÅÒ-221 /È.Ñ.Ñòðåïåòîâà/ ¾ ¿ 2017ã. Íîðìîêîíòðîëåð ,êàíä.ôèç.-ìàò. íàóê,äîöåíò /Ä.Å.Øàôðàíîâ/ ¾ ¿ 2017ã. ×åëÿáèíñê 2017 ÓÄÊ517.9 ÑòðåïåòîâàÈ.Ñ. ×èñëåííîåðåøåíèåîáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿîäíîéìàòåìàòè÷åñêîéìîäåëèãèä- ðîäèíàìèêè/È.Ñ.Ñòðåïåòîâà.×åëÿáèíñê,2017.29ñ. Âûïóñêíàÿêâàëèôèêàöèîííàÿðàáîòàïîñâÿùåíààíàëèòè÷åñêîìóè÷èñëåííîìóèññëåäî- âàíèþçàäà÷èäëÿóðàâíåíèÿäâèæåíèÿãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíîñòüþ.Äîêàçàíà òåîðåìàîñóùåñòâîâàíèèðåøåíèÿîáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿóðàâíåíèÿäâèæåíèÿ ãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíîñòüþ.Ðàçðàáîòàíàëãîðèòìíàõîæäåíèÿðåøåíèÿîá- ðàòíîéñïåêòðàëüíîéçàäà÷èäëÿìàòåìàòè÷åñêîéìîäåëèäâèæåíèÿãðóíòîâûõâîä.Ðåàëè- çîâàíàïðîãðàììà,ïîçâîëÿþùàÿâÿâíîìâèäåîïðåäåëèòüïðèáëèæåííîåðåøåíèåçàäà÷è. Ïðèâîäÿòñÿâû÷èñëèòåëüíûåýêñïåðèìåíòû. Áèáëèîãðàôèÿ:37íàçâàíèé;1èëëþñòðàöèÿ;1ïðèëîæåíèå. Îãëàâëåíèå Îáîçíà÷åíèÿèñîêðàùåíèÿ................................. 4 Ââåäåíèå........................................... 5 1Ïðåäâàðèòåëüíûåñâåäåíèÿ............................. 8 2Íåêîòîðûåñâîéñòâà÷èñëîâîéïîñëåäîâàòåëüíîñòè................ 11 3Ïîñòàíîâêàçàäà÷è................................... 13 4Îñíîâíîåñïåêòðàëüíîåòîæäåñòâî.......................... 15 5Ñóùåñòâîâàíèåïîòåíöèàëà.............................. 18 6Àëãîðèòì÷èñëåííîãîðåøåíèÿîáðàòíîéñïåêòðàëüíîéçàäà÷è......... 21 7Âû÷èñëèòåëüíûåýêñïåðèìåíòû........................... 23 Çàêëþ÷åíèå.......................................... 24 Ñïèñîêëèòåðàòóðû..................................... 25 Ïðèëîæåíèå1........................................ 29 3 Îáîçíà÷åíèÿèñîêðàùåíèÿ 1.Ìíîæåñòâà,êàêïðàâèëî,îáîçíà÷àþòñÿçàãëàâíûìèáóêâàìèãîòè÷åñêîãîàëôàâèòà, êðîìå: N ìíîæåñòâîíàòóðàëüíûõ÷èñåë; N 0 = f 0 g[ N ; R ìíîæåñòâîäåéñòâèòåëüíûõ÷èñåë; R + = f a 2 R : a� 0 g ; R = f 0 g[ R + ; C ìíîæåñòâîêîìïëåêñíûõ÷èñåë; L p ( ) ïðîñòðàíñòâàËåáåãà; W l p ( ) ïðîñòðàíñòâàÑîáîëåâàèò.ä. 2.Ýëåìåíòûìíîæåñòâîáîçíà÷àþòñÿñòðî÷íûìèáóêâàìèëàòèíñêîãîèãðå÷åñêîãîàëôà- âèòîâ,êðîìåîïåðàòîðîâ,êîòîðûåîáîçíà÷àþòñÿçàãëàâíûìèáóêâàìèëàòèíñêîãîàëôàâèòà. 3.Îáëàñòüîïðåäåëåíèÿîïåðàòîðà A îáîçíà÷àåòñÿ÷åðåç dom A ,åãîÿäðî÷åðåç ker A , îáðàç÷åðåç im A ,ëèíåéíàÿîáîëî÷êàìíîæåñòâà U ÷åðåç span U . 4.Ìíîæåñòâàîïåðàòîðîâîáîçíà÷àþòñÿðóêîïèñíûìèçàãëàâíûìèáóêâàìèëàòèíñêîãî àëôàâèòà,íàïðèìåð: L ( X ; Y ) ìíîæåñòâàëèíåéíûõíåïðåðûâíûõîïåðàòîðîâ,îïðåäåëåííûõíà X èäåéñòâó- þùèõâïðîñòðàíñòâî Y , L ( X ; X ) L ( X ) ; C l ( X ; Y ) ìíîæåñòâàëèíåéíûõçàìêíóòûõîïåðàòîðîâ,îïðåäåëåííûõíàïðîñòðàíñòâå X èäåéñòâóþùèõâïðîñòðàíñòâî Y , C l ( X ; X ) = C l ( X ) . 5.Ñèìâîëàìè I è O îáîçíà÷àþòñÿòîæäåñòâåííûéèíóëåâîéîïåðàòîðû,îáëàñòèîïðåäå- ëåíèÿêîòîðûõñëåäóþòèçêîíòåêñòà. 6.Âñåêîíòóðûîðèåíòèðîâàíûäâèæåíèåì"ïðîòèâ÷àñîâîéñòðåëêè"èîãðàíè÷èâàþò îáëàñòü,ëåæàùóþ"ñëåâà"ïðèòàêîìäâèæåíèè. 4 Ââåäåíèå Îáðàòíûåçàäà÷èâîçíèêàþòâñëó÷àå,êîãäàíåîáõîäèìîâîññòàíîâèòüâíóòðåííèåïà- ðàìåòðûñðåäûèçäàííûõ,ïîëó÷åííûõãðàíè÷íûìèèçìåðåíèÿìè.Íàïðèìåð,îïðåäåëèòü ìåñòîèìîùíîñòüïðîèçîøåäøåãîçåìëåòðÿñåíèÿïîèçìåðåíèÿìêîëåáàíèéíàïîâåðõíîñòè çåìëè.Âýòîìñëó÷àåíåîáõîäèìîïîëó÷èòüíåêîòîðóþêîñâåííóþèíôîðìàöèþîáèññëåäó- åìîìîáúåêòå,îïðåäåëÿåìóþïðèðîäîéýòîãîîáúåêòà.Âïîëíååñòåñòâåííîâîçíèêàåòâîïðîñ îáîïðåäåëåíèèîïåðàòîðàïîåãîíåêîòîðûìñïåêòðàëüíûìõàðàêòåðèñòèêàì,íàïðèìåð: ñïåêòðó(ïðèðàçëè÷íûõãðàíè÷íûõóñëîâèÿõ),ñïåêòðàëüíîéôóíêöèè,äàííûìðàññåÿíèÿ èò.ä. Äàííàÿäèññåðòàöèÿïîñâÿùåíàèññëåäîâàíèþîáðàòíîéçàäà÷èñëåäóþùåãîâèäà (ÎÇ) Ïîèçâåñòíûìñîáñòâåííûì÷èñëàì f  n g 1 n =1 ,îïå- ðàòîðà T èèçâåñòíûì L -ñîáñòâåííûì÷èñëàì f  n g 1 n =1 âîçìóùåííîãîîïåðàòîðà T + P ,íàéòèïî- òåíöèàë p . Èññëåäîâàíèåîáðàòíûõñïåêòðàëüíûõçàäà÷îáû÷íîñâÿçàíîñòðåìÿîñíîâíûìèýòàïà- ìè: 1)Âûÿñíèòü,êàêèåñïåêòðàëüíûåäàííûåîäíîçíà÷íîîïðåäåëÿþòîïåðàòîð,èäîêàçàòüñî- îòâåòñòâóþùèåòåîðåìûåäèíñòâåííîñòè; 2)Ðåøèòüîáðàòíóþçàäà÷ó:ðàçðàáîòàòüìåòîäàðåøåíèÿ,ïîñòðîèòüàëãîðèòìâîññòàíîâ- ëåíèÿîïåðàòîðàïîðàññìàòðèâàåìûìñïåêòðàëüíûìäàííûì; 3)Íàéòèõàðàêòåðèñòè÷åñêèåñâîéñòâàðàññìàòðèâàåìûõñïåêòðàëüíûõäàííûõ,ïîëó÷èòü íåîáõîäèìûåèäîñòàòî÷íûåóñëîâèéðàçðåøèìîñòèîáðàòíîéçàäà÷è. Öåëüþ ðàáîòûÿâëÿåòñÿðåøåíèåîáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿîäíîéìàòåìàòè- ÷åñêîéìîäåëèãèäðîäèíàìèêèâ n -ìåðíîìñëó÷àå. Äëÿäîñòèæåíèÿöåëèáûëèïîñòàâëåíûñëåäóþùèå çàäà÷è : -èññëåäîâàòüìåòîäûðåøåíèÿîáðàòíûõñïåêòðàëüíûõçàäà÷; -ðàçðàáîòàòüìåòîäèêóâîññòàíîâëåíèÿïîòåíöèàëàâñïåêòðàëüíîéçàäà÷åäëÿìàòåìàòè÷å- ñêîéìîäåëèãèäðîäèíàìèêè; -äîêàçàòüòåîðåìóîñóùåñòâîâàíèèïîòåíöèàëà; -ðåàëèçîâàòüïðîãðàììó,ïîçâîëÿþùóþîïðåäåëèòüâÿâíîìâèäåïðèáëèæåííûéïîòåíöèàë âîçìóùåííîãîîïåðàòîðà-ïðîâåñòè÷èñëåííûéýêñïåðèìåíò. Àêòóàëüíîñòüèññëåäîâàíèÿ. Ïîñòîÿííîðàñòóùèéèíòåðåñêýòîéòåìåîáóñëîâëåí ïîÿâëåíèåìâñåíîâûõïðèëîæåíèéâðàçëè÷íûõîáëàñòÿõíàóê,âòîì÷èñëå,âìàòåìàòèêå, ôèçèêå,òåîðèèîáîëî÷åê,ãèäðîäèíàìèêå,êâàíòîâîéìåõàíèêå,ñåéñìè÷åñêîéòîìîãðàôèè, 5 ìîëåêóëÿðíîéñïåêòðîñêîïèè,ãåîôèçèêåèò.ä. Òàê,ðåøàÿîáðàòíóþñïåêòðàëüíóþçàäà÷óäëÿîïåðàòîðàËàïëàñà,ìîæíîîïðåäåëèòü: -âíóòðèàòîìíûåñèëûïîèçâåñòíûìóðîâíÿìýíåðãèè(ò.å.ñïåêòðó); -êàêñäâèãàòüëîêàëèçàöèþîòäåëüíûõñîñòîÿíèéâïðîñòðàíñòâåèíàýíåðãåòè÷åñêîéøêà- ëå; -êàêèìïîòåíöèàëüíûìâîçìóùåíèåììîæíîóñòðàíèòüèçäèñêðåòíîãîñïåêòðàïðîèçâîëü- íûéóðîâåíü,íåòðîãàÿîñòàëüíûõ; -êàêïîðîäèòüíàçàäàííîììåñòåíîâûéóðîâåíüýíåðãèè; -êàêèçìåíèòüñêîðîñòüðàñïàäàîòäåëüíûõêâàçèñòàöèîíàðíûõñîñòîÿíèéèêâàíòîâûåïå- ðåõîäûìåæäóäèñêðåòíûìèñîñòîÿíèÿìè; -êàêóïðàâëÿòüïðîçðà÷íîñòüþêâàíòîâûõñèñòåì,ñîçäàâàòüòåõíîëîãèèïåðåñòðîéêèñèñòåì âìèêðîýëåêòðîíèêåèäð. Íàñåãîäíÿøíèéäåíüíàèáîëååïîäðîáíîèçó÷åíûîáðàòíûåñïåêòðàëüíûåçàäà÷èäëÿ îïåðàòîðàØòóðìàËèóâèëëÿ Ty = � y 00 + q ( x ) y: Âïåðûåïîñòàíîâêóîáðàòíîéçàäà÷èìîæíîâñòðåòèòüâñòàòüåÂ.À.Àìáàðöóìÿíà1929 ãîäà[32],âêîòîðîéîíïîêàçàë,÷òîäëÿêðàåâîéçàäà÷è 8 : � y 00 +(  � q ( x ) y =0 ; (0 6 x 6  ) ; y 0 (0)= y 0 (  )=0 ; ãäå q ( x ) -äåéñòâèòåëüíàÿíåïðåðûâíàÿôóíêöèÿ,èåñëè  n = n 2 , n =0 ; 1 ;::: ,òî q  0 . Äðóãèìèñëîâàìè,áûëîïîêàçàíî,÷òîåñëèñïåêòðóðàâíåíèÿ y 00 + y =0 ïðèòåõæåêðàå- âûõóñëîâèÿõñîõðàíèëñÿ,òîâîçìóùåíèÿáûòüíåìîãëî.Ò.å.òîëüêîïîñïåêòðóîïåðàòîðà áûëòî÷íîâîññòàíîâëåíïîòåíöèàë.ÂðàáîòåøâåäñêîãîìàòåìàòèêàÃ.Áîðãà[33]äîêàçàíî, ÷òîîïåðàòîðØòóðìà-Ëèóâèëëÿîäíîçíà÷íîîïðåäåëèòüîäíèìñïåêòðîìíåâîçìîæíî, ïîýòîìóäàííûéðåçóëüòàòÿâëÿåòñÿèñêëþ÷åíèåì.Âýòîéæåðàáîòåäîêàçàíî,÷òîäâàñïåê- òðàîïåðàòîðàØòóðìà-Ëèóâèëëÿ(ïðèðàçíûõãðàíè÷íûõóñëîâèÿõ)îïðåäåëÿþòîïåðàòîð ØòóðìàËèóâèëëÿîäíîçíà÷íî.ÏîçäíååðåçóëüòàòÁîðãàáûëòàêæåäîêàçàíÍ.Ëåâèíñî- íîì[34]ñïîìîùüþäðóãîãîìåòîäà. Äëÿøèðîêîãîêëàññàçàäà÷âûñøèõîïåðàòîðîââ÷àñòíûõïðîèçâîäíûõ,îïåðàòîðûïðå- îáðàçîâàíèÿïðè n� 2 èìåþòñëîæíóþñòðóêòóðó,âñðàâíåíèèñîïåðàòîðîìØòóðìà- Ëèóâèëëÿ,îáðàòíûåçàäà÷èäëÿòàêèõîïåðàòîðîâîêàçàëèñüçíà÷èòåëüíîòðóäíååäëÿèñ- ñëåäîâàíèÿ.Â÷àñòíîñòè,ìåòîäîïåðàòîðàïðåîáðàçîâàíèÿîêàçàëñÿíåýôôåêòèâåíïðèèñ- ñëåäîâàíèèýòîãîêëàññàçàäà÷. Áëàãîäàðÿèäååèñïîëüçîâàíèÿêîíòóðíûõèíòåãðàëîâáûëðàçðàáîòàíáîëååóíèâåðñàëü- íûéìåòîäñïåêòðàëüíûõîòîáðàæåíèé.ÂïåðâûåýòèèäåèïðèìåíèëÍ.Ëåâèíñîí[34]. 6 äàëüíåéøåì,ýòîòìåòîäïîëó÷èëðàçâèòèåâðàáîòàõÇ.Ë.Ëåéáåíçîíà[14],Â.À.Þðêî[31]è äð.Íàîñíîâåìåòîäàñïåêòðàëüíûõîòîáðàæåíèéáûëàäîêàçàíàîäíîçíà÷íàÿðàçðåøèìîñòü îáðàòíîéçàäà÷è,ïîëó÷åíîðåøåíèåèíåîáõîäèìûåèäîñòàòî÷íûåóñëîâèÿíàñïåêòðàëüíûå äàííûå. Èññëåäîâàíûîáðàòíûåçàäà÷èäëÿäèñêðåòíûõîïåðàòîðîâ,íåëèíåéíûõäèôôåðåíöè- àëüíûõóðàâíåíèé,äèôôåðåíöèàëüíûõîïåðàòîðîâñçàïàçäûâàíèåì,ìàòðè÷íûõîïåðàòîðîâ ØòóðìàËèóâèëëÿ,äèôôåðåíöèàëüíûõîïåðàòîðîâíàãðàôàõèäð.[31]. Âàæíûìêëàññîìîáðàòíûõçàäà÷ÿâëÿþòñÿîáðàòíûåçàäà÷èñïåêòðàëüíîãîàíàëèçàäëÿ óðàâíåíèéâ÷àñòíûõïðîèçâîäíûõ.Òàêèåçàäà÷èáûëèèññëåäîâàíûâðàáîòàõÀ.Ë.Áóõãåéìà [1],Ì.Ì.Ëàâðåíòüåâà[12],Ë.Ï.Íèæíèêà[21],À.È.Ïðèëåïêî[36],Â.Ã.Ðîìàíîâà[24]è äðóãèõ. Òåîðèÿîáðàòíûõñïåêòðàëüíûõçàäà÷äëÿîïåðàòîðàËàïëàñàñïîòåíöèàëîìïîëó÷èëà ðàçâèòèåâðàáîòàõÂ.À.Ñàäîâíè÷åãî,Â.Â.Äóáðîâñêîãîèèõó÷åíèêîâ[2]. ×èñëåííûåìåòîäûðåøåíèÿïðÿìûõèîáðàòíûõñïåêòðàëüíûõçàäà÷áûëèðàçâèòû Ñ.È.Êàä÷åíêî[17]. Âðàáîòàõ[9]Ã.À.Çàêèðîâîéáûëîïðîâåäåíîêà÷åñòâåííîåè÷èñëåííîåèññëåäîâàíèåîá- ðàòíîéñïåêòðàëüíîéçàäà÷èâñëó÷àå,êîãäàîïåðàòîðËàïëàñàçàäàííà N ìåðíîìïàðàë- ëåëåïèïåäåêðàåâûìèóñëîâèÿìèÄèðèõëåèëèÍåéìàíà,èìååòêðàòíûéñïåêòðèñòåïåíü ; çàâèñÿùóþîòðàçìåðíîñòèîáëàñòè,íàêîòîðîéðàññìàòðèâàåòñÿçàäà÷à.Àèìåííî,èññëå- äîâàíûñóùåñòâîâàíèåèåäèíñòâåííîñòüðåøåíèÿèâîññòàíîâëåíîâîçìóùåíèåâîáðàòíûõ ñïåêòðàëüíûõçàäà÷äëÿìàòåìàòè÷åñêèõìîäåëåéñâîçìóùåííîéñòåïåíüþîïåðàòîðàËà- ïëàñàñêðàòíûìñïåêòðîì.Âðàáîòå[11]ðåçîëüâåíòíûéìåòîäáûëïðèìåíåíäëÿðåøåíèÿ ïðÿìîéñïåêòðàëüíîéçàäà÷èäëÿóðàâíåíèÿÄçåêöåðàïîîòíîñèòåëüíîìóñïåêòðóíåâîçìó- ùåííîãîîïåðàòîðà. 7 1. Ïðåäâàðèòåëüíûåñâåäåíèÿ Îïðåäåëåíèå1.1. [4] Ðàâíîìåðíîéíîðìîéîïåðàòîðà A 2 R ( H ) íàçûâàåòñÿ÷èñëî k A k =sup k ' k =1 k A' k : Ïóñòü � ñïðÿìëÿåìûéêîíòóð,îãðàíè÷èâàþùèéîáëàñòü G � , �   ( A ) .Òîãäàîïåðàòîð P � = � 1 2 i Z � R ( ;A ) d (1 : 1) ÿâëÿåòñÿ ïðîåêòîðîì ,ïåðåñòàíîâî÷íûìñ A . Åñëè G � \  ( A )= f  k g n k =1 ; òî P � = n X k =1 P  k ;P  k P  j =0( j 6 = k ) ; (1 : 2) ãäå P  k ( k = 1 ;n ) ïðîåêòèðóþùèå H íàèíâàðèàíòíûåîòíîñèòåëüíî A ïîäïðîñòðàíñòâà P  k H ,âêàæäîìèçêîòîðûõâåñüñïåêòð A ñîñòîèòòîëüêîèçîäíîãî÷èñëà  k : Îïðåäåëåíèå1.2. [4] Êîðíåâîéêðàòíîñòüþîïåðàòîðà A ,îòâå÷àþùåéêîíòóðó � ,íàçû- âàåòñÿñóììààëãåáðàè÷åñêèõêðàòíîñòåéâñåõñîáñòâåííûõ÷èñåë  j ( j = 1 ;n ) îïåðàòîðà A; êîòîðûåïîïàäàþòâîáëàñòü G � ,ò.å.÷èñëî  � ( T )= n X j =1   j ( A )=dim P � H : Ëåììà1.1. [4] Ïóñòü P è Q äâàïðîåêòîðà.Åñëè k P � Q k 1 ,òî dim P H =dim Q H : Îïðåäåëåíèå1.3. [25] Îïðåäåëåííûéâñþäóâ H ëèíåéíûéîïåðàòîð A íàçûâàåòñÿâïîëíå íåïðåðûâíûì,åñëèîíïåðåâîäèòâñÿêîåîãðàíè÷åííîåìíîæåñòâîòî÷åêâòàêîåìíîæå- ñòâî,èçâñÿêîéáåñêîíå÷íîéïîñëåäîâàòåëüíîñòèêîòîðîãîìîæíîâûäåëèòüñõîäÿùóþñÿ êíåêîòîðîìóýëåìåíòó H (âñìûñëåìåòðèêèâ H )ïîäïîñëåäîâàòåëüíîñòü.×àñòîòàêèå îïåðàòîðûíàçûâàþòêîìïàêòíûìè. Ëåììà1.2. [25] Âïîëíåíåïðåðûâíûéîïåðàòîðîãðàíè÷åí. Îïðåäåëåíèå1.4. [25] Îïåðàòîð A ,äåéñòâóþùèéâñåïàðàáåëüíîìãèëüáåðòîâîìïðî- ñòðàíñòâå H ,íàçûâàåòñÿäèñêðåòíûì,åñëèñóùåñòâóåòíåêîòîðîåêîìïëåêñíîå÷èñëî  0 òàêîå,÷òî R (  0 ;A ) ÿâëÿåòñÿâïîëíåíåïðåðûâíûìîïåðàòîðîìâ H . Òåîðåìà1.1. [4] Ìíîæåñòâî S 1 ; ÿâëÿþùååñÿäâóñòîðîííèìèäåàëîìêîëüöà R ,ïðèòîì ñàìîñîïðÿæåííûì(ñèììåòðè÷åñêèì)èçàìêíóòûì,îáëàäàåòñëåäóþùèìèñâîéñòâàìè: 1) S 1 ëèíåéíîåìíîæåñòâî,ïðè÷åìåñëè A 2 R , B 2 S 1 ,òî AB 2 S 1 è BA 2 S 1 . 2) Åñëè A 2 S 1 ,òî A  2 S 1 : 3) S 1 çàìêíóòîâ R ( S 1 = S 1 ). Îòñþäàñëåäóåò,÷òîåñëè A 2 R ,òî k A k = k A k 1 : 8 Îïðåäåëåíèå1.5. [4] Ñîáñòâåííûå÷èñëà f s j ( A ) g 1 j =1 îïåðàòîðà ( A  A ) 1 = 2 2 S 1 íàçûâà- þòñÿ s ÷èñëàìèèëèñèíãóëÿðíûìè÷èñëàìèîïåðàòîðà A 2 S 1 : Î÷åâèäíî,åñëè A ñàìîñîïðÿæåííûé,òî s j ( A )= j  j ( A ) j ;j 2 N : Âñÿêèéâïîëíåíåïðåðûâíûéëèíåéíûéîïåðàòîð A äîïóñêàåòðàçëîæåíèåØìèäòà[4] A = dim R ( H ) X j =1 s j ( A )(  ;' j ) j ; (1 : 3) ãäå f ' j g è f j g íåêîòîðûåîðòîíîðìèðîâàííûåñèñòåìû,ïîëíûåâ H . Îïðåäåëåíèå1.6. [25] Ìíîæåñòâîâïîëíåíåïðåðûâíûõîïåðàòîðîâ,ñèíãóëÿðíûå÷èñëà êîòîðûõïðèíåêîòîðîì p� 0 îáðàçóþòñõîäÿùèéñÿðÿä 1 X j =1 s p j ( A ) ; îáîçíà÷èì S p : Îïåðàòîð A íàçûâàåòñÿÿäåðíûì,åñëè A 2 S 1 ; îïåðàòîð A íàçûâàåòñÿîïåðàòîðîì ÃèëüáåðòàØìèäòà,åñëè A 2 S 2 : Îïðåäåëåíèå1.7. [25] Ìíîæåñòâîîïåðàòîðîâ S p ïðè p  1 îáðàçóåòñèììåòðè÷íî íîðìèðîâàííûéèäåàëâàëãåáðåâñåõîãðàíè÷åííûõîïåðàòîðîâñíîðìîé k A k p = 1 X j =1 s p j ( A ) ! 1 =p : (1 : 4) Ïðè p =1 ýòàíîðìàíàçûâàåòñÿÿäåðíîé,ïðè p =2 íîðìîéÃèëüáåðòàØìèäòà (àáñîëþòíîéíîðìîé). Òåîðåìà1.2. Åñëè 0 p 1 p 2 1 è A 2 S p 1 ,òî A 2 S p 2 è k A k p 2 k A k p 1 ; (1 : 5) â÷àñòíîñòè, k A kk A k p , 1  p 1 . Òåîðåìà1.3. Åñëèîïåðàòîðû A j ( j = 1 ;n )ïðèíàäëåæàòïðîñòðàíñòâàì S p j è n X j =1 p � 1 j  1 ñîîòâåòñòâåííî,òîîïåðàòîð A = A 1 A 2 :::A n 2 S p ,ãäå p � 1 = n X j =1 p � 1 j , ïðè÷åì k A k p k A 1 k p 1 k A 2 k p 2  ::: k A n k p n : (1 : 6) Â÷àñòíîñòè,åñëè A 2 S p (1  p 1 ) è B 2 S q ,ãäå 1 p + 1 q =1 ,òî AB;BA 2 S 1 ; k AB k 1 k A k p k B k q ; k BA k 1 k A k p k B k q : (1 : 7) Òåîðåìà1.4. Åñëè B;C 2 S 2 ,òî BC 2 S 1 è X j j  j ( BC ) j 1 . 9 Òåîðåìà1.5. Åñëè B 2 S p (1  p 1 ) , A è C îãðàíè÷åííûåîïåðàòîðû,òî k ABC k p k A kk B k p k C k : Îïðåäåëåíèå1.8. [4] Ãîâîðÿò,÷òîîïåðàòîð A 2 R ( H ) èìååòêîíå÷íûéìàòðè÷íûé ñëåä,åñëèäëÿëþáîãîîðòîíîðìèðîâàííîãîáàçèñà f ' j g 1 j =1 ïðîñòðàíñòâà H ñõîäèòñÿ ðÿä 1 X j =1 ( A' j ;' j ) . Òåîðåìà1.6. [4] Äëÿòîãî,÷òîáûîïåðàòîð A 2 R ( H ) èìåëêîíå÷íûéìàòðè÷íûéñëåä, íåîáõîäèìîèäîñòàòî÷íî,÷òîáû A 2 S 1 .Åñëè A 2 S 1 ,òîñóììà 1 X j =1 ( A' j ;' j ) íåçàâèñèò îòâûáîðàîðòîíîðìèðîâàííîãîáàçèñà f ' j g 1 j =1 âïðîñòðàíñòâå H . Ýòàñóììàîáîçíà÷àåòñÿ÷åðåç SpA èíàçûâàåòñÿ( ìàòðè÷íûì ) ñëåäîì îïåðàòîðà A . Òåîðåìà1.7. [4] Åñëè A;B 2 S 1 ,òî 1 : Sp( A + B )= Sp A + Sp B; ; 2 C : 2 : Sp A  = Sp A: Òåîðåìà1.8. [4] Åñëè A 2 S 1 ;B 2 R ,ïðè÷åì AB 2 S 1 è BA 2 S 1 ; òî Sp( AB )=Sp( BA ) : Òåîðåìà1.9. [16] Åñëè A 2 S 1 ,òîìàòðè÷íûéñëåäîïåðàòîðà A ñîâïàäàåòñåãîñïåê- òðàëüíûìñëåäîì Sp A = X j  j ( A ) : (1 : 8) Òåîðåìà1.10. Åñëè A 2 S 1 ,òî j Sp A jk A k 1 : Òåîðåìà1.11. [25] Åñëè A 2 S 2 , B 2 R òî k AB k 2 k B kk A k 2 è k BA k 2 k B kk A k 2 Îïðåäåëåíèå1.9. [38] Ìíîæåñòâî  L ( T )= f  2 C :( L � T ) � 1 2L ( F ; U ) g íàçûâàåòñÿ ðåçîëüâåíòíûììíîæåñòâîìîïåðàòîðà T îòíîñèòåëüíîîïåðàòîðà L . Îïðåäåëåíèå1.10. [38] Îïåðàòîð-ôóíêöèÿ R 0 (  )=( L � T ) � 1 íàçûâàåòñÿ L -ðåçîëüâåíòîé îïåðàòîðà T: Îïðåäåëåíèå1.11. [38] Îïåðàòîð-ôóíêöèÿ R (  )=( L � T � P ) � 1 íàçûâàåòñÿ L -ðåçîëüâåíòîé îïåðàòîðà T + P: 10 2. Íåêîòîðûåñâîéñòâà÷èñëîâîéïîñëåäîâàòåëüíîñòè Ïóñòü f  t g 1 t =1 ÷èñëîâàÿïîñëåäîâàòåëüíîñòü,êîòîðàÿóäîâëåòâîðÿåòàñèìïòîòèêå  t  Ct ,ãäå C = 4  ab : Ëåììà2.1. [11] Ââåäåìñëåäóþùèåîáîçíà÷åíèÿ: r t = 1 2 min t f  t +1 �  t ;  t �  t � 1 g ;t� 1; r 1 = 1 2 (  2 �  1 ); r 0 =inf t r t ; a t = f  :Re  =  t +1 +  t 2 g ; � r t = f  2 C :Re  2 a t g ; r t = f  2 C : j  t �  j = r t g ; r t = f  : j  t �  j r 0 g ; = 1 T t =1 r t : Òîãäàñïðàâåäëèâûñëåäóþùèåíåðàâåíñòâà: :::  a t � 1   t � r t  t  t + r t  a t   t +1 � r t +1  t +1  t +1 + r t +1  a t +1  :::; èâêëþ÷åíèÿ r t  r t , � r t  r t , r t  ;t 2 N : Ëåììà2.2. [11] Ïðè t  1 ;a t � 1  Re  a t èìååòìåñòîîöåíêà 1 X j =1 1 j  �  j j 2  1 r 2 t � 1 +  +2 r t : Äîêàçàòåëüñòâî. 1 X j =1 1 j  �  j j 2 1 j  �  t j 2 + + 1 X j = t +2 1 (  j � a t ) 2 + 1 ( a t � 1 �  t � 1 ) + 1 (  t +1 � a t ) 2 + t � 2 X j =1 + 1 ( a t � 1 �  j ) 2 : Äëÿäàííîãî t íàéäóòüñÿ k è s òàêèå÷òî,  t =  ks .Îáîçíà÷èì k 2 + s 2 =~ r 2 : 1 X j = t +2 1 (  2 � a t ) 2 = X p 2 + q 2 � ~ r 2 1 (  pq � a t � 1 ) 2 = Z 1 ~ r Z p ~ r 2 � p 2 0 dpdq (( p 2 + q 2 ) � a t � 1 ) 2 = = Z 1 ~ r Z  2 0 dd' (  2 � a t � 1 ) 2 =  4 frac 1~ r 2 � at � 1   4 1  t � a t � 1   2 r t : 11 Ïðîâåäåìñëåäóþùóþîöåíêó. t � 2 X j =1 1 ( a t � 1 �  j ) 2 = t � 1 X j =1 1 ( a t �  j ) 2 = t � 1 X j =1 1 ( a t �  pq ) 2 = = X ( p 2 + q 2 ) ~ r 2 1 ( a t � ( p 2 + q 2 )) 2  Z ~ r 0 Z p ~ r 2 � p 2 0 dpdq ( a t � ( p 2 + q 2 )) 2 = Z ~ r 0 Z  2 0 dd' ( a t �  2 ) 2    4  a t � ( a t �  t ) ( a t �  t ) a t    2 r t Ñîáèðåìâñåîöåíêèèïîëó÷èì: 1 X j =1 1 j  �  j j 2 1 r 2 t +  +2 r t : Ëåììàäîêàçàíà . 12 3. Ïîñòàíîâêàçàäà÷è Ïîäãðóíòîâûìèâîäàìèïîíèìàþòñÿñâîáîäíûå(ãðàâèòàöèîííûå)âîäûïåðâîãîîòïî- âåðõíîñòèÇåìëèñòàáèëüíîãîâîäîíîñíîãîãîðèçîíòà,êîòîðûéçàêëþ÷åíâðûõëûõîòëîæå- íèÿõèëèâåðõíåéòðåùèíîâàòîé÷àñòèêîðåííûõïîðîä,çàëåãàþùåãîíàïåðâîìîòïîâåðõíî- ñòèâîäîóïîðíîìñëîå,âûäåðæàííîìïîïëîùàäè.Îáëàñòüèõïèòàíèÿñîâïàäàåòñîáëàñòüþ ðàñïðîñòðàíåíèÿâîäîïðîíèöàåìûõïîðîä.Âåðõíÿÿãðàíèöàçîíûíàñûùåíèÿíàçûâàåòñÿ óðîâíåìèëèçåðêàëîìãðóíòîâûõâîä.Ïîðîäà,íàñûùåííàÿâîäîé,íàçûâàåòñÿâîäîíîñíûì ãîðèçîíòîì.Åãîìîùíîñòüîïðåäåëÿåòñÿðàññòîÿíèåìïîîòçåðêàëàãðóíòîâûõâîääîâîäî- óïîðà.Îíàèçìåíÿåòñÿâïðîñòðàíñòâåèâîâðåìåíè.Ïèòàíèåãðóíòîâûõâîäïðîèñõîäèòçà ñ÷åòèíôèëüòðàöèèàòìîñôåðíûõîñàäêîâ,èíîãäàçàñ÷åòèíôèëüòðàöèèâîäðåêèäðóãèõ ïîâåðõíîñòíûõâîäîåìîâ,òàêæåïîäïèòûâàíèÿèçáîëååãëóáîêèõâîäîíîñíûõãîðèçîíòîâ[5]. Äâèæåíèåãðóíòîâûõâîäïîä÷èíÿåòñÿñèëåòÿæåñòèèîñóùåñòâëÿåòñÿââèäåïîòîêîâïî ñîîáùàþùèìñÿïîðàìèëèòðåùèíàì.Çåðêàëîãðóíòîâûõâîääîèçâåñòíîéñòåïåíèïîâòî- ðÿåòðåëüåôïîâåðõíîñòè,èãðóíòîâûåïîòîêèäâèæóòñÿîòïîâûøåííûõó÷àñòêîâêïîíè- æåííûì(ðåêàì,îçåðàì,ìîðÿì),ãäåïðîèñõîäèòèõðàçãðóçêàââèäåðîäíèêîâèëèñêðûòûì ñóáàêâàëüíûìðàññðåäîòî÷åííûìñïîñîáîì(íàïðèìåð,ïîäâîäàìèðóñåëðåê,äíîìîçåðè ìîðåé).Òàêèåîáëàñòèíàçûâàþòñÿîáëàñòÿìèðàçãðóçêèèëèäðåíèðîâàíèÿ.Ãðóíòîâûéïî- òîê,íàïðàâëåííûéêìåñòàìðàçãðóçêè,îáðàçóåòêðèâîëèíåéíóþïîâåðõíîñòü,íàçûâàåìóþ äåïðåññèîííîé.Òå÷åíèåãðóíòîâîéâîäûíàçûâàåòñÿôèëüòðàöèåé.Îíàçàâèñèòîòíàêëîíà çåðêàëàãðóíòîâûõâîäèëèîòãèäðàâëè÷åñêîãî(íàïîðíîãî)ãðàäèåíòà,àòàêæåîòâîäîïðî- íèöàåìîñòèãîðíûõïîðîä.Âáîëüøèíñòâåñâîåìãðóíòîâûåâîäûèìåþòñâîáîäíóþïîâåðõ- íîñòüèíåïîñðåäñòâåííóþñâÿçüñàòìîñôåðîé[5].Áîëüøîéïðàêòè÷åñêèéèíòåðåñâòåîðèè äâèæåíèÿãðóíòîâûõâîäïðåäñòàâëÿåòóðàâíåíèå 8 : (  � ) u t =  u �  2 u + p ( x ) u + f; u j @ = u j @ =0 ; (3 : 1) äâèæåíèÿãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíîñòüþ,êîòîðîåìîäåëèðóåòýâîëþöèþñâîáîä- íîéïîâåðõíîñòèôèëüòðóþùåéñÿæèäêîñòè,ãäå = " + k kh 0 a ; = 2( " + k ) k 2 H 2 0 ; = h 0 3 a : Èññëåäîâàíèåýòîãîóðàâíåíèÿìîæíîâñòðåòèòüâðàáîòàõìíîãèõàâòîðîâ.Òàê,áûëèèñ- ñëåäîâàíûíåñòàöèîíàðíûåìîäåëèñâîáîäíîéïîâåðõíîñòèôèëüòðóþùåéñÿæèäêîñòè[30], ¾êâàçèáàíàõîâ¿àíàëîãîäíîðîäíîéçàäà÷èÄèðèõëåâîãðàíè÷åííîéîá-ëàñòèñãëàäêîéãðà- íèöåéäëÿëèíåéíîãîóðàâíåíèÿÄçåêöåðà[37],îñóùåñòâîâàíèèýêñïîíåíöèàëüíûõäèõî- òîìèéðåøåíèéýâîëþöèîííîãîóðàâíåíèÿÄçåêöåðàñîáîëåâ-ñêîãîòèïàâêâàçèñîáîëåâûõ ïðîñòðàíñòâàõ[35].×òîêàñàåòñÿñïåêòðàëüíûõçàäà÷,òîïåðâûåïîïûòêèèññëåäîâàíèÿ, èçâåñòíûåàâòîðàì,áûëèïðåäïðèíÿòûâðàáîòå[7]. 13 Çàäàäèìîïåðàòîðû T;L : U �! F ôîðìóëàìè T =  �  2 ; = n X j =1 @ 2 @x 2 j ;L =  �  ; (3 : 2) ïðè÷åì U = f u 2 W k +2 p ( ): u ( x )=0 ;x 2 @ g ; F = f u 2 W k p ( ) g ; 1 p 1 ;k =0 ; 1 ;::: dom T = f u 2 W k +4 ( ): u ( x )=0 g\ U ; Ïóñòü = f x =( x 1 ;x 2 ;:::;x N ):0  x j  a j ;j =1 ;:::N g ;a j � 0 Ïóñòü P îïåðàòîðóìíîæåíèÿíàôóíêöèþ p 2 C 1 ( ) : Ðàññìîòðèìîïåðàòîð T + P .Îáîçíà÷èì÷åðåç f  n g 1 n =1 =  L ( T + P ) ,ãäå  n çàíóìåðîâàíû âïîðÿäêåíåâîçðàñòàíèÿèõäåéñòâèòåëüíûõ÷àñòåéñó÷åòîìàëãåáðàè÷åñêîéêðàòíîñòè. Ñîáñòâåííûå÷èñëàîïåðàòîðà T :  n =  n  n �  n �  ; (3 : 3) ãäå f  n g 1 n =1 =  () ñîáñòâåííûå÷èñëàîäíîðîäíîéçàäà÷èÄèðèõëåäëÿîïåðàòîðàËàïëà- ñà,çàíóìåðîâàííûåïîíåâîçðàñòàíèþ. Ðàññìîòðèìîäíóèçêëàññè÷åñêèõçàäà÷âòåîðèèâîçìóùåíèé,çàäà÷óâîññòàíîâëåíèÿ êîýôôèöèåíòàóðàâíåíèÿâñëåäóþùåéïîñòàíîâêå: 8 � � � � � � : ( T + P ) u =  n Lu; u j @ = u j @ ;  u = u: (3 : 4) Íåîáõîäèìîïîèçâåñòíûìñîáñòâåííûì÷èñëàì f  n g 1 n =1 ,îïåðàòîðà T èèçâåñòíûì L -ñîáñòâåííûì÷èñëàì f  n g 1 n =1 âîçìóùåííîãîîïåðàòîðà T + P ,íàéòèïîòåíöèàë p . 14 4. Îñíîâíîåñïåêòðàëüíîåòîæäåñòâî Ðàññìîòðèìîïåðàòîðû T è L ,çàäàííûåôîðìóëàìè: T =  �  2 ; = n X j =1 @ 2 @x 2 j ;L =  �  ; (4 : 1) Äëÿîïåðàòîðà L èìååì L' n =(  � ) ' n =(  �  n ) ' n = n (  �  n ) ' n ; 6 =  n 0 ; =  n ; ñëåäîâàòåëüíî R L 0 (  ) ' n = n ' n  �  n ; 6 =  n ; 0 ; =  n ; (4 : 2) R 0 (  ) ' n = n ' n (  �  n )(  �  n ) ; 6 =  s ' s  2 �  ; =  s : (4 : 3) Èñõîäÿèçôîðìóë(2.2.1),(2.2.2)è(2.2.3)äàëååáóäåìïîëàãàòü,÷òî  6 =  n ; 6 =  n ; 8 n 2 N : Îïåðàòîðû R 0 (  ) ;R L 0 (  ) ; ãäå  2  ( T ) ; ÿâëÿþòñÿÿäåðíûìè,òàêêàêðÿäû 1 X n =1 1  �  n ; 1 X n =1 (  �  n )  �  n ñõîäÿòñÿ.Ïóñòü n = f  : j  �  n j = r n g ; r n = 1 2 min n fj  n +1 �  n j ; j  n �  n � 1 jg ; r 0 =inf n r n : Ëåììà4.1. Åñëèïðè  2 n k PR 0 (  ) k = q 1 ; òîñïðàâåäëèâîðàâåíñòâî LR (  )= LR 0 (  )+ 1 X k =1 [ R 0 (  ) P ] k LR 0 (  )(4 : 4) Äîêàçàòåëüñâòî. Ðàññìîòðèìòîæäåñòâî L � T � P =( I � PR 0 (  ))( L � T ) : Òàêêàê k PR 0 (  ) k 1 ,òîñóùåñòâóåòëèíåéíûéîãðàíè÷åííûéîïåðàòîð R (  )=( L � T � P ) � 1 = R 0 (  )( I � PR 0 (  )) � 1 : Îòñþäàñëåäóåò,÷òî R (  )= R 0 (  ) B (  ) ,ãäå B (  ) íåêîòîðûéîãðàíè÷åííûéîïåðàòîð. Ïîñêîëüêó T äèñêðåòíûéîïåðàòîð,òî R 0 (  ) ÿâëÿþùèéñÿâïîëíåíåïðåðûâíûì,ñëåäîâà- òåëüíî, R (  ) òàêæåâïîëíåíåïðåðûâåí,ò.å.îïåðàòîð T + P ÿâëÿåòñÿäèñêðåòíûì. 15 Èçïîñëåäíåãîñîîòíîøåíèÿòàêæåñëåäóåò,÷òî R (  ) ÿäåðíûéîïåðàòîð,äëÿêîòîðîãî áóäåòñïðàâåäëèâîðàçëîæåíèåâñõîäÿùèéñÿïîíîðìåðÿä R (  )= R 0 (  )+ 1 X k =1 [ R 0 (  ) P ] k R 0 (  ) ; Äîìíîæèìïðåäûäóùååòîæäåñòâîíà L ñëåâàèïîëó÷èìòðåáóåìîåðàâåíòñâî. Ëåììà4.2. Ïóñòü  2  L ( T ) ; òîãäàâûïîëíÿåòñÿñëåäóþùàÿîöåíêà k LR 0 (  ) k 1  ( ; L ( T )) ; (4 : 5) çäåñü  ( ; L ( T )) îçíà÷àåòðàññòîÿíèåîòòî÷êè  äî L -ñïåêòðàîïåðàòîðà T . Äîêàçàòåëüñòâî. Ïóñòü  2  L ( T ) : Èç[38]èçâåñòíîïðåäñòàâëåíèå L -ðåçîëüâåíòûîïåðàòîðà T ââèäåðÿäà Íåéìàíà R 0 (  )= R 0 (  ) 1 X k =0 (  �  ) k ( LR 0 (  )) Î÷åâèäíî,÷òîðÿäâïðàâîé÷àñòèàáñîëþòíîñõîäèòñÿ,ïîêðàéíåéìåðåäëÿòåõ  ,êîòîðûå óäîâëåòâîðÿþòóñëîâèþ j  �  j 1 k LR 0 (  ) k : Îòñþäàïîëó÷èì k LR 0 (  ) k 1  ( ; L ( T )) Ëåììàäîêàçàíà . ÐàññìîòðèìíîðìóðàçíîñòèïðîåêòîðîâÐèññà 1 2 i Z n ( R (  ) � R 0 (  )) d  Z n k R 0 (  ) P kk R (  ) kj d j Z n k R 0 (  ) L kk R (  ) kj d j 1 ; ïîýòîìóâñåêîðíåâûåïîäïðîñòðàíñòâàîïåðàòîðà T + P èìåþòòàêóþæåðàçìåðíîñòü,÷òî èîïåðàòîð T .Ñëåäîâàòåëüíî,ñïåêòðîïåðàòîðà T + P áóäåòîäíîêðàòíûì.Ðàññìîòðèì ðÿä(4.1).Óìíîæèìïðàâóþèëåâóþ÷àñòèïîëó÷åííîãîðàâåíñòâàíà  2 i èïðîèíòåãðèðóåì äàííîåðàâåíñòâîïîêîíòóðó n .Ïîëó÷èì 1 2 i Z n LR (  ) d = 1 2 i Z n LR 0 (  ) d + 1 X k =1 1 2 i Z n  [ R 0 (  ) P ] k LR 0 (  ) d: Âîñïîëüçóåìñÿÿäåðíîñòüþîïåðàòîðîâ T è T + P èíàéäåììàòðè÷íûéñëåäîáåèõ÷àñòåé ïîëó÷åííîãîðàâåíñòâà. 16 Sp 1 2 i Z n LR 0 (  ) d = Sp 1 2 i Z n  1 X k =1 P k  �  k d =  n SpP n =  n 1 X s =1 ( P n ' s ;' s )= =  n 1 X s =1  ( ' s ;' n ) ' n ;' s  =  n 1 X s =1  ( ' s ;' n )( ' n ;' s )  =  n : Àíàëîãè÷íî Sp 1 2 i Z n LR (  ) d =  n : Âû÷èñëèìïåðâóþïîïðàâêóòåîðèèâîçìóùåíèé: 1 n = Sp 1 2 i Z n R 0 (  ) PLR 0 (  ) d = 1 2 i Z n Sp ( R 0 (  ) PLR 0 (  )) d = = 1 2 i Z n  1 X s =1 ( R 0 (  ) PLR 0 ' s ;' s ) d = = 1 2 i Z n  1 X s =1 ( PR 2 0 (  ) L' s ;' s ) d = 1 2 i Z n  1 X s =1 ( P' s ;' s ) (  �  s ) 2 (  �  s ) d = = 1 2 i 1 X s =1 ( P' s ;' s ) Z n  (  �  s ) 2 (  �  s ) d = = 1 2 i ( P' n ;' n ) Z n  (  �  n ) n (  �  n ) d = = 1 2 i ( P' n ;' n ) Z n d (  �  n ) 2 (  �  n ) = ( P' n ;' n )  �  n : Ïîëó÷èìñïåêòðàëüíîåòîæäåñòâî  n =  n + ( P' n ;' n )  �  n + n ; (4 : 6) ãäå n = 1 X k =2 1 2 i Z n  [ R 0 (  ) P ] k LR 0 (  ) d: 17 5. Ñóùåñòâîâàíèåïîòåíöèàëà Ðÿä 1 X n =1 r n  max  2 r n k R 0 (  ) k 2 2  2 ! 1 2 ñõîäèòñÿ,îáîçíà÷èìåãîñóììó÷åðåç s .Ïóñòüäàëåå r 2 (0 ; min f r 0 ; 1 s g ) Ëåììà5.1. Åñëè k P j k r; 0 rr 0 j =1 ; 2 ,òî j n ( p 1 ) � ( p 2 ) j rr n k P 1 � P 2 k max  2 n k R 0 (  ) k 2 2 : Äîêàçàòåëüñòâî. Âåäåìîáîçíà÷åíèå R j (  )=( T + P j � E ) � 1 ;j =1 ; 2 : Óìíîæèìðÿä R (  )= 1 X k =1 R 0 (  )( P j R 0 (  )) k ; 2 íà ( P j R 0 (  )) 2 ; ïîëó÷èì R (  )( P j R 0 (  )) 2 = 1 X k =2 ( � 1) k R 0 (  )( P j R 0 (  )) k ; 2 : Îáîçíà÷èì÷åðåç ( k ) n k -óþïîïðàâêó: ( k ) n ( p )= ( � 1) k 2 i Z n Sp  ( PR 0 (  )) k  LR 0 (  ) d = ( � 1) k +1 2 ik Z n Sp [ PR 0 (  )] k d; n ( p )= 1 X k =2 ( k ) n ( p ) : Îöåíèìðàçíîñòè k -õïîïðàâîê, k  2 : j ( k ) n ( p 1 ) � ( k ) n ( p 2 ) j = j ( � 1) k +1 2 ik Z n Sp [ P 1 R 0 (  )] k d � � ( � 1) k +1 2 ik Z n Sp [ P 2 R 0 (  )] k d j = 1 2 k Z n Sp  ( R 0 (  ) P 1 ) k � ( R 0 (  ) P 2 ) k  d   1 2 k Z n k ( R 0 (  ) P 1 ) k � ( R 0 (  ) P 2 ) k kj d j  r n n max  2 n k ( R 0 (  ) P 1 ) k � ( R 0 (  ) P 2 ) k k  r n n max  2 n k � 1 X s =0 ( R 0 (  ) P 1 ) s ( P 1 + P 2 ) R 0 (  )( R 0 P 2 ) k � s � 1   r n n max  2 n k � 1 X s =0 k P 1 � P 2 k  r 2  k � 1 k R 0 (  ) k 2 2 k R 0 (  ) k k � 2 ! = = r n k P 1 � P 2 k  r 2  k � 1 max  2 n � k R 0 (  ) k 2 2 k R 0 (  ) k k � 2  : 18 Îöåíèììîäóëüðàçíîñòè j n ( p 1 ) � n ( p 2 ) j r n k P 1 � P 2 k r 2 max  2 n k R 0 (  ) k 2 2 1 X k =0  r 2  k max  2 n k R 0 (  ) k k   rr n k P 1 � P 2 k max  2 n k R 0 (  ) k 2 2 : Ëåììàäîêàçàíà . Òåîðåìà5.1. Åñëèäëÿêîìïëåêñíîéïîñëåäîâàòåëüíîñòè ~  n âûïîëíÿåòñÿíåðàâåíñòâî 1 X n =1 j  n �  n j r 2 (1 � ! ) ; òîñóùåñòâóåò p 2 L 2 (0 ; ) ,òàêîé÷òîäëÿëþáîãî n 2 N âûïîëíÿåòñÿ ~  n =  n ; ãäå  n 2  L ( T ) : Äîêàçàòåëüñòâî. Âïðîñòðàíñòâå L 2 (0 ; ) ðàññìîòðèìóðàâíåíèåîòíîñèòåëüíî p p = ( p ) � 0 ; (5 : 1) ãäå 0 =( � 1) N p 2 N V 1 X n =1 (~  n �  n )(  �  n ) ' ( n k ) ; (5 : 2) ( p )=( � 1) N p 2 N V 1 X n =1 (~  n �  n )(  �  n ) n ( p ) ' k n ; (5 : 3) Ââåäåìâðàññìîòðåíèåñëåäóþùóþñèñòåìóôóíêöèé: v n = r 2 N + q V N Y j =1 cos  2 m 2 j x j a j  ; ãäå m =( m 1 ;:::;m N ) , m j 2f 0 g[ N , q -÷èñëîíåíóëåâûõèíäåêñîââìóëüòèèíäåêñå m . Ââåäåìîïåðàòîð A : L 2 (0 ; ) ! L 2 (0 ; ) ; îïðåäåëÿåìûéðàâåíñòâîì Ap = 0 � ( p ) : Òàêêàê k Ap k L 2 k 0 k + k ( p ) k r 2 (1 � ! )+ r 2 ! = r 2 ; òîîïåðàòîð A îòîáðàæàåòçàìêíóòûéøàð U (0 ; r 2 ) âñåáÿ.Ïîêàæåì,÷òîîïåðàòîð A áóäåò ñæèìàþùèì. 19 k Ap 1 � Ap 2 k L 2 = k ( p 1 ) � ( p 2 ) k = p 2 N V 1 X n =1 j n ( p ) � n ( p 2 ) j 2 ! 1 2  p 2 N V  rs k p 1 � p 2 k L 2 = ! k p 1 � p 2 k L 2 : Óðàâíåíèå(5.1)èìååòåäèíñòâåííîåðåøåíèå p .Îïðåäåëèìîïåðàòîð P ,äåéñòâóþùèéâ L 2 (0 ; ) ,ñëåäóþùèìîáðàçîì: P' ( x )= p ( x ) ' ( x ) ,ãäå p ðåøåíèåóðàâíåíèÿ(5.1).Ïîêàæåì, ÷òîðåøåíèå p èåñòüèñêîìûéïîòåíöèàë.Óìíîæèìñêàëÿðíîóðàâíåíèå(5.1)íàôóíêöèþ v n ,ïîëó÷èì ( p;v n )=( � 1) N p 2 N V [(~  n �  n )(  �  n ) � n ( p )(  �  n )] : (5 : 4) Ïðåîáðàçóåì ( P' n ;' n )= 2 N V Z n p ( x 1 ;:::;x N ) N Y j =1 sin 2  ( m j x j ) a j  dx 1 :::dx N = = ( � 1) N V Z n p ( x 1 ;:::;x N ) N Y j =1 cos  (2 m j x j ) a j  dx 1 :::dx N = = ( � 1) N p 2 N V ( p;v n ) Ïîäñòàâèìïîñëåäíååðàâåíñòâîâñïåêòðàëüíîåòîæäåñòâî(4.6):  n =  n +~  n �  n � n ( p )+ n ( p ) )  n =~  n : Òåîðåìàäîêàçàíà . 20 6. Àëãîðèòì÷èñëåííîãîðåøåíèÿîáðàòíîéñïåêòðàëüíîé çàäà÷è Íàïðàêòèêå÷àñòîâîçíèêàåòíåîáõîäèìîñòüâóïðàâëåíèèïîòåíöèàëàìèñïîìîùüþ ñïåêòðàëüíûõïàðàìåòðîâ.Ïðèýòîìâîçíèêàåòïðîáëåìàäàæåïðèñàìîìîïðåäåëåíèèïî- òåíöèàëà.Òðóäíîñòüçàäà÷èîïðåäåëåíèÿïîòåíöèàëàñîñòîèòâòîì,÷òîíåñóùåñòâóåòóíè- âåðñàëüíûõàëãîðèòìîâèìåòîäîâåå÷èñëåííîãîðåøåíèÿ. Íàîñíîâåòåîðåòè÷åñêèõðåçóëüòàòîâáûëàðàçðàáîòàíàïðîãðàììà,ïîçâîëÿþùàÿîïðå- äåëèòüâÿâíîìâèäåïðèáëèæåííûéïîòåíöèàë,òàêîé,÷òîñïåêòðâîçìóùåííîãîîïåðàòîðà áóäåòñîâïàäàòüñäàííîéïîñëåäîâàòåëüíîñòüþ. Ðàññìîòðèìóðàâíåíèåäâèæåíèÿãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíîñòüþ,êîòîðîåìî- äåëèðóåòýâîëþöèþñâîáîäíîéïîâåðõíîñòèôèëüòðóþùåéñÿæèäêîñòè: (  � ) u t =  u �  2 u + p ( x ) u + f; ãäå = " + k kh 0 a ; = 2( " + k ) k 2 H 2 0 ; = h 0 3 a : Ðàññìîòðèììàòåìàòè÷åñêóþìîäåëüäâèæåíèÿãðóíòîâûõâîä: 8 : (  � ) u t =  u �  2 u + p ( x ) u + f; u j @ = u j @ =0 ; (6 : 1) Çàäàäèìîïåðàòîðû T;L : U �! F ôîðìóëàìè T =  �  2 ; = n X j =1 @ 2 @x 2 j ;L =  �  ; (6 : 2) ïðè÷åì U = f u 2 W k +2 p ( ): u ( x )=0 ;x 2 @ g ; F = f u 2 W k p ( ) g ; 1 p 1 ;k =0 ; 1 ;::: domT = f u 2 W k +4 ( ): u ( x )=0 g\ U ; Ïóñòü = f x =( x 1 ;x 2 ;:::;x N ):0  x j  a j ;j =1 ;:::N;a j � 0 g Ïóñòü P îïåðàòîðóìíîæåíèÿíàôóíêöèþ p 2 C 1 ( ) : Ðàññìîòðèìîïåðàòîð T + P .Îáîçíà÷èì÷åðåç f  n g 1 n =1 =  L ( T + P ) ,ãäå  n çàíóìåðîâàíû âïîðÿäêåíåâîçðàñòàíèÿèõäåéñòâèòåëüíûõ÷àñòåéñó÷åòîìàëãåáðàè÷åñêîéêðàòíîñòè. Ñîáñòâåííûå÷èñëàîïåðàòîðà T :  n =  n  n �  n �  ; (6 : 3) ãäå f  n g 1 n =1 =  () ñîáñòâåííûå÷èñëàîäíîðîäíîéçàäà÷èÄèðèõëåäëÿîïåðàòîðàËàïëà- ñà,çàíóìåðîâàííûåïîíåâîçðàñòàíèþ. 21 Ðàññìîòðèìîäíóèçêëàññè÷åñêèõçàäà÷âòåîðèèâîçìóùåíèé,çàäà÷óâîññòàíîâëåíèÿ êîýôôèöèåíòàóðàâíåíèÿâñëåäóþùåéïîñòàíîâêå: 8 � � � � � � : ( T + P ) u =  n Lu; u j @ = u j @ =0 ;  u = u: (6 : 4) Íåîáõîäèìîïîèçâåñòíûìñîáñòâåííûì÷èñëàì f  n g 1 n =1 ,îïåðàòîðà T èèçâåñòíûì L -ñîáñòâåííûì÷èñëàì f  n g 1 n =1 âîçìóùåííîãîîïåðàòîðà T + P ,íàéòèïîòåíöèàë p . Àëãîðèòìíàõîæäåíèÿðåøåíèÿîáðàòíîéñïåêòðàëüíîéçàäà÷è (6 : 4) ñâîäèòñÿêñëåäóþ- ùèìøàãàì: Øàã1. Ââîäïàðàìåòðîâñðåäû: , ,  , k , h 0 . Øàã2. Çàäàíèåñîáñòâåííûõ÷èñåëîïåðàòîðàËàïëàñà{  n },ñîáñòâåííûõ÷èñåëíåâîçìó- ùåííîãî{  n }èâîçìóùåííîãî{  n }îïåðàòîðîâ. Øàã3. Âû÷èñëåíèåïîïðàâîêòåîðèèâîçìóùåíèé. Øàã4. Âû÷èñëåíèåêâàäðàòàñêàëÿðíîãîïðîèçâåäåíèÿèñóììèðîâàíèååãîðåçóëüòàòîâ. Øàã5. Âûâîäôîðìóëûïîòåíöèàëàèåãîãðàôèê. Âïðèëîæåíèè1ïðåäñòàâëåíàîáîáùåííàÿñõåìààëãîðèòìàðàáîòûýòàïîâïðîãðàììû. Äëÿðåàëèçàöèèðàçðàáîòàííîãîàëãîðèòìàèñïîëüçîâàëèñüâñòðîåííûåôóíêöèèèñòàí- äàðòíûåîïåðàòîðûïðîãðàììíîãîïàêåòàMaple17.0.Áûëàðàçðàáîòàíàðîãðàììà"×èñëåí- íîåèññëåäîâàíèåîáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿîäíîéìàòåìàòè÷åñêîéìîäåëèãèäðî- äèíàìèêè"ïðåäíàçíà÷åíàäëÿ÷èñëåííîãîèññëåäîâàíèÿîáðàòíîéñïåêòðàëüíîéçàäà÷è (6 : 4) íà n ìåðíîìïàðàëëåëåïèïåäåâçàâèñèìîñòèîòçàäàííûõïàðàìåòðîâ.Äàííàÿïðîãðàììà ïîçâîëÿåòíàõîäèòüïðèáëèæåííîåðåøåíèåîáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿîäíîéìàòå- ìàòè÷åñêîéìîäåëèãèäðîäèíàìèêèèñòðîèòüãðàôèê.Ãðàôè÷åñêèåèçîáðàæåíèÿïîëó÷åíû ïðèïîìîùèïàêåòàplots.Ïðîãðàììàýêñïëóàòèðóåòñÿíàïåðñîíàëüíîìêîìïüþòåðåïîä óïðàâëåíèåìîïåðàöèîííîéñèñòåìûMicrosoftWindows. Ëîãè÷åñêàÿñòðóêòóðàïðîãðàììû Ïðîãðàììàâêëþ÷àåòâñåáÿñëåäóþùèåýòàïû: Ââîäïàðàìåòðîâñðåäû; Çàäàíèåñîáñòâåííûõ÷èñåëîïåðàòîðàËàïëàññà; Çàäàíèåñîáñòâåííûõ÷èñåëíåâîçìóùåííîãîèâîçìóùåííîãîîïåðàòîðîâ; Âû÷èñëåíèåïîïðàâîêòåîðèèâîçìóùåíèé; Âû÷èñëåíèåêâàäðàòàñêàëÿðíîãîïðîèçâåäåíèÿèñóììèðîâàíèååãîðåçóëüòàòîâ; Âûâîäôîðìóëûïîòåíöèàëàèåãîãðàôèê; 22 7. Âû÷èñëèòåëüíûåýêñïåðèìåíòû Ðàáîòóïðîãðàììûïðîèëëþñòðèðóåìíàïðèìåðåçàäà÷èâäâóìåðíîìñëó÷àå. 8 : (  � ) u t =  u �  2 u + p ( x ) u + f; u j @ = u j @ =0 : (8 : 1) Îïåðàòîðû T;L : U �! F çàäàíûôîðìóëàìè T =  �  2 ; = n X j =1 @ 2 @x 2 j ;L =  �  ; Ðåäóöèðóåìçàäà÷ó (8 : 1) êçàäà÷å: 8 � � � � � � : ( T + P ) u =  n Lu; u j @ = u j @ ;  u = u: (8 : 2) Ïðèìåð1. Ïóñòüäàíûïåðâûå4 L -cîáñòâåííûõ÷èñåëâîçìóùåííîãîîïåðàòîðà: f  11 g =28 : 53095300 ; f  12 g =43 : 05988707 ; f  13 g =87 : 26078746 ; f  21 g =102 : 0387722 ; Ôîðìóëàïðèáëèæåííîãîïîòåíöèàëà,âîññòàíîâëåííûéâïðîãðàììåïîïåðâûì÷åòûðåì ÷ëåíàìïîñëåäîâàòåëüíîñòè: ~ p = � 16 � (0 : 120 e � 1(4 : 41 � (5 = 4)  2 )) p (2) cos (2  x ) cos ( y ) � � (0 : 120 e � 1(4 : 41 � (17 = 4)  2 )) p 2 cos (2 x ) cos (2 y ) � � (0 : 120 e � 1(4 : 41 � 2 Pi 2 )) p 2 cos (4 x ) cos ( y ) � (0 : 120 e � 1(4 : 41 � 5  2 )) p 2 cos (4 x ) cos (2 y ) : 23 Ïðèìåð2. Ïóñòüäàíûïåðâûå9 L -cîáñòâåííûõ÷èñåëâîçìóùåííîãîîïåðàòîðà: f  11 g =28 : 53055300 ; f  12 g =43 : 05948707 ; f  13 g =67 : 57670640 ; f  21 g =87 : 26038746 ; f  22 g =102 : 0383722 ; f  23 g =126 : 6829350 ; f  31 g =185 : 8631229 ; f  32 g =200 : 6617561 ; f  33 g =225 : 3278884 : Ïðèáëèæåííûéïîòåíöèàë,âîññòàíîâëåííûéâïðîãðàììåïîïåðâûìäåâÿòè÷ëåíàìïî- ñëåäîâàòåëüíîñòè f  n g èçîáðàæåííàðèñ.1 Ðèñ.1:Âîññòàíîâëåííûéïîòåíöèàë. Çàêëþ÷åíèå Âðàìêàõðàáîòûèññëåäîâàíàçàäà÷àäëÿóðàâíåíèÿäâèæåíèÿãðóíòîâûõâîäñîñâîáîä- íîéïîâåðõíîñòüþ.Äîêàçàíàòåîðåìàîñóùåñòâîâàíèèðåøåíèÿîáðàòíîéñïåêòðàëüíîéçàäà- ÷èäëÿóðàâíåíèÿäâèæåíèÿãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíîñòüþ.Ïîëó÷åíàôîðìóëà äëÿïðèáëèæåííîãîâîññòàíîâëåíèÿïîòåíöèàëà.Ðàçðàáîòàíàëãîðèòìíàõîæäåíèÿðåøåíèÿ îáðàòíîéñïåêòðàëüíîéçàäà÷èäëÿìàòåìàòè÷åñêîéìîäåëèäâèæåíèÿãðóíòîâûõâîä.Ðåà- ëèçîâàíàïðîãðàììà,ïîçâîëÿþùàÿâÿâíîìâèäåîïðåäåëèòüïðèáëèæåííîåðåøåíèåçàäà÷è. Ïðîâåäåíûâû÷èñëèòåëüíûåýêñïåðèìåíòû. 24 Ñïèñîêëèòåðàòóðû 1. Áóõãåéì,À.Ë.Ââåäåíèåâòåîðèþîáðàòíûõçàäà÷/À.Ë.Áóõãåéì.Íîâîñèáèðñê:Íàóêà, 1988.181ñ. 2. Âåëèêèõ,À.Ñ.Îáðàòíûåçàäà÷èñïåêòðàëüíîãîàíàëèçà:àâòîðåô.äèñ.êàíä.ôèç.-ìàò. íàóê/À.Ñ.Âåëèêèõ.Ìàãíèòîãîðñê,1999.16ñ. 3. Ãàâè÷,È.Ê.Ãèäðîãåîäèíàìèêà:Ó÷åáíèêäëÿâóçîâ/È.Ê.Ãàâè÷Ì.:Íåäðà,1988. 349ñ. 4. Ãîõáåðã,È.Ö.Ââåäåíèåâòåîðèþëèíåéíûõíåñàìîñîïðÿæåííûõîïåðàòîðîâ/È.Ö.Ãîõ- áåðã,È.Ã.Êðåéí.//Ì.:Íàóêà,1965.448ñ. 5. Äçåêöåð,Å.Ñ.Îáîáùåííûåóðàâíåíèÿäâèæåíèÿãðóíòîâûõâîäñîñâîáîäíîéïîâåðõíî- ñòüþ/Å.Ñ.Äçåêöåð//ÄÀÍÑÑÑÐ.1972.Ò.202,5.Ñ.10311033. 6. Äóáðîâñêèé,Â.Â.Âîññòàíîâëåíèåïîòåíöèàëàïîñîáñòâåííûìçíà÷åíèÿìðàçíûõçàäà÷ /Â.Â.Äóáðîâñêèé//ÓÌÍ.1997.Ñ.155156. 7. Äóáðîâñêèé,Â.Â.Òåîðåìàîåäèíñòâåííîñòèðåøåíèÿîáðàòíûõçàäà÷ñïåêòðàëüíî- ãîàíàëèçà/Â.Â.Äóáðîâñêèé//Äèôôåðåíö.óðàâíåíèÿ.1997.Ò.33,3. Ñ.421422. 8. Äóáðîâñêèé,Â.Â.Òåîðåìàîñóùåñòâîâàíèèðåøåíèÿîáðàòíîéçàäà÷èñïåêòðàëüíîãî àíàëèçàäëÿñòåïåíèîïåðàòîðàËàïëàñà/Â.Â.Äóáðîâñêèé,À.Ñ.Âåëèêèõ//Ýëåêòðî- ìàãíèòíûåâîëíûèýëåêòðîííûåñèñòåìû.1998.Ò.3,5.C.69. 9. Çàêèðîâà,Ã.À.Âîññòàíîâëåíèåïîòåíöèàëàâîáðàòíîéñïåêòðàëüíîéçàäà÷åäëÿîïåðà- òîðàËàïëàñàñêðàòíûìñïåêòðîì//×åëÿáèíñê.:ÂåñòíèêÞÓðÃÓ.Ñåðèÿ"Ìàòåìàòè- ÷åñêîåìîäåëèðîâàíèåèïðîãðàììèðîâàíèå".2010.35.Ñ.25-28. 10. Çàêèðîâà,Ã.À.Çàäà÷àâîññòàíîâëåíèÿïîòåíöèàëàäëÿóðàâíåíÿýâîëþöèèñâîáîäíîé ïîâåðõíîñòèôèëüòðóþùåéñÿæèäêîñòè/Ã.À.Çàêèðîâà,Å.Â.Êèðèëëîâ//Þæíî- Óðàëüñêàÿìîëîäåæíàÿøêîëàïîìàòåìàòè÷åñêîìóìîäåëèðîâàíèþñáîðíèêòðóäîââñå- ðîññèéñêîéíàó÷íî-ïðàêòè÷åñêîéêîíôåðåíöèè×åëÿáèíñê:ÈçäàòåëüñêèéöåíòðÞÓð- ÃÓ.2014.Ñ.59-62. 11. ÇàêèðîâàÃ.À.,ÊèðèëëîâÅ.Â.ÐåãóëÿðèçîâàííûéL-ñëåäîäíîãîâîçìóùåííîãîîïåðàòîðà //Ã.À.Çàêèðîâà,Å.Â.Êèðèëëîâ//Îäåññà.:ÂiñíèêÎä.íàö.óí-òó.Ìàò.iìåõ.2013. Ò.18,âèï.2(18).Ñ.7-13. 12. Ëàâðåíòüåâ,Ì.Ì.Íåêîððåêòíûåçàäà÷èìàòåìàòè÷åñêîéôèçèêèèàíàëèçà/Ì.Ì.Ëàâ- ðåíòüåâ,Â.Ã.Ðîìàíîâ,Ñ.Ï.Øèøàòñêèé.Ì.:Íàóêà,1980.286ñ. 25 13. Ëåâèòàí,Á.Ì.Îáðàòíûåçàäà÷èØòóðìàËèóâèëëÿ/Á.Ì.Ëåâèòàí.Ì.:Íàóêà,1984. 240ñ. 14. Ëåéáåíçîí,Ç.Ë.Îáðàòíàÿçàäà÷àñïåêòðàëüíîãîàíàëèçàäëÿäèôôåðåíöèàëüíûõîïå- ðàòîðîââûñøèõïîðÿäêîâ/Ç.Ë.Ëåéáåíçîí//Òðóäûìîñê.ìàòåì.î-âà,1966.15. Ñ.70144. 15. Ëåîíòüåâ,À.Ô.Îöåíêàðîñòàðåøåíèÿîäíîãîäèôôåðåíöèàëüíîãîóðàâíåíèÿïðèáîëü- øèõçíà÷åíèÿõïàðàìåòðà/À.Ô.Ëåîíòüåâ//Ñèá.Ìàò.Æóð.1960.3.Ñ.456 487. 16. Ëèäñêèé,Â.Á.Íåñàìîñîïðÿæåííûåîïåðàòîðû,èìåþùèåñëåä/Â.Á.Ëèäñêèé// ÄÀÍÑÑÑÐ.1959.Ò.1253.Ñ.485487. 17. Êàä÷åíêî,Ñ.È.×èñëåííûéìåòîäðåøåíèÿîáðàòíûõçàäà÷,ïîðîæäåííûõâîçìóùåííûì ñàìîñîïðÿæåííûìîïåðàòîðîì//ÂåñòíèêÞÓðÃÓ.Ñåðèÿ"Ìàòåìàòè÷åñêîåìîäåëèðî- âàíèåèïðîãðàììèðîâàíèå".2013.Ò6,4.Ñ.15-25. 18. Ïîëóáàðèíîâà-Êî÷èíà,Ï.ß.Òåîðèÿäâèæåíèÿãðóíòîâûõâîä/Ï.ß.Ïîëóáàðèíîâà- Êî÷èíà.Ì.:Íàóêà,1977.664ñ. 19. Ìàð÷åíêî,Â.À.Íåêîòîðûåâîïðîñûòåîðèèäèôôåðåíöèàëüíîãîîïåðàòîðàâòîðîãîïî- ðÿäêà/Â.À.Ìàð÷åíêî//ÄÀÍÑÑÑÐ.1950.Ò.72,3.C.457460. 20. Ìàð÷åíêî,Â.À.ÑïåêòðàëüíàÿòåîðèÿîïåðàòîðîâØòóðìàËèóâèëëÿ/Â.À.Ìàð÷åíêî. Êèåâ:ÍàóêîâàÄóìêà,1972.220ñ. 21. Íèæíèê,Ë.Ï.Îáðàòíûåçàäà÷èðàññåÿíèÿäëÿãèïåðáîëè÷åñêèõóðàâíåíèé/Ë.Ï.Íèæ- íèê.Êèåâ.:Íàóêîâàäóìêà,1991.232ñ. 22. Ïîâçíåð,À.ß.ÎäèôôåðåíöèàëüíûõóðàâíåíèÿõòèïàØòóðìàËèóâèëëÿíàïîëóîñè /À.ß.Ïîâçíåð//Ìàòåì.ñá.1948.23(65):1.C.352. 23. Ðèññ,Ô.Ëåêöèèïîôóíêöèîíàëüíîìóàíàëèçó/Ô.Ðèññ,Á.Ñ.-Íàäü.Ì.:Èçä-âî èíîñòð.ëèò.,1954.499ñ. 24. Ðîìàíîâ,Â.Ã.Îáðàòíûåçàäà÷èìàòåìàòè÷åñêîéôèçèêè/Â.Ã.Ðîìàíîâ.Ì.:Íàóêà, 1984.263ñ. 25. Ñàäîâíè÷èé,Â.À.Òåîðèÿîïåðàòîðîâ/Â.À.Ñàäîâíè÷èé.Ì.:Âûñøàÿøêîëà,1999. 368ñ. 26. Ñàäîâíè÷èé,Â.À.Ðåãóëÿðèçîâàííûéñëåäîïåðàòîðàñÿäåðíîéðåçîëüâåíòîé,âîçìó- ùåííîãîîãðàíè÷åííûì/Â.À.Ñàäîâíè÷èé,C.Â.Êîíÿãèí,Â.Å.Ïîäîëüñêèé//Äîêëàäû ÐÀÍ.2000.Ò.373,1.Ñ.2628. 26 27. ÑâèðèäþêÃ.À.,ÑóõàíîâàÌ.Â.Ðàçðåøèìîñòüçàäà÷èÊîøèäëÿëèíåéíûõñèíãóëÿð- íûõóðàâíåíèéýâîëþöèîííîãîòèïà//Äèôôåðåíö.óðàâíåíèÿ.1992.Ò.28,3. Ñ.323330 28. ÑåäîâÀ.È.,ÇàêèðîâàÃ.À.Îñóùåñòâîâàíèèèåäèíñòâåííîñòèðåøåíèÿîáðàòíîéçàäà÷è ñïåêòðàëüíîãîàíàëèçàäëÿñòåïåíèîïåðàòîðàËàïëàñàíàïàðàëëåëåïèïåäå//Âåñòíèê ÌàÃÓ.Ìàòåìàòèêà.2006.Âûï.9Ñ.145-49. 29. Òèõîíîâ,À.Í.Îåäèíñòâåííîñòèðåøåíèÿçàäà÷èýëåêòîðàçâåäêè/À.Í.Òèõîíîâ, À.À.Ñàìàðñêèé//ÄÀÍÑÑÑÐ.1949.Ò.69,6.Ñ.797800. 30. Òîðøèíà,Î.À.Ñîáñòâåííûå÷èñëàâîçìóùåííîãîîïåðàòîðàËàïëàñà-Áîõíåðà/Î.À. Òîðøèíà//ÂåñòíèêÍàóêàèñîâðåìåííîñòü.Ñåðèÿ:Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåè ïðîãðàììèðîâàíèå.2013.Ò.2.26. 31. Þðêî,Â.À.Ââåäåíèåâòåîðèþîáðàòíûõñïåêòðàëüíûõçàäà÷.Ì.:ÔÈÇÌÀÒËÈÒ, 2007.384ñ. 32. Ambarzumian,V.A.  U bereineFragederEigengwerttheorie/V.A.Ambarzumian //Zeits.f.Phisik.1929.53.S.690695. 33. Borg,G.EineUmkehrungderSturmLiouvilleschenEigenwertaufgabe/G.Borg//Acta Math.1946.Bd.78,1.S.190. 34. Levinson,N.TheinverseSturmLiouvilleproblem/N.Levinson//Math.Tidssk.1949. P.2530. 35. MarkK.Canoneheartheshapeofadrum?//AmericanMathematicalMonthly.1966. Vol.73.Is.4,Prt.2:PapersinAnalysis.P.76-86. 36. Ïóáëèêàöèèàâòîðàïîòåìåäèññåðòàöèè 39. Ïðèëîæåíèå1 Ñõåìààëãîðèòìàïðîãðàììû"×èñëåííîåèññëåäîâàíèåîáðàòíîéñïåêòðàëüíîéçàäà÷è äëÿîäíîéìàòåìàòè÷åñêîéìîäåëèãèäðîäèíàìèêè". 29

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