dx dt = f(t,x), t [0,T], (1) g ( x(0),x(t) ) = T, (2)

ISSN 16820525 Ì À Ò Å Ì À Ò È Ê À Ë Û Æ Ó Ð Í À Ë Ì À Ò Å Ì À Ò È Å Ñ Ê È É Æ Ó Ð Í À Ë M A T H E M A T I C A L J O U R N A L 2010 òîì 10 1 35 Èçäàåòñÿ ñ 2001 ãîäà Èíñòèòóò ìàòåìàòèêè ÌÎ è Í ÐÊ Àëìàòû

Ìèíèñòåðñòâî îáðàçîâàíèÿ è íàóêè Ðåñïóáëèêè Êàçàõñòàí ÈÍÑÒÈÒÓÒ ÌÀÒÅÌÀÒÈÊÈ ÌÀÒÅÌÀÒÈ ÅÑÊÈÉ ÆÓÐÍÀË Òîì 10 1 35 2010 Ïåðèîäè íîñòü 4 íîìåðà â ãîä Ãëàâíûé ðåäàêòîð Ì.Ò.Äæåíàëèåâ Çàìåñòèòåëè ãëàâíîãî ðåäàêòîðà: Ä.Á.Áàçàðõàíîâ Ì.È.Òëåóáåðãåíîâ Ðåäàêöèîííàÿ êîëëåãèÿ: Ë.À.Àëåêñååâà Ã.È.Áèæàíîâà Ð.Ã.Áèÿøåâ Í.Ê.Áëèåâ Â.Ã.Âîéíîâ Í.Ò.Äàíàåâ Ä.Ñ.Äæóìàáàåâ À.Ñ.Äæóìàäèëüäàåâ Ò.Ø.Êàëüìåíîâ À.Æ.Íàéìàíîâà Ì.Î.Îòåëáàåâ È.Ò.Ïàê Ì.Ã.Ïåðåòÿòüêèí Ñ.Í.Õàðèí À.Ò.Êóëàõìåòîâà îòâåòñòâåííûé ñåêðåòàðü Ø.À.Áàëãèìáàåâà òåõíè åñêèé ñåêðåòàðü Àäðåñ ðåäêîëëåãèè è ðåäàêöèè: 050010 ã.àëìàòû óë.ïóøêèíà 125 ê. 304 Òåëåôîí 8-7272-72-01-66 journal@math.kz http://www.math.kz Æóðíàë çàðåãèñòðèðîâàí â Ìèíèñòåðñòâå êóëüòóðû èíôîðìàöèè è îáùåñòâåííîãî ñîãëàñèÿ Ðåñïóáëèêè Êàçàõñòàí: Ñâèäåòåëüñòâî 1915-Æ îò 17 àïðåëÿ 2001ã. c Èíñòèòóò ìàòåìàòèêè ÌÎÍ ÐÊ 2010ã.

ÑÎÄÅÐÆÀÍÈÅ Òîì 10 1 35 2010 Èçîëèðîâàííûå îãðàíè åííûå íà âñåé îñè ðåøåíèÿ ñèñòåì íåëèíåéíûõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé À. Ä. Àáèëüäàåâà Ä.Ñ. Äæóìàáàåâ...................................................... 5 Ðàçðåøèìîñòü îáðàòíîé çàäà è ìàãíèòíîé ãèäðîäèíàìèêè äëÿ âÿçêîé íåñæèìàåìîé æèäêîñòè Ó. Ó. Àáûëêàèðîâ Ñ. Å. Àéòæàíîâ...................................................... 13 Ïðèáëèæåíèå ôóíêöèé êëàññà Áåñîâà ïîëèíîìàìè ïî îáîáùåííîé ñèñòåìå Õààðà Ã. Àêèøåâ............................................................................... 23 Äèôôåðåíöèàëüíàÿ àëãåáðà áèêâàòåðíèîíîâ. Ïðåîáðàçîâàíèÿ Ëîðåíöà áèâîëíîâûõ óðàâíåíèé Ë. À. Àëåêñååâà........................................................................... 33 Ìíîãîïåðèîäè åñêèå ðåøåíèÿ êâàçèëèíåéíûõ ãèïåðáîëè åñêèõ ñèñòåì äèôôåðåíöèàëüíûõ óðàâíåíèé â àñòíûõ ïðîèçâîäíûõ À. Ó. Áåêáàóîâà Ê. Ê. Êåíæåáàåâ Æ. À. Ñàðòàáàíîâ.................................42 Îïòèìèçàöèÿ áèëèíåéíîé áèîëîãè åñêîé ìîäåëè ñ òðåìÿ ôóíêöèÿìè óïðàâëåíèÿ ñ ó åòîì çàïàçäûâàíèÿ Ë. Õ. Æóíóñîâà...........................................................................47 Î ñõîäèìîñòè îäíîãî àëãîðèòìà íàõîæäåíèÿ ïðèáëèæåííîãî ðåøåíèÿ ïîëóïåðèîäè åñêîé êðàåâîé çàäà è äëÿ ñèñòåì íàãðóæåííûõ ãèïåðáîëè åñêèõ óðàâíåíèé Æ. Ì. Êàäèðáàåâà....................................................................... 52 Ìåòîäû ìàòåìàòè åñêîé ìîðôîëîãèè â àíàëèçå ôîòîñôåðíîãî ôîíîâîãî ìàãíèòíîãî ïîëÿ ñîëíöà Ë. Ì. Êàðèìîâà.......................................................................... 60 Ðåçóëüòàòû èñëåííîãî ìîäåëèðîâàíèÿ êîëåáàíèé ñåéñìè åñêîãî ìàÿòíèêà Í. È. Ìàðòûíîâ Ì. À. Ðàìàçàíîâà Æ. Ñ. Ñóéìåíáàåâà È. Î. Ôåäîðîâ............... 70 Î ñõîäèìîñòè îäíîãî àëãîðèòìà ìåòîäà ïàðàìåòðèçàöèè Ñ. Ì. Òåìåøåâà........................................................................... 83 Î ðàçðåøèìîñòè íåêîòîðûõ çàäà äëÿ óðàâíåíèÿ Ëàïëàñà Á. Ò. Òîðåáåê Á. Õ. Òóðìåòîâ........................................................... 93 Î ñóììèðîâàíèè êîýôôèöèåíòîâ Ôóðüå Ë. Ï. Ôàëàëååâ.......................................................................... 104

Ìàòåìàòè åñêèé æóðíàë. Àëìàòû. 2010. Òîì 10. 1 35. C. 8392 ÓÄÊ 519.624 Î ÑÕÎÄÈÌÎÑÒÈ ÎÄÍÎÃÎ ÀËÃÎÐÈÒÌÀ ÌÅÒÎÄÀ ÏÀÐÀÌÅÒÐÈÇÀÖÈÈ Ñ. Ì. Òåìåøåâà Èíñòèòóò Ìàòåìàòèêè ÌÎèÍ ÐÊ 050010 Àëìàòû Ïóøêèíà 125 anar@math.kz dzhumabaev@list.ru Äëÿ ðåøåíèÿ íåëèíåéíîé äâóõòî å íîé êðàåâîé çàäà è äëÿ ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ïðåäëàãàåòñÿ äâóõïàðàìåòðè åñêîå ñåìåéñòâî àëãîðèòìîâ ìåòîäà ïàðàìåòðèçàöèè.  òåðìèíàõ ôóíêöèé ïðàâîé àñòè äèôôåðåíöèàëüíîãî óðàâíåíèÿ è ãðàíè íîãî óñëîâèÿ óñòàíîâëåíû äîñòàòî íûå óñëîâèÿ ñõîäèìîñòè àëãîðèòìîâ è ñóùåñòâîâàíèÿ èçîëèðîâàííîãî ðåøåíèÿ ðàññìàòðèâàåìîé çàäà è.  ðàáîòå ðàññìàòðèâàåòñÿ íåëèíåéíàÿ äâóõòî å íàÿ êðàåâàÿ çàäà à ãäå f : [0 T] R n R n g : R n R n R n íåïðåðûâíû. dx dt = ftx t [0T] 1 g x0xt = T 2 Ðåøåíèåì çàäà è 1 2 ÿâëÿåòñÿ íåïðåðûâíî äèôôåðåíöèðóåìàÿ íà [0 T] âåêòîð-ôóíêöèÿ x t óäîâëåòâîðÿþùàÿ íà [0T] äèôôåðåíöèàëüíîìó óðàâíåíèþ 1 ïðè ýòîì â òî êàõ t = 0 t = T óðàâíåíèþ 1 óäîâëåòâîðÿþò îäíîñòîðîííèå ïðîèçâîäíûå ẋ ïðàâ.0 ẋ ëåâ.t è èìåþùàÿ â òî êàõ t = 0 t = T çíà åíèÿ x 0 x T äëÿ êîòîðûõ ñïðàâåäëèâî ðàâåíñòâî 2. Êðàåâûå çàäà è äëÿ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ðàçëè íûìè ìåòîäàìè èññëåäîâàíû ìíîãèìè àâòîðàìè [111].  ñòàòüå [12] çàäà à 1 2 èññëåäîâàëàñü ìåòîäîì ïàðàìåòðèçàöèè. Ïðåäëîæåíû àëãîðèòìû íàõîæäåíèÿ ðåøåíèÿ çàäà è 1 2 è óñòàíîâëåíû äîñòàòî íûå óñëîâèÿ èõ ñõîäèìîñòè. Êàæäûé øàã àëãîðèòìà ñîñòîèò èç äâóõ ïóíêòîâ: a ðåøåíèå ñèñòåìû íåëèíåéíûõ óðàâíåíèé îòíîñèòåëüíî ââåäåííûõ ïàðàìåòðîâ b ðåøåíèå çàäà è Êîøè ïðè íàéäåííûõ çíà åíèÿõ ïàðàìåòðîâ.  íàñòîÿùåé ðàáîòå ïðåäëàãàåòñÿ äâóõïàðàìåòðè åñêîå ñåìåéñòâî àëãîðèòìîâ ðåøåíèÿ çàäà è 1 2 ãäå â ïóíêòå b íåò íåîáõîäèìîñòè â ðåøåíèè çàäà è Êîøè. Keywords: nonlinear two-point boundary value problem parametrization method sucient conditions for the existence of an isolated solution 2000 Mathematics Subject Classication: 34A45 c Ñ. Ì. Òåìåøåâà 2010.

84 Ñ. Ì. Òåìåøåâà Âîçüìåì íàòóðàëüíîå èñëî N N è ðàçîáüåì ïðîìåæóòîê [0 T] ñ øàãîì h = T/N : [0T = N [r 1h rh. Ñóæåíèå ôóíêöèè xt íà èíòåðâàë [r 1h rh îáîçíà èì åðåç r=1 x r t è çàäà ó 1 2 ñâåäåì ê ìíîãîòî å íîé êðàåâîé çàäà å: dx r dt = ftx r t [r 1hrh r = 1 : N 3 g x 1 0 lim Nt t Nh 0 = 0 4 lim t sh 0 x st = x s+1 sh s = 1 : N 1 5 ãäå 5 óñëîâèÿ ñêëåèâàíèÿ ðåøåíèÿ âî âíóòðåííèõ òî êàõ ðàçáèåíèÿ èíòåðâàëà [0 T]. Îòìåòèì òî óñëîâèÿ ñêëåèâàíèÿ ðåøåíèÿ 5 è äèôôåðåíöèàëüíûå óðàâíåíèÿ 3 îáåñïå èâàþò òàê æå è íåïðåðûâíîñòü ïðîèçâîäíûõ â òî êàõ ðàçáèåíèÿ èíòåðâàëà [0 T]. åðåç C[0T]h R nn îáîçíà èì ïðîñòðàíñòâî ñèñòåì ôóíêöèé x[t] = x 1 tx 2 t... x N t ãäå x r : [r 1hrh R n íåïðåðûâíà è èìååò êîíå íûé ëåâîñòîðîííèé ïðåäåë lim x rt ïðè âñåõ r = 1 : N ñ íîðìîé x[ ] 2 = max t rh 0 r=1:n sup t [rh x r t. Ââåäåì îáîçíà åíèå λ r =x r ïðîèçâåäåì çàìåíó u r t = x r t λ r t [rh r = 1 : N è îò çàäà è 3-5 ïåðåéäåì ê ýêâèâàëåíòíîé ìíîãîòî å íîé êðàåâîé çàäà å ñ ïàðàìåòðàìè: du r dt = ftλ r + u r t [r 1hrh r = 1 : N 6 u r r 1h = 0 r = 1 : N 7 g λ 1 λ N + lim Nt t T 0 = 0 8 λ s + lim t sh 0 u st λ s+1 = 0 s = 1 : N 1. 9 Ðåøåíèåì çàäà è 69 ÿâëÿåòñÿ ïàðà λ u [t] ñ ýëåìåíòàìè λ = λ 1 λ 2 N...λ R nn u [t] = u 1 tu 2 t...u N t C[0T]h R nn ãäå íåïðåðûâíî äèôôåðåíöèðóåìàÿ è îãðàíè åííàÿ íà [rh ôóíêöèÿ u rt óäîâëåòâîðÿåò äèôôåðåíöèàëüíîìó óðàâíåíèþ 6 ïðè âñåõ t [r 1hrh ïðè t = r 1h óðàâíåíèþ 6 óäîâëåòâîðÿåò ïðàâîñòîðîííÿÿ ïðîèçâîäíàÿ ôóíêöèè u rt âûïîëíÿåòñÿ óñëîâèå u rr 1h = 0 r = 1 : N è äëÿ λ 1 λ N + lim t Nh 0 u N t λ s + lim t sh 0 u st λ s+1 s = 1 : N 1 èìåþò ìåñòî ðàâåíñòâà 8 9. Åñëè λ u [t] ðåøåíèå çàäà è 69 òî x t = { λ r + u rt ïðè t [r 1hrh r = 1 : N λ N + lim t T 0 u N t ïðè t = T áóäåò ðåøåíèåì çàäà è 1 2. Ïóñòü òåïåðü xt ðåøåíèå çàäà è 1 2. Tîãäà ïàðà λ ũ[t] ñ ýëåìåíòàìè λ = λ1 λ 2... λ N R nn ũ[t] = ũ 1 t ũ 2 t...ũ N t ãäå λ r = x r 1h ũ r t = xt x r 1h t [r 1hrh r = 1 : N áóäåò ðåøåíèåì çàäà è 6-9. Ïðè ôèêñèðîâàííîì çíà åíèè λ r çàäà à Êîøè 6 7 äëÿ âñåõ r = 1 : N ýêâèâàëåíòíà èíòåãðàëüíîìó óðàâíåíèþ Âîëüòåððà âòîðîãî ðîäà: u r t = fτλ r + u r τdτ t [r 1hrh. 10 Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

Î ñõîäèìîñòè îäíîãî àëãîðèòìà ìåòîäà ïàðàìåòðèçàöèè 85 Ïîäñòàâèâ âìåñòî u r τ ñîîòâåòñòâóþùóþ ïðàâóþ àñòü 10 è ïîâòîðÿÿ ýòîò ïðîöåññ ν ðàç äëÿ u r t ïîëó èì ñëåäóþùåå ïðåäñòàâëåíèå: u r t = f τ1 τ 1 λ r + f τν 1 τ 2 λ r +... + f τ ν λ r + u r τ ν dτ ν... dτ 2 dτ1 t [r 1hrh r = 1 : N. 11 Îïðåäåëèâ èç 11 lim u rt r = 1 : N ïîäñòàâèâ èõ â 8 9 ïðåäâàðèòåëüíî óìíîæàÿ t rh 0 8 íà h = T/N > 0 ïîëó èì ñèñòåìó íåëèíåéíûõ óðàâíåíèé îòíîñèòåëüíî λ r R n : Nh h g λ 1 λ N + f τν 1 τ 1 λ N +... + f τ ν λ N + u N τ ν dτ ν... dτ 1 = 0 N 1h N 1h sh λ s + f τν 1 τ 1 λ s +... + f τ ν λ s + u s τ ν dτ ν... dτ 1 λ s+1 = 0 s = 1 : N 1 s 1h s 1h êîòîðóþ çàïèøåì â âèäå Q νh λ u = 0 λ R nn. 12 Óñëîâèå A. Ñóùåñòâóþò N N h = T/N ν N òàêèå òî ñèñòåìà íåëèíåéíûõ óðàâíåíèé Q νh λ 0 = 0 èìååò ðåøåíèå λ 0 = λ 0 1 λ0 2...λ0 N R nn è ñèñòåìà ôóíêöèé u 0 [t] = u 0 1 t u0 2 t...u0 N t ñ êîìïîíåíòàìè u 0 r t = f τ1 τ 1 λ 0 r + f τν 1 τ 2 λ 0 r +... + ïðèíàäëåæèò ïðîñòðàíñòâó C[0 T]h R nn. Âîçüìåì èñëà ρ λ > 0 ρ u > 0 ρ x > 0 è ñîñòàâèì ìíîæåñòâà: f τ ν λ 0 r dτν... dτ 2 dτ1 t [r 1hrh r = 1 : N 13 S λ 0 ρ λ = { λ = λ1 λ 2...λ N R nn : λ λ 0 = max r=1:n λ r λ 0 r < ρ λ } S u 0 [t]ρ u = { u[t] C[0 T]h R nn : u[ ] u 0 [ ] 2 < ρ u } Ω s λ 0 s u 0 s th ρ x = { tx : t [s 1hsh x λ 0 s u 0 } s t < ρ x s = 1 : N 1 Ω N λ 0 N u0 N th ρ x = { tx : t [N 1hNh] x λ 0 N u0 N t < ρ x} G 0 1ρ x = N 1 s=1 Ω s λ 0 s u 0 s th ρ x Ω N λ 0 N u0 N th ρ x G 0 2ρ λ ρ x = { v w R 2n : v λ 0 1 < ρ λ w λ 0 N lim t T 0 u0 N t < ρ x}. Óñëîâèå B. Ôóíêöèè f g ñîîòâåòñòâåííî â G 0 1 ρ x G 0 2 ρ λρ x íåïðåðûâíû èìåþò ðàâíîìåðíî íåïðåðûâíûå àñòíûå ïðîèçâîäíûå f x g v g w è âûïîëíÿþòñÿ íåðàâåíñòâà: f xtx L 0 g vv w L 1 g wv w L 2 ãäå L 0 L 1 L 2 ïîñòîÿííûå. Ïðåäïîëîæèì òî âûïîëíåíî óñëîâèå A. Tîãäà çà íà àëüíîå ïðèáëèæåíèå ðåøåíèÿ çàäà è 69 âîçüìåì ïàðó λ 0 u 0 [t] è ïîñëåäîâàòåëüíîñòü λ k u k [t] k = 1 2... íàéäåì ïî ñëåäóþùåìó àëãîðèòìó: Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

86 Ñ. Ì. Òåìåøåâà Øàã 1. à Èç óðàâíåíèÿ Q νh λu 0 = 0 íàéäåì λ 1 = λ 1 1 λ1 2...λ1 N R nn ; á íà [r 1h rh r = 1 : N îïðåäåëèì ôóíêöèè u 1 r t ïî ôîðìóëàì: u 1 r t = f τν 1 τ 1 λ 1 r +... + f τ ν λ 1 r + u 0 r τ ν dτ ν... dτ 1. 14 Øàã 2. à Èç óðàâíåíèÿ Q νh λu 1 = 0 íàéäåì λ 2 = λ 2 1 λ2 2...λ2 N R nn ; á íà [r 1h rh r = 1 : N îïðåäåëèì ôóíêöèè u 2 r t ïî ôîðìóëàì: u 2 r t = f τν 1 τ 1 λ 2 r +... + f τ ν λ 2 r + u 1 r τ ν dτ ν... dτ 1. 15 Ïðîäîëæàÿ ïðîöåññ íà k -îì øàãå íàéäåì ïàðó λ k u k [t] k = 0 1 2... óäîâëåòâîðÿþùóþ ðàâåíñòâàì Q νh λ k u k 1 = 0 è u r k t = f τν 1 τ 1 λ r k +... + f τ ν λ k r + u k 1 r τ ν dτ ν... dτ 1. 16 Îòëè èå ïðåäëàãàåìûõ àëãîðèòìîâ îò àëãîðèòìîâ [12] çàêëþ àåòñÿ â ïóíêòå b ãäå ôóíêöèè u k r t îïðåäåëÿþòñÿ ïî ôîðìóëå 16. Äîñòàòî íûå óñëîâèÿ îñóùåñòâèìîñòè ñõîäèìîñòè àëãîðèòìà îäíîâðåìåííî îáåñïå èâàþùèå ñóùåñòâîâàíèå èçîëèðîâàííîãî ðåøåíèÿ ìíîãîòî å íîé êðàåâîé çàäà è ñ ïàðàìåòðàìè 69 óñòàíàâëèâàåò ñëåäóþùàÿ Tåîðåìà 1. Ïóñòü ñóùåñòâóþò ν N N N h = T/N ρ λ > 0 ρ u > 0 ρ x > 0 ïðè êîòîðûõ âûïîëíÿþòñÿ óñëîâèÿ A B ìàòðèöà ßêîáè Q νhλ u λ u[t] Sλ 0 ρ λ Su 0 [t]ρ u è èìåþò ìåñòî íåðàâåíñòâà: 1 Qνh λ u 1 γν h 2 q ν h = hl 0 ν ν hl 0 j 1 + γ ν hmax1hl 2 < 1 3 γ ν h Q νh λ 0 u 0 + q νh 1 q ν h γ νhmax1hl 2 hl 0 ν u 0 [ ] 2 < ρ λ q ν h 4 1 q ν h u0 [ ] 2 < ρ u ν 1 hl 0 5 j { hl0 j } ρ λ + ρ u max + b ν < ρ x j=0:ν 1 j=0 0 ïðè ν = 1 ãäå t 2ν 2 b ν = max sup fτλ 0 hl 0 j r dτ ïðè ν 2. r=1:n t [rh j=1 j=ν îáðàòèìà äëÿ âñåõ Tîãäà îïðåäåëÿåìàÿ ïî àëãîðèòìó ïîñëåäîâàòåëüíîñòü ïàð λ k u k [t] k = 0 1 2... ïðèíàäëåæèò Sλ 0 ρ λ Su 0 [t]ρ u ñõîäèòñÿ ê λ u [t] ðåøåíèþ çàäà è 69 â Sλ 0 ρ λ Su 0 [t]ρ u è ñïðàâåäëèâû îöåíêè: u [ ] u k [ ] 2 q νh k 1 q ν h q νh u 0 [ ] 2 k = 12... 17 λ λ k q νh k 1 q ν h γ νhmax1hl 2 hl 0 ν u 0 [ ] 2 k = 1 2... 18 Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

Î ñõîäèìîñòè îäíîãî àëãîðèòìà ìåòîäà ïàðàìåòðèçàöèè 87 Ïðè åì ëþáîå ðåøåíèå çàäà è 69 â Sλ 0 ρ λ Su 0 [t]ρ u èçîëèðîâàíî. Äîêàçàòåëüñòâî. Âîçüìåì ëþáóþ ïàðó λu[t] Sλ 0 ρ λ Su 0 [t]ρ u. Â ñèëó óñëîâèÿ B èìåþò ìåñòî íåðàâåíñòâà: λ r + 1+ τν + ρ λ λ r + u r t λ 0 r u 0 r t λ r λ 0 r + u r t u 0 r t < ρ λ + ρ u 19 f τ 1 λ r +u r τ 1 dτ 1 λ 0 r u 0 L 0 dτ f τ ν+1 λ 0 r f τ1 τ 1 λ 0 r + λ 1 r λ 0 r + r t λ r λ 0 r + f τν 1 τ 2 λ 0 r +... + dτν+1... dτ 2 τ1 L 0 u r τ u 0 r τ dτ+ f τ 2 λ 0 +... + 1 L 0 t r 1h j + ρ u L 0 t r 1h + sup j=0 f τ ν λ r +u r τ 1 dτ 1 f τ ν λ 0 r dτν... dτ 2 dτ1 τν 1 t [rh L 0 τ1 f τ 2 λ 0 r +...+ fτ ν λ 0 r dτ ν... dτ 2 dτ1 fτλ 0 r dτ L 0t r 1h ν t [r 1hrh r = 1 : N 20 λ r + f τ1 τ 1 λ r + f τ 2 λ r + u r τ 2 dτ 2 dτ1 λ 0 r u 0 r t 2 L 0 t r 1h j L 0 t r 1h 2 ρ λ + ρ u + sup 2! j=0 ν+1 j=ν t [rh fτλ 0 r dτ L 0 t r 1h j t [r 1hrh r = 1 : N 21 λ r + f τν 2 τ 1 λ r +... + f τ ν 1 λ r + u r τ ν 1 dτ ν 1... dτ 1 λ 0 r u 0 r t ν 1 ρ λ j=0 L 0 t r 1h j... + ρ u L 0 t r 1h ν 1 ν 1! 2ν 2 j=ν + sup t [rh fτ λ 0 r dτ L 0 t r 1h j t [r 1hrh r = 1 : N. 22 Ââèäó 1922 íåðàâåíñòâà 5 òåîðåìû äëÿ âñåõ t [r 1hrh è r = 1 : N ïàðû t λr + u r t t λr + t λr + tλr + fτ 1 λ r + u r τ 1 dτ 1 f τ 1 λ r + τ1 f τ 1 λ r +... + fτ 2 λ r + u r τ 2 dτ 2 dτ1... τν 2 fτ ν 1 λ r + u r τ ν 1 dτ ν 1... dτ 1 Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

88 Ñ. Ì. Òåìåøåâà ïðèíàäëåæàò ìíîæåñòâó G 0 1 ρ x. Ðåøåíèå çàäà è 69 íàéäåì ïî àëãîðèòìó. Ñîãëàñíî óñëîâèÿì òåîðåìû îïåðàòîð Q νh λu 0 â Sλ 0 ρ λ óäîâëåòâîðÿåò âñåì ïðåäïîëîæåíèÿì òåîðåìû 1 èç [12]. Âîçüìåì íåêîòîðîå èñëî ε 0 > 0 ïðè êîòîðîì áóäóò èìåòü ìåñòî íåðàâåíñòâà: ε 0 γ ν h 1 2 γ ν h Q νh λ 0 u 0 + q νh 1 ε 0 γ ν h 1 q ν h max1hl 2 hl 0 ν u 0 [ ] 2 < ρ λ. Q νh λu 0 / Sλ 0 ρ λ δ 0 0ρ λ ] Q νhλu 0 Q νh λu 0 < ε 0 λ λ Sλ 0 ρ λ λ λ < δ 0. { α α 0 = max 1 γ νh Q νh λ 0 u 0 + q νh δ 0 1 q ν h max1hl 2 hl 0 ν } u 0 [ ] 2 λ 10 = λ 0 λ 1m+1 = λ 1m 1 α Qνh λ 1m u 0 1Qνh λ 1m u 0 m = 0 1 2... λ 1 Sλ 0 ρ λ Q νh λu 0 = 0 λ 1 λ 0 γ ν h Q νh λ 0 u 0. u 1 [t] u 1 r t u 0 r t t [r 1hrh r = 1 : N u 1 r t u 0 r t ν hl 0 j j=1 f τ 1 λ 1 r +... + f τ 1 λ 0 r +... + τν 1 τν 1 λ 1 λ 0 + max r=1:n f τ ν λ 1 r + u 0 r τ ν dτ ν... dτ 1 f τ ν λ 0 r dτν... dτ 1 sup u 0 r t hl 0 ν t [rh ν hl 0 j γ ν h Q νh λ 0 u 0 + hl 0 ν u 0 [ ] 2 t [r 1hrh r = 1 : N. j=1 λ 0 R nn A Q νh λ 0 0 = 0 Q νh λ 0 u 0 = Q νh λ 0 u 0 Q νh λ 0 0 max1hl 2 hl 0 ν u 0 [ ] 2. u 1 [ ] u 0 [ ] 2 q ν h u 0 [ ] 2. u 1 [t] Su 0 [t]ρ u. Q νh λu Q νh λ 1 u 0 = 0 γ ν h Q νh λ 1 u 1 = γ ν h Q νh λ 1 u 1 Q νh λ 1 u 0 γ ν hmax1hl 2 hl 0 ν u 1 [ ] u 0 [ ] 2. Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

Î ñõîäèìîñòè îäíîãî àëãîðèòìà ìåòîäà ïàðàìåòðèçàöèè 89 Åñëè λ Sλ 1 ρ 1 +ε 1 ãäå ρ 1 = γ ν hmax1hl 2 hl 0 ν u 1 [ ] u 0 [ ] 2 è èñëî ε 1 > 0 óäîâëåòâîðÿåò íåðàâåíñòâó: ε 1 + γ ν h Q νh λ 0 u 0 + q νh 1 q ν h γ νhmax1 hl 2 hl 0 ν u 0 [ ] 2 < ρ λ òî â ñèëó óñëîâèé 2 3 òåîðåìû è ñîîòíîøåíèé 23 26 ïîëó èì òî λ λ 0 λ λ 1 + λ 1 λ 0 < ρ 1 + ε 1 + λ 1 λ 0 γ ν h max1 hl 2 hl 0 ν u 1 [ ] u 0 [ ] 2 + ε 1 + γ ν h Q νh λ 0 u 0 < < q νh 1 q ν h γ νhmax1hl 2 hl 0 ν u 0 [ ] 2 + ε 1 + γ ν h Q νh λ 0 u 0 < ρ λ ñëåäîâàòåëüíî Sλ 1 ρ 1 + ε 1 Sλ 0 ρ λ. Îïåðàòîð Q νh λ u 1 â ìíîæåñòâå Sλ 1 ρ 1 + ε 1 óäîâëåòâîðÿåò âñåì óñëîâèÿì òåîðåìû 1 [12] ïîýòîìó èòåðàöèîííûé ïðîöåññ λ 20 = λ 1 λ 2m+1 = λ 2m 1 α Qνh λ 2m u 1 1 Qνh λ 2m u 1 m = 0 1 2... ñõîäèòñÿ ê èçîëèðîâàííîìó ðåøåíèþ λ 2 Sλ 1 ρ 1 +ε 1 óðàâíåíèÿ Q νh λu 1 = 0 è èìååò ìåñòî îöåíêà: èëè ó èòûâàÿ 27 ïîëó èì òî Ïî λ 2 = λ 2 1 λ2 2...λ2 ñèñòåìû ôóíêöèé u 2 [t]. Tîãäà λ 2 λ 1 γ ν h Q νh λ 1 u 1 λ 2 λ 1 γ ν hmax1hl 2 hl 0 ν u 1 [ ] u 0 [ ] 2. N R nn èñïîëüçóÿ ôîðìóëû 15 îïðåäåëèì êîìïîíåíòû u 2 r t u 1 r t q ν h max sup u 1 r t u 0 r t t [r 1hrh r = 1 : N r=1:n t [rh ò.å. u 2 [ ] u 1 [ ] 2 q ν h u 1 [ ] u 0 [ ] 2. Ïðåäïîëàãàÿ òî ïàðà λ k 1 u k 1 [t] Sλ 0 ρ λ Su 0 [t] ρ u íàéäåíà è óñòàíîâëåíû îöåíêè: λ k 1 λ k 2 γ ν hmax1hl 2 hl 0 ν u k 2 [ ] u k 3 [ ] 2 28 u k 1 [ ] u k 2 [ ] 2 q ν h u k 2 [ ] u k 3 [ ] 2 29 Q νh λ k 2 u k 2 max1hl 2 hl 0 ν u k 2 [ ] u k 3 [ ] 2 30 k -îå ïðèáëèæåíèå λ k ïî ïàðàìåòðó íàõîäèì èç óðàâíåíèÿ: Q νh λu k 1 = 0 λ R nn. 31 Èñïîëüçóÿ ðàâåíñòâî Q νh λ k 1 u k 2 = 0 óñòàíàâëèâàåì ñïðàâåäëèâîñòü íåðàâåíñòâà: γ ν h Q νh λ k 1 u k 1 γ ν hmax1hl 2 hl 0 ν u k 1 [ ] u k 2 [ ] 2. 32 Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

90 Ñ. Ì. Òåìåøåâà Âîçüìåì ρ k 1 = γ ν h Q νh λ k 1 u k 1 è ïîêàæåì òî Sλ k 1 ρ k 1 + ε 1 Sλ 0 ρ λ. Äëÿ ëþáîãî λ Sλ k 1 ρ k 1 + ε 1 â ñèëó 26 28 2932 âûïîëíÿåòñÿ ñëåäóþùåå: λ λ 0 λ λ k 1 + λ k 1 λ k 2 +... + λ 2 λ 1 + λ 1 λ 0 ρ k 1 +ε 1 +γ ν hmax1hl 2 hl 0 ν u k 2 [ ] u k 3 [ ] 2 +...+ u 1 [ ] u 0 [ ] 2 + λ 1 λ 0 γ ν hmax1hl 2 hl 0 ν q ν h k 1 +q ν h k 2 +...+q ν h u 0 [ ] 2 +γ ν h Q νh λ 0 u 0 + + ε 1 < q νh 1 q ν h γ νhmax1hl 2 hl 0 ν u 0 [ ] 2 + γ ν h Q νh λ 0 u 0 + ε 1 < ρ λ ò.å. Sλ k 1 ρ k 1 + ε 1 Sλ 0 ρ λ. Tàê êàê îïåðàòîð Q νh λu k 1 â Sλ k 1 ρ k 1 + ε 1 óäîâëåòâîðÿåò âñåì óñëîâèÿì òåîðåìû 1 èç [12] òî â Sλ k 1 ρ k 1 +ε 1 ñóùåñòâóåò ðåøåíèå λ k óðàâíåíèÿ 31 è ñïðàâåäëèâà îöåíêà: λ k λ k 1 γ ν h Q νh λ k 1 u k 1. 33 Ïî ôîðìóëàì 16 îïðåäåëèì êîìïîíåíòû ñèñòåìû ôóíêöèé u k [t] è óñòàíîâèì îöåíêè: u k [ ] u k 1 [ ] 2 q ν h u k 1 [ ] u k 2 [ ] 2. 34 Åñëè ρ k = γ ν h Q νh λ k u k = 0 òî Q νh λ k u k = 0 ò.å. h g λ k 1 λk N + Nh N 1h f τ 1 λ k N +... + τν 1 N 1h f τ ν λ k N + uk N τ ν dτ ν... dτ 1 = 0 sh λ k s + f τν 1 τ 1 λ k s +...+ s 1h s 1h è ïàðà λ k u k [t] ÿâëÿåòñÿ ðåøåíèåì çàäà è 69. Ñîãëàñíî 32 33 óñòàíîâèì òî f τ ν λ k s +u k s τ ν dτ ν... dτ 1 λ k s+1 = 0 s = 1 : N 1 λ k λ k 1 γ ν hmax1hl 2 hl 0 ν u k 1 [ ] u k 2 [ ] 2. 35 Èç 34 35 è óñëîâèÿ 2 òåîðåìû ñëåäóåò òî ïîñëåäîâàòåëüíîñòü ïàð λ k u k [t] ïðè k ñõîäèòñÿ ê ðåøåíèþ λ u [t] çàäà è 69 ïðè åì â ñèëó íåðàâåíñòâ 4 5 òåîðåìû ïàðû λ k u k [t] k = 1 2... è λ u [t] ïðèíàäëåæàò Sλ 0 ρ λ Su 0 [t]ρ u. Ââèäó 34 35 ëåãêî ïîêàçàòü òî ïðè âñåõ k = 12... è p N u k+p [ ] u k [ ] 2 q νh k 1 q ν h q νh u 0 [ ] 2 λ k+p λ k 2 q νh k 1 q ν h γ νhmax1hl 2 hl 0 ν u 0 [ ] 2. Ïåðåõîäÿ ê ïðåäåëó ïðè p p N â ïîñëåäíèõ íåðàâåíñòâàõ ïîëó èì îöåíêè 17 è 18 òåîðåìû. Ïîêàæåì èçîëèðîâàííîñòü ðåøåíèÿ. Ïóñòü ïàðà λ ũ[t] ðåøåíèå çàäà è 69 ïðèíàäëåæàùåå Sλ 0 ρ λ Su 0 [t]ρ u. Tîãäà ñóùåñòâóþò èñëà δ 1 > 0 δ2 > 0 òàêèå òî Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

Î ñõîäèìîñòè îäíîãî àëãîðèòìà ìåòîäà ïàðàìåòðèçàöèè 91 λ λ 0 + δ 1 < ρ λ ũ[ ] u 0 [ ] 2 + δ 2 < ρ u. Åñëè λ S λ δ 1 u[t] Sũ[t] δ 2 òî â ñèëó íåðàâåíñòâ λ λ 0 λ λ + λ λ 0 δ 1 + λ λ 0 < ρ λ u[ ] u 0 [ ] 2 u[ ] ũ[ ] 2 + ũ[ ] u 0 [ ] 2 δ 2 + ũ[ ] u 0 [ ] 2 < ρ u λ S λ δ 1 u[t] Sũ[t] δ 2 ò.å. S λ δ 1 Sλ 0 ρ λ Sũ[t] δ 2 Su 0 [t]ρ u. Âîçüìåì èñëî ε > 0 òàêîå òî εγ ν h < 1 γ ν h ν 1 εγ ν h max1hl hl 0 j 2 j=1 hl0 ν + 1 âûòåêàåò åå ðàâíîìåðíàÿ íåïðåðûâ- Èç óñëîâèÿ B Q νhλu è ñòðóêòóðû ìàòðèöû ßêîáè íîñòü â S λ δ 1 Sũ[t] δ 2. Ïîýòîìó ñóùåñòâóåò δ 0min{ δ 1 δ 2 }] ïðè êîòîðîì Q νhλu Q νh λ ũ < 1. 36 < ε äëÿ âñåõ λ u S λδ Sũ[t]δ. Çàìåòèì òî åñëè λũ[t] ðåøåíèå çàäà è 6-9 òî Q νh λũ = 0 ïðè ëþáîì ν N. Ïóñòü ˆλû[t] S λδ Sũ[t]δ äðóãîå ðåøåíèå çàäà è 69. Tàê êàê Q νh λũ = 0 è Q νh ˆλû = 0 òî èç ðàâåíñòâ ñëåäóåò òî λ = λ Qνh λ ũ 1Qνh Qνh λ ũ 1Qνh λũ ˆλ = ˆλ ˆλ û λ ˆλ Qνh λũ 1 1 Qνh ˆλ + θ λ = ˆλ ũ Q νh λ ũ dθ λ 0 ˆλ Qνh λũ 1Qνh ˆλ ũ Q νh ˆλû îòêóäà λ ˆλ γ νh 1 εγ ν h Q νhˆλũ Q νh ˆλ û γ νh 1 εγ ν h maxl 2h1 hl 0 ν ũ[ ] û[ ] 2. 37 Tàê êàê ôóíêöèè ũ r t û r t ÿâëÿþòñÿ ðåøåíèÿìè çàäà è Êîøè 6 7 ñîîòâåòñòâåííî ïðè λ r = λ r è λ r = ˆλ r òî íà [r 1hrh äëÿ âñåõ r = 1 : N : ũ r t û r t = f τ 1 λ τ1 r + f τ 2 λ τν 1 r +...+ f τ ν λ r +ũ r τ ν dτ ν... dτ 2 dτ1 f τ 1 ˆλ τ1 r + f τ 2 ˆλ τν 1 r +... + f τ ν ˆλ r + û r τ ν dτ ν... dτ 2 dτ1 îòêóäà ν L 0 t r 1h j λ r ˆλ r + j=1 ũ[ ] û[ ] 2 sup t [rh ũ r t û r t L 0t r 1h ν ν hl 0 j λ ˆλ hl 0 ν + ũ[ ] û[ ] 2 j=1 Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35. 38

92 Ñ. Ì. Òåìåøåâà Â ñèëó 37 èç 38 ñëåäóåò òî γ ν h ν ũ[ ] û[ ] 2 1 εγ ν h max1 hl hl 0 j 2 j=1 hl0 ν + 1 ũ[ ] û[ ] 2. ũ[t] = û[t] λ = ˆλ x k t k = 01 2... x k t = λ k r + u k r t t [r 1hrh r = 1 : N x k T = λ k N + lim t T 0 uk N t Sx 0 tρ x x : [0T] R n xt x 0 t < ρ x t [rh r = 1 : N xt x 0 T < ρ x. Tåîðåìà 2. Ïóñòü ñóùåñòâóþò h > 0 : Nh = T ρ λ > 0 ρ u > 0 ρ x > 0 ïðè êîòîðûõ âûïîëíÿþòñÿ óñëîâèÿ A B ìàòðèöà ßêîáè Q 1hλu îáðàòèìà äëÿ âñåõ λ u[t] S λ 0 ρ λ S u 0 [t]ρ u è ñïðàâåäëèâû íåðàâåíñòâà 15 òåîðåìû 1. Tîãäà ïîñëåäîâàòåëüíîñòü ôóíêöèé x k t k = 12... ñîäåðæèòñÿ â Sx 0 tρ x ñõîäèòñÿ ê x t Sx 0 tρ x ðåøåíèþ çàäà è è ñïðàâåäëèâî íåðàâåíñòâî x t x k t q νh k 1 q ν h q ν h + γ ν hmax1hl 2 hl 0 ν Ïðè åì ëþáîå ðåøåíèå çàäà è â Sx 0 tρ x èçîëèðîâàíî. u 0 [ ] 2 t [0T]. Öèòèðîâàííàÿ ëèòåðàòóðà 1. Øàìàíñêèé T.Å. Ìåòîäû èñëåííîãî ðåøåíèÿ êðàåâûõ çàäà íà ÝÖÂÌ. Êèåâ 1963..1. 2. Áåëëìàí Ð. Êàëàáà Ð. Êâàçèëèíåàðèçàöèÿ è íåëèíåéíûå êðàåâûå çàäà è. Ì. 1968. 3. Keller H.B. Numerical methods for two-point boundary-value problems. Blaisdell 1968. 4. Roberts S.M. Shipman J.S. Two-point boundary-value problems: Shooting methods. N.Y.: Elsevier 1972. 5. Áàõâàëîâ Í.Ñ. èñëåííûå ìåòîäû. Ì. 1973. 6. Keller H.B. White A.B. //SIAM J. Numer. Anal. 1975. Vol.12 No 5. P. 791802. 7. Ñîâðåìåííûå èñëåííûå ìåòîäû ðåøåíèÿ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé. Ïîä ðåä. Õîëëà Äæ. Óàòòà Äæ. Ì. 1979. 8. Áàáåíêî Ê.È. Îñíîâû èñëåííîãî àíàëèçà. Ì. 1986. 9. Êèãóðàäçå È.T. Êðàåâûå çàäà è äëÿ ñèñòåì îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé. "Ñîâðåìåííûå ïðîáëåìû ìàòåìàòèêè. Íîâåéøèå äîñòèæåíèÿ. T. 30. Èòîãè íàóêè è òåõí. ÂÈÍÈÒÈ ÀÍ ÑÑÑÐ". Ì. 1987. 10. Ñàìîéëåíêî À.Ì. Ðîíòî Í.È. èñëåííî-àíàëèòè åñêèå ìåòîäû èññëåäîâàíèÿ ðåøåíèé êðàåâûõ çàäà. Êèåâ 1986. 11. Äæóìàáàåâ Ä.Ñ. //Æ. âû èñë. ìàòåì. è ìàòåì. ôèç. 1989. T. 29 1. Ñ. 5066. 12. Äæóìàáàåâ Ä.Ñ. Tåìåøåâà Ñ.Ì. //Æ. âû èñë. ìàòåì. è ìàòåì. ôèç. 2007. T. 47 1. Ñ. 3963. Ïîñòóïèëà â ðåäàêöèþ 12.02.2010ã. Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

114 Abstracts Ìàðòûíîâ Í.È. Ðàìàçàíîâà Ì.À. ѳéìåíáàåâà Æ.Ñ. Ôåäîðîâ È.Î. Ñåéñìèêàëû ìàÿòíèêòi òåðáåëiñòåðiíi ñàíäû ³ëãiëåóiíi íºòèæåëåði // Ìàòåìàòèêàëû æóðíàë. 2010. Ò. 10. 1 35. Á. 70 82. Ñåéñìèêàëû ìàÿòíèê òåðáåëiñòåðií ñàíäû ³ëãiëåói æ³ðãiçiëäi. Îíû íºòèæåëåði êåëåøåê æåð ñiëêiíó äàéûíäàëó êåçå iíäå ñåéñìèêàëû ìàÿòíèêòi á ðàëó á ðûøûíû íàéçàøà æºíå øû àíà òºðiçäi ìiíåç- ëû ûí ò³ñiíäiðóãå ì³ìêiíäiê áåðåäi. "Àëåì" ðàëäàðûíû íà òû æàçóëàðûíà ñºéêåñ êåëåòií ñèíòåòèêàëû øû àíà òàð ³ëãiëåíãåí. šäåáèåòòåð òiçiìi 11. ÓÄÊ: 519.624 2000 MSC: 34A45 Temesheva S.M. On a convergence of the algorithm of parametrization method // Mathematical journal. 2010. Vol. 10. 1 35. P. 83 92. Two parametrical family of algorithms of parametrization method is oered for solving nonlinear two-points boundary value problem of ordinary dierential equations' systems. The sucient conditions of isolated solution of considered problem are estabilished in the terms of right hand side's function of dierential equation and boundary conditions. References 12. ÓÄÊ: 519.624 Òåìåøåâà Ñ.Ì. // Ìàòåìàòèêàëû æóðíàë. 2010. Ò. 10. 1 35. Á. 83 92. 2000 MSC: 34A45 Ïàðàìåòðëåó ºäiñiíi áið àëãîðèòìiíi æèíà òûëû û æàéûíäà àðàïàéûì äèôôåðåíöèàëäû òå äåóëåð æ³éåñiíi áåéñûçû åêi í³êòåëi øåòòiê åñåáií øåøó ³øií ïàðàìåòðëåó ºäiñiíi àëãîðèòìäåðiíi åêi ïàðàìåòðëi ºóëåòi ñûíûëàäû. Äèôôåðåíöèàëäû òå äåóäi î æà ûíäà û ôóíêöèÿ æºíå øåêàðàëû øàðò òåðìèíäåðiíäå àðàñòûðûëûï îòûð àí åñåïòi î øàóëàí àí øåøiìiíi áàð áîëóûíû æºíå àëãîðèòìäåðäi æèíà òûëû ûíû æåòêiëiêòi øàðòòàðû òà àéûíäàëäû. šäåáèåòòåð òiçiìi 12. ÓÄÊ: 517.95 Torebek B.T. Turmetov B.Kh. equation // Mathematical journal. 2010. Vol. 10. 1 35. P. 93 103. 2000 MSC: 34K06 34K10 45J05 On the solvability of some problems for the Laplace In this paper we study the properties of some integro-dierential operators in the class of harmonic functions. As an application of these operators operator boundary value problems in the unit ball are considered. References 12. ÓÄÊ: 517.95 2000 MSC: 34K06 34K10 45J05 Ò ðåáåê Á.Ò. Ò³ðìåòîâ Á.Õ. Ëàïëàñ òå äåói ³øií êåéáið åñåïòåðäi øåøiëiìäiëiãi òóðàëû // Ìàòåìàòèêàëû æóðíàë. 2010. Ò. 10. 1 35. Á. 93 103. Á ë æ ìûñòà ãàðìîíèÿëû ôóíêöèÿëàð êëàñûíäà êåéáið èíòåãðî-äèôôåðåíöèàëäû îïåðàòîðëàðäû àñèåòòåði çåðòòåëãåí. Îñû îïåðàòîðëàðäû ïàéäàëàíóûìåí áiðëiê øàðäà û îïåðàòîðëû øåòòiê åñåïòåð àðàñòûðûë àí. Ìàòåìàòè åñêèé æóðíàë 2010. Òîì 10. 1 35

ÌÀÒÅÌÀÒÈ ÅÑÊÈÉ ÆÓÐÍÀË Òîì 10 1 35 2010 Ãëàâíûé ðåäàêòîð: Ì.Ò.Äæåíàëèåâ Çàìåñòèòåëè ãëàâíîãî ðåäàêòîðà: Ä.Á.Áàçàðõàíîâ Ì.È.Òëåóáåðãåíîâ Ðåäàêöèîííàÿ êîëëåãèÿ: Ë.À.Àëåêñååâà Ã.È.Áèæàíîâà Ð.Ã.Áèÿøåâ Í.Ê.Áëèåâ Â.Ã.Âîéíîâ Í.Ò.Äàíàåâ Ä.Ñ.Äæóìàáàåâ À.Ñ.Äæóìàäèëüäàåâ Ò.Ø.Êàëüìåíîâ À.Æ.Íàéìàíîâà Ì.Î.Îòåëáàåâ È.Ò.Ïàê Ì.Ã.Ïåðåòÿòüêèí Ñ.Í.Õàðèí À.Ò.Êóëàõìåòîâà îòâåòñòâåííûé ñåêðåòàðü Ø.À.Áàëãèìáàåâà òåõíè åñêèé ñåêðåòàðü Àäðåñ ðåäêîëëåãèè è ðåäàêöèè: 050010 Àëìàòû óë.ïóøêèíà 125 ê.304 òåë.: 87272-72-01-66 journal@math.kz http://www.math.kz Ïîäïèñàíî â ïå àòü 05.04.2010ã. Òèðàæ 300 ýêç. Îáúåì 117 ñòð. Ôîðìàò 62 94/16 ñì Áóìàãà îôñåòíàÿ 1 ã.àëìàòû óë. Êóðìàíãàçû/Ìàóëåíîâà 110/81 Òåë./ôàêñ: 2-72-60-11 2-72-61-50 e-mail: print express@bk.ru