Lineer olmayan schrödinger denkleminin enerji korumalı yöntemle çözümü ve model indirgeme yönteminin uygulanması
Tam metin
(2)
(3)
(4)
(5)
(6)
(7).
(8) . . !"# $%#%&'! !(#)*#+# (#, !"# .//)*/+/ (#,.
(9) "#( !00/" -. !" 12013 4(567 (7 (, 87779 2 !" 12013 4(567 (7 :&)!% 877779 #+!( &(# '!)!(#3 4(567 (7 !,0!% 87779 5;7 (7 !(% < 87779 (*7 5;7 (7
(10) #=0!% < 87779 (*7 5;7 (7 !)0, 8 7779 -.
(11)
(12)
(13)
(14)
(15)
(16)
(17) !" #$%$&'()*+ "
(18)
(19) . . #
(20) . " ""
(21)
(22) . #
(23) . "
(24) " ,
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32) -. . . . /$01$%+
(33) . !
(34) 2 3(3*4$
(35) . /
(36) "
(37)
(38)
(39)
(40). #
(41)
(42)
(43) " . /
(44) "
(45)
(46)
(47) " -. . . & &)152+ "" . $
(48) * 6 )%+
(49) "
(50) "
(51) 7 28! ! ""
(52)
(53) " &
(54)
(55) " " 9 $##:-3*13+ "
(56)
(57) . 9 ;<=>. 4 $##:-3*13. .
(58) >. . . . . . . >. =.
(59) 4 ; =. &
(60) - . ?. ; ;. & ! . & . ==. ; >. & ! . & . =>. - . ;<. > =. : @ $
(61) A:@$B -. ;<. > ;. $
(62) - A& 2B. ;?. > ; =. $
(63) -. ;C. > ; ;. & & $
(64) -. ;D. > ; >. $
(65) - . ;E. > ; ?. 1 : 5 $
(66) -. ;F. > ; C. 5 ! 1 : 5 $
(67) -. . ?. ;F.
(68) 8 9 ? =. ><. ! ! *1& . ><. ? = =. *1& # ' . >>. ? = ;. *1& : @ $
(69) - 38
(70). ? = >. >?. *1& $
(71) - 3
(72) >D. .
(73) ? = ?. *1& 5 $
(74) - . $8 . ? ;. >E. 5 ! *1& ? ; =. ?;. 5 ! *1& : @ $
(75) -8 3
(76). ? >. ?D ?F. 5 *1& & ? > =. 5 *1& & :@$ - 3
(77) C<. ? > ;. 5 *1& & & $
(78) -8 3
(79). ? ?. C;.
(80)
(81) $ ? ? =. C?. 5 *1& &
(82)
(83) $ . ? C =. C ;. CD. DF. 3 $
(84)
(85) A3$B -. DG. C = =. 3$ 9 $
(86)
(87)
(88) A9$B. DG. C = ;. 3$ - ( !
(89) & 3
(90). E;. C = >. 5 2 ) ) . EC. *1& 3$ - 3
(91). EE. C ; =. *1& 5 3$ - . $ . F;. C >. 5 ! *1& 3$ - 3
(92). G<. C ?. 5 *1& & 3$ - 3
(93). G;. C ? =. 5 *1& & 5 3$ - . $ . G?. @ . GE. D < ;. ! ! *1& . GE. D < >. 5 *1& &. =<C. D < ?. 5 ! *1& . =<F. A . ===. . ==;. >. ==F. .
(94) >. . ? =. 5 *1&
(95)
(96) . D =. '. (Δt = 0.1). 6.0.2 p
(97) H A B . . H AB D ;. '. 6.0.2 q. GF.
(98) H A B . H AB D >. '. GF. 6.0.2 ψ
(99) H A B . H AB D ?. '. DE. 6.0.2. GG. !
(100) H A B 8. H AB. GG. D C. '. 6.0.2 T = 10. D D. '. 6.0.2 T = 500. D E. '. 6.0.2 T = 10. D F. '. 6.0.2 T = 500. D G. '. 6.0.3. * "". =<>. D =< '. 6.0.3. )I
(101) H. =<?. D == '. 6.0.3. I
(102) H I
(103) H. "" "". =<< =<< =<= =<=. * H A B
(104) . H AB. =<?. D =; '. 6.0.4. * "". =<D. D => '. 6.0.4. )I
(105) H. =<E. D =? '. 6.0.4. * H A B
(106) . H AB D =C '. 6.0.5. D =D '. 6.0.5. =<E. )I
(107) H. =<G. * H A B
(108) . H AB. =<G. .
(109) #.!($#%!$# $%#%&$& B#)#0 )1 4(5E(01 !" 1201 !" &(& .! (#,#. 3 3 3 3 3 3. $%B/) &)%&( #.!($#%!$# C! :#)#0)!(# %!0%#=D:#)E#$'( %!0%#= 4(567 (7 (, 5=%5( D !00/" -.
(110) :/ %!"*! )#!!( 5)0' F,(G*#E!( 8 9 *!=)!0# !!(H# =5(/0 G"!))#+#! $,#I 5(%)0 .!=%G( )1 89 #)! ;G"&)0&2%&(7 '(1F 05*!) #*#(E!0! 'G%!0# 5)(= /'E/ *#= '(1210 89 'G%!0# /'E/)(= B#( .! #=# B5'/%)/ .! #=#)# #;# !)*! !*#)! $'1$) $5/;)(1 'G%!0#')! !)*! !*#)! $'1$) $5/;)( ;5= '=1 5)*/+/ .! ,% )#"# $5/F/ !)*! !*#)!)!()! /'/0)/ 5)*/+/ EG(&)0!=%!*#(7 'G%!0## J#
(111) 0#)%5 'I1$11 .! $#$%!0# !!(H#$## =5(/*/+/ EG(&)0!=%!*#(7 '(1F #=#)# #;# *+1)10 )#"# 'I1)012%1(7 ,%( !)#0!)!(. 3. #!!( 5)0' F,(G*#E!( *!=)!0#K I!(#'5*#= .! $5)#%5 ;G"&0)!(K 5(%)0 .!=%G( )1 'G%!0#K 5(% 5=% 'G%!0#K *#= '(1210 'G%!0#K *+1)10 )#"#K :#)#0 )1 '1$) 5*/ 3 .
(112) #.!($#%' $%#%/%!. F#!F! 4(5E(00! 4(5E(00!. /I!(.#$5( !E(!! L(*!* * %!. 3 3 3 3 3 3. $%B/) &)%&( #.!($#%' $%#%/%! 56 F#!F! %,!0%#F$ * 50I/%!( %,!0%#F$ 4(567 (7 (, 4,77 D -. C .
(113) M 4 . %,#$ %,!$#$ %,! !!(E' I(!$!(.#E .!(E! .!F%5( N!)* 8C9 #%!E(%5( L$ II)#!* %5 %,! 5)#!( F,(G*#E!( 8 9 !O/%#5 * %,! *#$F(!%#"!* 05*!) #$ (!*/F!* B' I(5I!( 5(%,5E5) *!F50I5$#%#5 8497 /0!(#F) (!$/)%$ 65( 5! * %L5 *#0!$#5) * F5/I)!* L#%, I!(#5*#F * $5)#%5 $5)/%#5$ F5N(0 %,! F5.!(E! (%!$ 56 %,! 4 (!*/F!* 05*!)7 ,! (!*/F!* 05*!) I(!$!(.!$ %,!
(114) 0#)%5# $%(/F%/(! * #$ )$5 !!(E' I(!$!(.#E7 C5( F5/I)!* *#$I!($#5 )'$#$ L$ )$5 F((#!* 5/%7. !'L5(*$. 3 5)#!( F,(G*#E!( !O/%#5K .!(E! .!F%5( N!)* 0!%,5*K 0#*DI5#% 0!%,5*K I(5I!( 5(%,5E5) *!F50I5$#%#5K *#$I!($#5 )'$#$. F#!F! 5*! 3 .
(115) . 2 H "
(116)
(117)
(118)
(119) " " "" 8
(120) . ! "
(121) . . &
(122)
(123) " " "" " 8. . &
(124)
(125) " "
(126)
(127) . "
(128)
(129)
(130) $
(131)
(132) 8
(133)
(134)
(135)
(136) I
(137) " .
(138)
(139)
(140). !
(141) 8.
(142)
(143)
(144)
(145) 8
(146) " . .
(147) " . JC> ?<K. %8# . %8# . %8# "
(148) 8 . !
(149)
(150) .
(151)
(152).
(153)
(154) JC>K. & 8. . J>>K &
(155)
(156) I . :
(157)
(158) 8. # . 8. =.
(159) J=D ;EK. /
(160)
(161) A:@$B I 8.
(162)
(163)
(164) & I
(165)
(166)
(167)
(168)
(169)
(170) J=<K. 3
(171)
(172) . I
(173)
(174) " 9 8
(175) I . )I . . !
(176)
(177)
(178) " 8
(179) "
(180) . 5
(181) .
(182) "" 8 . ! .
(183) !
(184) "
(185)
(186) . ) .
(187)
(188)
(189)
(190) A3$B ! "
(191)
(192) 8
(193) . 3$ .
(194)
(195). 1 & 3$ .
(196). ! 3$
(197)
(198)
(199) 8. JC?K !
(200) &
(201)
(202)
(203) . 3$ :@$ 8. ' " 3$
(204)
(205) 8. . 2
(206)
(207)
(208)
(209). .
(210)
(211)
(212)
(213)
(214)
(215) "
(216) 5 " .
(217)
(218)
(219) 7 "" " I
(220) $
(221)
(222) "" " & 8
(223)
(224)
(225) . !
(226)
(227) & 8.
(228)
(229)
(230) . ;. & .
(231) *1&
(232)
(233)
(234)
(235) 8 ! " 3$
(236)
(237) . /
(238)
(239) 3$
(240) 8.
(241) $
(242)
(243) "
(244) . 3$ :@$ . "
(245)
(246)
(247)
(248)
(249)
(250) 8
(251) . >.
(252) .
(253)
(254) &
(255)
(256) . 8
(257)
(258)
(259) . & .
(260)
(261)
(262)
(263)
(264)
(265)
(266) . !
(267)
(268) .
(269)
(270)
(271) " .
(272)
(273) . y˙ = f (y),. y(0) = y0 ,. ! " "
(274) . y ∈ Rn. A; =B.
(275) . 10 7 7 7 .
(276)
(277) . yn+1 = yn + hf (yn ).
(278) h ϕh : yn → yn+1 !.
(279) " . #
(280) "
(281) "
(282) " $%&' 10 7 77 ( .
(283)
(284) . yn+1 = yn + hf (yn+1).
(285) $%&' ?.
(286) 10 7 7-7 )
(287)
(288)
(289) . . yn+1 = yn + hf
(290)
(291) . h ↔ −h. yn + yn+1 2. . : L. yn ↔ yn+1. .
(292)
(293)
(294)
(295) " J;DK. 10 7 77 u˙ = a(u, v),. v˙ = b(u, v).
(296) .
(297)
(298)
(299) . un+1 = un + ha(un+1 , vn ),. vn+1 = vn + hb(un+1 , vn ). un+1 = un + ha(un , vn+1 ),. vn+1 = vn + hb(un , vn+1 ). "
(300) $%&' 10 7 7?7 * + y0 ϕt(y0) = y(t)
(301) ϕt : U → R2d. (U ⊂ R2d ).
(302) "
(303) " ,- $%&'. 10 7 7@7 %.
(304) + y c
(305) I(y(t)) = c. .
(306) /
(307) " I(y) " !
(308) $%&' (!= 7 7A7 0
(309)
(310) 1. 2. 3
(311) !4
(312) .
(313) . .
(314)
(315) u(t)
(316) + " / v(t) " 2
(317) u˙ = u(v − 2),. v˙ = v(1 − u).
(318) . "
(319) #
(320) I(u, v) = ln u − u + 2 ln v − v . 5 0=. v−2 d 1−u u˙ − v˙ = I(u, v) u v dt. 6 t ∈ R I(u(t), v(t))
(321)
(322) 7 I "
(323) $%&' C.
(324) " f : Rm → Rm x(t) ˙ = f (x) !
(325) " c1 , c2, · · · , cm
(326) 10 7 77. x(t) ∈ Rm. d (c1 x1 (t) + c2 x2 (t) + · · · + cm xm (t)) = 0 dt. A; ;B.
(327) . " %% ++2 x ∈ Rm cT f (x) = 0. A; >B.
(328)
(329) 8 xn+1 = xn + hf (xn)
(330) %9 " cT xn+1 = cT xn + hcT f (xn ) = cT xn. .
(331) ! 3 /
(332)
(333) / 2 +
(334) ! 2
(335) $%:' M
(336) . " f : Rm → Rm x(t) ˙ = f (x) !
(337) " C ∈ Rm×m
(338)
(339) xT Cx
(340) . ! +
(341) xT Cx 2
(342) "
(343) 5 10 7 77. x(t) ∈ Rm. 0=. d T (x Cx) = x˙ T Cx + xT C x˙ = 2xT C x˙ = 2xT Cf (x) dt. # C = C T " x(t) ˙ = f (x)
(344) xT Cx . 2 "
(345) ++2 x ∈ Rm f xT Cf (x) = 0.
(346) D.
(347) xn+1 = xn + hf ( xn +x2 n+1 ).
(348) . . fnmid. := f. #
(349)
(350)
(351)
(352) . xn + xn+1 2. .
(353) ,
(354) . xTn+1 Cxn+1 = xTn Cxn
(355)
(356) . $
(357) . xTn+1 Cxn+1 = (xn + hfnmid )T C(xn + hfnmid ) = xTn Cxn + 2hxTn Cfnmid + h2 fnmid Cfnmid T.
(358) . $
(359) . xn =. xn + xn+1 1 mid − hfn 2 2.
(360)
(361) . . xTn+1 Cxn+1. xn + xn+1 1 mid − hfn = + 2h 2 2 T xn + xn+1 = xTn Cxn + 2h Cfnmid 2. T. xTn Cxn. . Cfnmid + h2 fnmid Cfnmid T. ! .
(362)
(363) .
(364)
(365) . xTn+1 Cxn+1 = xTn Cxn !
(366) )
(367) J;CK. M
(368)
(369)
(370)
(371)
(372) . R2d. R2d. " 8. "
(373) I 8. ξ p , ξ q , τ p , τ q ∈ Rd. " . E. (p, q).
(374) . ξ = (ξ p , ξ q )T. .
(375) τ = (τ p , τ q )T.
(376) " . 0 ≤ s ≤ 1}
(377)
(378) d = 1.
(379) A. ⎛. B. or.area(P ) = det ⎝
(380)
(381) . d>1. ⎞ ξ. p. ξ. q. τ. p. τ. q. ⎠ = ξ pτ q − ξ q τ p.
(382) . w(ξ, τ ) :=. d . ⎛ det ⎝. i=1. ⎛. ξip. τip. ξiq. τiq. ⎞ ⎠=. d . (ξip τiq − ξiq τip ). J = ⎝. ⎞. 0. I. −I 0. ⎠. . A; ?B. i=1.
(383)
(384) . A B . P = {tξ + sτ |0 ≤ t ≤ 1,. I, d. !. w(ξ, τ ). . " . A; ?B . w(ξ, τ ) = ξ T Jτ
(385) J;DK. ⎛. 10 7 7 7. A : R2d → R2d.
(386) " J. =⎝. 0. ⎞ I. ⎠. . −I 0. w(Aξ, Aτ ) =. AT JA = J "
(387) + ξ, τ ∈ R2d w(ξ, τ )
(388) "
(389)
(390) " $%&'. d = 1
(391) w(ξ, τ )
(392)
(393)
(394)
(395) A A.
(396) .
(397) .
(398)
(399). ,. A
(400) " "" " 8.
(401)
(402) . M
(403)
(404)
(405) . 10 7 7 7 U ⊂ R2d g : U → R2d .
(406) " + x ∈ U . g (x)T Jg (x) = J. w(g (x)ξ, g (x)τ ) = w(ξ, τ ) F.
(407) g (x) ;
(408) +
(409)
(410) # ⎛. J =⎝. 0. ⎞. I. −I 0. ⎠. " I ∈ Rd× $%&' d. &
(411)
(412)
(413)
(414)
(415) & " @ JDCK % JC=K 6+ J;;K
(416)
(417) " " . & 8& N 4 JC>K 1 N. % J?<K
(418)
(419) . %8# %8# !
(420) %8#
(421)
(422) J?K. !8.
(423)
(424) %8# "
(425)
(426)
(427) . $
(428) %8# .
(429) .
(430)
(431)
(432) J;D C>K. !5(!0 7 7 7 )
(433)
(434)
(435) "
(436)
(437)
(438) !.
(439) $%&'. . =F>?
(440)
(441)
(442) 8
(443)
(444)
(445)
(446) . O
(447)
(448)
(449) .
(450)
(451) " " . $
(452) + " .
(453)
(454)
(455)
(456) """ , *P ) 1+
(457)
(458)
(459) "
(460)
(461)
(462) J;DK. 2 . H . . ! 8.
(463)
(464)
(465)
(466) . ! " .
(467) 8.
(468) $ " " JD?K
(469)
(470)
(471) & . 8 . . F, G G. " "
(472)
(473).
(474) " "". {F, G}. " . !
(475) . M. " 8. " "
(476)
(477)
(478) .
(479) 2 ! <
(480) {., .}
(481) 5. 10 7 7 -7 •. F, G, L M. =. 7 . {aF + bG, L} = a {F, L} + b {G, L} {F, aG + bL} = a {F, G} + b {F, L} •. > ?
(482) . {F, G} = − {G, F } •. 1 8#
(483). {F, G.L} = {F, G} .L + G. {F, L} . •. ..
(484)
(485) . O !. {{F, G} , L} + {{L, F } , G} + {{G, L} , F } = 0. M. "
(486)
(487) . . M. M. 8. "
(488)
(489)
(490) . < / P : M → R / 2 " + H ∈ M {P, H} = 0 # P @ "
(491) 2 + 10 7 7 7. m = 2n n M = Rm. M.
(492) . (p, q) = (p1 , · · · , pn , q 1 , · · · , q n ). " .
(493)
(494) . n ∂F ∂G ∂F ∂G {F, G} = − ∂q i ∂pi ∂pi ∂q i i=1 i, j = 1, · · · , n. . =<. A; CB.
(495) i i. q , p = 1,. i j. q ,p = 0. (i = j) ,. i j. p , p = 0,. !
(496)
(497) . i j. q ,q = 0. A; DB. #
(498) . # 8 4
(499) . F, G ∈ M. . {F, G} = 0. ' .
(500)
(501) . 4 . < " H . 10 7 7 ?7. M. : M → R. 2 . XH (F ) = {F, H} = − {H, F }. 2 "
(502) 6
(503) "
(504) .
(505) . m = 2n + l. "
(506)
(507). p, q
(508) A; DB j k i k i k. r ,r = p ,r = q ,r =.
(509) . 0,. i = 1, · · · , n,. r. . p. . q.
(510) . j, k = 1, · · · , l.
(511)
(512) . A; CB .
(513)
(514)
(515)
(516) .
(517). n ∂H ∂ ∂H ∂ XH = − i i i ∂q i ∂p ∂q ∂p i=1
(518)
(519) . !"# $!"" %!% % P =. P dx ∈ F. . F. " 8.
(520)
(521) " . P, L ∈ F. " .
(522) {P, L} =. δP.J .δLdx ==. A; EB.
(523) 6
(524)
(525)
(526) ; = => . $ A; EB
(527) . J. ". "
(528) .
(529) %A <
(530)
(531) %..9 J (x)
(532) 6
(533)
(534) 10 77 7. J (x). !00 777 %A 2
(535)
(536) " J (u) q × q
(537) !.
(538) J (u)
(539)
(540)
(541)
(542)
(543) $&B' (
(544) . P =. P dx. L =. . Ldx. . P, L.
(545)
(546) . .
(547).
(548). δP.J (u).δLdx = −. δL.J (u).δPdx.
(549)
(550) .
(551). δP.(J (u) + J ∗ (u)).δLdx = 0. J ∗ (u), J (u) . B
(552)
(553) 8. A. . J (u) ""
(554)
(555) . '
(556)
(557) .
(558)
(559)
(560) 0
(561) "
(562) " J (u)
(563) 6
(564)
(565) $&B'. 5/; 77-7 u. J (u)/ q × q. u(x, t) ∈ Rq ,. t∈R. " . . δH ∂u = J (u) ∂t δu. Ω.
(566)
(567) . H(x, un )dx. Ω ⊂ Rp × R,. dx = dx1 dx2 · · · dxp. . . J (u). " . . "". .
(568) " . u(x). " " . . H(u) =. H(u). δH δu. @ ".
(569) 8. 6 . d H[u + δu] − H[u] = H[u + δu]|=0 A; FB →0 Δx.
(570) dx1 δH δH dx = , δu = δu δu(x) δu x0. δH[u; δu] := lim. =;.
(571) H[u] =.
(572) . δH[u, δu] =. x1 ∂H. δu + ∂u. x0. . x1 x0. H(x, u, ux, · · · )dx 8. ∂H δux ∂ux. !. +. ∂H δuxx ∂uxx. p=q=1. ∂H δH = − ∂x δu ∂u. . ∂H ∂ux. +···. . ". . −. ∂x2. ∂H ∂uxx. +···. . , ". k = 1, · · · , q. δH δH ∂ = − δuk δuk ∂xl l=1 p. . . ∂H ∂uk,l. +···.
(573)
(574) . 10 777. . x, u. J (u)/ q × q. J . δC = 0 δu.
(575) 6
(576)
(577) 6
(578) C ∈ F . & !" $!"" %!% % . .
(579)
(580)
(581)
(582)
(583) . . dr k = 0, dt. ∂H dq i = i, dt ∂p. k = 1, · · · , l. ∂pi dpi = − i, dt ∂q. i = 1, · · · , n. A; GB. A; =<B. A; =<B *P H .
(584)
(585) . H. r
(586) 4
(587) . . .
(588)
(589) . r. 8. ' .
(590)
(591)
(592) . ! . .
(593)
(594)
(595)
(596)
(597) 4. =>.
(598)
(599)
(600) . x = (x1 , x2 , · · · , xm )
(601)
(602) .
(603) . H(x) . m. . " . XH. M. . .
(604).
(605)
(606) . XH =. m . ξ i (x). i=1 !. ξ (x) H i. ∂ ∂xi.
(607)
(608)
(609)
(610) . #
(611) 8.
(612)
(613)
(614)
(615)
(616)
(617)
(618). XH = {., H} = − {H, .} =. m . ξ i (x). i=1
(619)
(620) . !. XH (xc ) = {xc , H} = ξ c (x), . ∂ ∂xi. c ∈ {1, · · · , m}. ! . m i ∂F x ,H {F, H} = ∂xi i=1
(621)
(622) . F, G. 8.
(623)
(624) . m m i j ∂F ∂G x ,x {F, H} = ∂xi ∂xj i=1 j=1. A; ==B. . Jij = {xi , xj } ,. i, j = 1, · · · , m. J(x) = (Jij (x)) m × m.
(625)
(626)
(627) .
(628) A; ==B 8.
(629)
(630) . {F, H} = ∇F.J∇H . =?. A; =;B.
(631)
(632) %.%
(633)
(634) ! 2 "
(635) <
(636) + . !00 7-7 7. J(x) = (Jij (x)). •. =. 7 . •. 3
(637) ( . •. >
(638) Jij (x) = −Jji(x),. i, j = 1, · · · , m. •. ; / +. m. x ∈ M. . l=1. Jil ∂x∂ l Jjk + Jkl ∂x∂ l Jij + Jjl ∂x∂ l Jki = 0,. i, j, k = 1, · · · , m. 10 7-77. x = (x1 , x2 , · · · , xm ),.
(639)
(640) !. M ⊂ Rm. H(x) 6
(641) / J(x)
(642) . dx = J(x)∇H(x) dt. " x˙ = {x, H}.
(643) 6
(644)
(645) J(x)
(646)
(647)
(648)
(649) 6
(650)
(651) / xC <
(652) y(t) ∈ R2d. . y(t) ˙ = f (y(t)). . H = H(p1 , p2 , · · · , pd , q1 , q2 , · · · , qd ),. pi = pi (t),. qi = qi (t) i = 1, · · · , d. . " . . p˙ i = − . ∂H , ∂qi. q˙i =. ∂H ∂pi. A; =>B. A; =>B
(653) . p˙ = −Hq (p, q),. q˙ = Hp (p, q). =C. A; =?B.
(654)
(655) . H = H(p, q). !. .. . t+. " . " . ∂H dq ∂H dp ∂H ∂H ∂H ∂H dH = + =− + =0 dt ∂q dt ∂p dt ∂p ∂q ∂q ∂p . 8
(656) . (!= 7-7-7 p(t) ˙ = − sin q(t) q(t) ˙ = p(t). A; =CB.
(657) 6
(658)
(659) . H : R2 → R2 1 H(p, q) = p2 − cos q 2. 6
(660) Hq = sin q " Hp = p p˙ = −Hq (p, q) q˙ = Hp (p, q). . 4
(661)
(662) +
(663) /
(664)
(665) " " # 2
(666) 2 p(0) = p0, q(0) = q0 ϕt : R2 → R2
(667) " . ⎛. ϕt ⎝. ⎞. p0 q0. ⎛. ⎠=⎝. ⎞. p(t). ⎠. q(t).
(668) > %... g(x) ϕt (x) / x = [p, q]T t = 0 ϕt (x)
(669) " ;
(670) /
(671) %... 2
(672) (
(673) "
(674) . P (t) = ∂ϕ 2
(675) ∂x t. 0. ⎛ d ⎝ dt. ∂p ∂p0. ∂p ∂q0. ∂q ∂p0. ∂q ∂q0. ⎞. ⎛. ⎠=⎝ ⎛. P (t) = ⎝. ∂f1 ∂p0. ∂f1 ∂q0. ∂f2 ∂p0. ∂f2 ∂q0. 0 − cos q 1. 0 =D. ⎞⎛ ⎠⎝. ∂p ∂p0. ∂p ∂q0. ∂q ∂p0. ∂q ∂q0. ⎞ ⎠ P (t). ⎞ ⎠.
(676) d P (t)T JP (t) = P (t)T JP (t) + P (t)T JP (t) dt ⎞⎛ ⎛ 0. = P (t)T ⎝. 1. 0. ⎠⎝. ⎞. 1. ⎠ P (t). −1 0 − cos q 0 ⎞⎛ ⎞ 0 1 0 − cos q ⎠⎝ ⎠ P (t) + P (t)T ⎝ −1 0 1 0 ⎛⎛ ⎞ ⎛ ⎞⎞ −1 0 1 0 ⎠+⎝ ⎠⎠ P (t) = 0 = P (t)T ⎝⎝ 0 − cos q 0 − cos q ⎛. . >" 0
(677) P (t)T JP (t) /
(678) %... 6 t ϕt
(679) " .
(680)
(681) $%:' (!= 7-77 8 %99
(682)
(683) 2 !.
(684)
(685) "
(686) %.:
(687)
(688) . 2 ⎛ ⎞ ⎛ ⎞ ⎝. pn+1 qn+1. ⎠=⎝. pn − h sin qn qn + hpn. ⎠.
(689) . g : (pn, qn) → (pn+1, qn+1) ⎛. ∂(pn+1 , qn+1 ) ⎝ = ∂(pn , qn ). ∂pn+1 ∂pn ∂qn+1 ∂qn. ∂pn+1 ∂qn ∂qn+1 ∂qn. ⎞. ⎛. ⎠=⎝. 1 −h cos qn 1. h. ⎞ ⎠.
(690) "
(691) %... ⎛ ⎝. 2. 0. 1 + h cos qn. 2. −1 − h cos qn. 0. ⎞ ⎠.
(692) . /
(693) !
(694) 2
(695)
(696) " ++2 h
(697) $%:' !5(!0 7-7?7 < .DEE H(p, q),.
(698) ! " # +
(699) t ϕt
(700) " 6 !
(701)
(702) .
(703)
(704) $%&' =E. U ⊂ R2.
(705) (
(706) . A; =?B O . ⎛ ⎝. −Hpq −Hqq Hpp. ⎞ ⎠. Hqp.
(707) . ⎛. 0 −1. J −1 = ⎝. 1. 0. ⎞ ⎠. . ⎞. ⎛ ∇2 H = ⎝ " . J −1 ∇2 H. Hpp Hqp. ⎠. Hpq Hqq.
(708) ' . P (t) = J −1 ∇2 HP (t).
(709)
(710) . d P (t)T JP (t) = P (t)T JP (t) + P (t)T JP (t) dt = P (t)T ∇2 HJ −1 JP (t) + P (t)T JJ −1 ∇2 HP (t) = 0. . (!= 7-7@7. !. JJ −1 = I. . J −1 J = I.
(711) . ? !F !. ψ(−20, t) = ψ(20, t). . ∂2ψ ∂2ψ = − α sin(ψ) ∂t2 ∂x2. A; =DB. 7 / 6
(712)
(713) H=. 1 2 1 π + 2 2. " 6
(714)
(715) J. ⎛ =⎝. . ∂ψ ∂x. 0. . 2. + α(1 − cos(ψ)) dx 1. −1 0. ⎞ ⎠. ∂u = J δH 6! ∂t δu.
(716)
(717) # u = ⎝
(718) 5 =F. ⎞. ⎛ ψ π. ⎠. " π = ∂ψ 6
(719) ∂t.
(720) ⎛ ⎝. ψ˙ π˙. ⎞ ⎠ = J ∇H. # 6
(721) 1 2 1 π + H= (ψj+1 − ψj ) + α(1 − cos(ψj )) 2 j 2(Δx)2 j=0 ⎛ ⎞ 0 I ⎠
(722)
(723) 6
(724)
(725) overlineJ = ⎝ −I 0 N −1 .
(726) . " u(x, 0) = 6sech2x 2 (
(727) -2! 4 . (!= 7-7A7. u(−20, t) = u(20, t). ∂u ∂u ∂ 3 u = −6u − ∂x ∂x ∂x3. 6
(728)
(729) H=. 1 2 3 (ux ) − u dx 2. " 6
(730)
(731) S = ∂x∂ 6
(732) . ? 6
(733) H=. N −1 j=0. 1 (uj+1 − uj )2 − u3j 2(Δx)2. . " 6
(734)
(735)
(736)
(737) . ⎞. ⎛. 0 −1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 0 −1 1 ⎜ ⎟ ⎜ ⎟ J = ⎟ ⎜ ⎟ 2Δx ⎜ ⎟ ⎜ ⎜ 1 0 −1 ⎟ ⎠ ⎝ −1 1 0. . # 6
(738)
(739) . =G. du dt. = J ∇H. .
(740) . ' (
(741)
(742)
(743) ) & *+ , *-
(744) % )I
(745) . )I. H 4 6 1P+
(746)
(747) =G;F
(748)
(749)
(750)
(751) J=EK. &
(752)
(753)
(754) . I
(755) J?>K 8
(756) " J;?K
(757)
(758)
(759) . $
(760) .
(761) "
(762)
(763) J=GK. . I 8.
(764)
(765)
(766)
(767)
(768) .
(769)
(770) -
(771) "
(772)
(773) " "8 "
(774) "
(775) A:@$B
(776)
(777) . & . ! 8. -" :@$ , Q 8.
(778) "
(779)
(780) %8# . # . ". :@$
(781)
(782)
(783) J;E =DK. )I .
(784) !8 . $ 8. !8 JGK .
Benzer Belgeler
approximately 1.7-fold, and the bleeding time returned to baseline within 60 minutes of cessation of magnesium sulfate infusion.On the other hand, platelet thrombi formation was
[r]
Kaybını bir türlü kabullenmediği sevgili annesi, biricik babası ona ku cak açıyor, büyük aşkı Nâzım elinde çiçeklerle bekliyor.. Fikret Mualla ve atlet Haydar Aşan
Bu çalmada ise yapsnda karbonhidrat, protein ve ekerleri barndran ve bu bakmlardan çok sayda biyokütleye model tekil eden msr model biyokütle olarak seçilmi ve su
Tables give the exact value , approximate value for compact finite difference method, approximate value for restrictive Taylor approximation and absolute error for
Anahtar kelimeler: Yaklaşık Çözüm, Newton Metodu, Freshe Türevi, Gato Türevi Bu çalışmada Lineer olmayan diferansiyel denklemlerin yaklaşık çözümünde Newton
Bu çal••man•n amac•; ilaç üretiminde, uzun, orta ve k•sa dönemlerde kar• maksimize ederken miat ve ürün geçi• k•s•tlar•n• da ele alarak, en iyi üretim plan•n•