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Lineer olmayan schrödinger denkleminin enerji korumalı yöntemle çözümü ve model indirgeme yönteminin uygulanması

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(1)T.C. ĐSTANBUL KÜLTÜR ÜNĐVERSĐTESĐ FEN BĐLĐMLERĐ ENSTĐTÜSÜ. LĐNEER OLMAYAN SCHRÖDĐNGER DENKLEMĐNĐN ENERJĐ KORUMALI YÖNTEMLE ÇÖZÜMÜ VE MODEL ĐNDĐRGEME YÖNTEMĐNĐN UYGULANMASI. DOKTORA TEZĐ Canan AKKOYUNLU. Anabilim Dalı: Matematik-Bilgisayar Programı: Matematik. Tez Danışmanı: Prof. Dr. Erhan GÜZEL. TEMMUZ 2013.

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(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 .       

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