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→ − ∂f + div S = I, ∂t → (#&%)!*7 − S C A>B 2& f C $!) !"#$(* 03)$+!* '$ % )# ! $( ' # () # ' (-4'7 I C !# ")!$7 &%*:9!/ 4&!)(4!"#$3: '* )# 6 ,(-'()!* -(&./7 *'%* #* 03)$+!/ (&!3#( ! '4)! ($ $($ -(&.7 #3 !7 *'%* #* 14)47 03)$+!* (#&%)!* &%8)( , 6 )#)( $ D%4) 3 ,E4( da3 7 ! 3('))! @$$(5%()$( )38) -(!#( 6 &%* 03)$+!! 4πa2 f 7 )#))/ $ D%4) 3 da 1 ∂ 2 ∂f =− 2 a · S + I(a, t), AB ∂t a ∂a 2& S C % )# 6 $( ' # ()# ' (-4' &%* #* $($ ∂f + Af, S = −B AB ∂a A C $D00!+!) 7 &%*:9!/ #$# 6 # (7 B C $D00!+!) &!003-!! -(&./ (-4(4 DV (a − ak ), a2 DV T ak B= , 8παa3 A= AFB AB !) !$( ,(-'()!* )'/ 0(- >>> 2& D C $D00!+!) &!003-!! A&%*:9!/ &'& '9# '( !- $38(:92 (# '(B7 C 4%$3%*)/ ,E4 '9# '(7 c C $)+) (+!* '9# '( ' (# '7 c0 C $)+) ( +!* )(#9))2 (# '(7 = c − c0 C $)+) (+!)) #9)!7 T C 4( 3(7 ak C $! !"#$!/ (&!3#7 # ' # '3:9!/ 4!)!4343 (')'#)/ 03)$+!! (#&%)!*7 α C $D00!+!) '1)# )2 )( *8)!* G(&.! (&!3#4 a > ak *'%*: #* 3# /"! '4! (, =>? ()(%! !"#$! %3") # (+!)() .)! 3('))!* @$$(5%()$( AB # 3" 4 #%&3:9!1 39)!/ C "!#% ,(-3:9!1#* -(&./ # *)) A!) )#!')# 6 !# ")!$( ) -('!#! ' 4)! ! #9)!*BH C )3$%(+!* ) ,&)* (# ' A-($) #1())!* (# '))2 '9# '( ) (##4( !'( #*B7 #9)! (# '( # *))H C $ "- $! !"#$!/ (&!3# # *))/ &$! !"#$/ ,%(# ! .)! G%6&'!"( +!)(%6) (')'#)/ 03)$+!! f0 (a) A!# >B V R f0 (a) = C exp − T , AIB 2& C C $D00!+!) +!)(%6)# !7 T C 4( 3(7 R C 4!)!4(%6)(* (, (7 $ 3: ),1&!4 -( ( ! 6 &%* #-&()!* -(&.( (&!3#( a R=− 8παa3 + 4παa2 , 3ak AB &)($ ' ,94 #%3"( % )# 6 $( C 03)$+!* '4)! !" # $ % 32 !,%!8)!7 &3#$(:9 ()(%! !"#$ .)!7 (##4 ) ' (, ;!0.! +( ! <%-'( =? )( # (&!! $(%#+)+!!7 $2&( # )6 #9)!* (# '( 4(%(7 -)(")! &!003-!))2 "%)( ' '(8)!! &%* % )# ! $( AB 4)2 4)6. )#)27 ! !4 48) ),"6 ($ 39)! -'%* )(/ ! (#!4 !"#$ .)! &%* 03)$+!! (#&%)!* (-4(4 A' &'!8)/7 #'*-())/ # $! !"#$!4 (&!3#4 #!# 4 #" (B A!# B f (u, t) = n(t) · 2& n(t) C 3,'(:9(* 03)$+!* '4)!7 u2 exp[−1/(1 − 2u/3)] , (u + 3)7/3 (3/2 − u)11/3 u = r/rkr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a, t) 7 &%*:9(* #$# 6 ,(-'()!* -(&./7 -('!#! #9)!* (# '(7 ! # 34)6.)!4 #9)!* 2)(+!* -(&./ 34)6.( #* 4 27 4!)!434 (, ,(-'()!: -(&./ # ' # '3 a = 0 7 ' #)')4 '-)!$(: -(&.! 4(%1 (-4'7 $ 4 )("!)(: (# ! -( #" &!003-!! !- (# '( ! (# '* 6#* %* # 48) #"! ( 67 " ,%6.!)# ' ')'6 !) !$( ,(-'()!* )'/ 0(- >> ,(-'('.!1#* -(&./ !4: K)3%'/L (-47 I(a, t) +!)(%6)( δ 503)$+!! !($( &())/ (, !)* ( #%&3:9(* 4&%6 !# ")!$( Ω2 I(a, t) = I0 · exp − · δ3 (a), (t)2 A>B 2& Ω C 1(($ ) #9)!7 ! $ 4 48 !#1&! 6 2)(+!* -(&./7 I0 C #$# 6 ,(-'()!* -(&./ K)3%'2L (&!3#(7 ((4 #!# 4 ! 0 Ω A 0 C )("(%6) $)+) (+!)) #9)!B $%!"# ' -(&./7 &(# (:9!1 & $! ! "#$2 (-4(7 4(%7 -3%6 ( 4 $)$3)+!! +##' )3$%(+!! ! (# ')!* ,3& (')'#)(* 03)$+!* (#&%)!* ! 0 Ω !#1&! !) )#!') ,(-'()! -( &./7 -)("! %6) "!#% $ 1 &(# ( & $! !"#$2 (-4(7 " !'&! $ # 3 )'/ 0(- ! ' &(%6)/.4 $ $(%#+)+!! %* "!#%))2 .)!* +%#,(-) ''&)! ,-(-4)1 4))1 A'4* τ ! (&!3# r B G( 4(#. (, (&!3#( r0 ',4 (&!3# -(&.(7 '1)# )(* D)2!* $ 2 (')( T r0 = (T /8πα)1/2 &# ()'$( !'&))1 '(8)!/ ' AB ! ,(-'()!* $ ,-(-4)4 4))4 !'&* $ 3('))!: 1 ∂ ∂f = 2 ∂τ r ∂r 1 ∂f −f r ∂r δ(r) r −1 + I0 · exp −Ω20 x2 (t) , rk r2 AB 2& rk = (V c0 )/(4π0 r03) C $! !"#$!/ (&!3# -(&.( A# ' # '3 (')'#!: -)( # $38(:9!4 (# '4 ! $(%#+)+!! =?7 x(t) = 0/(t) 7 x(t = 0) = 1 7 Ω0 = Ω/0 ($)+7 #1())! %)2 $%!"# '( (# '))2 '9# '( '(8( #* 3('))!4 + q = 0 , AB 2& q = ρ 43 πa3f (a)da3 C 4(##( ,(-'('.!1#* -)7 $ (* (')( )3%: ' )("(%6)/ 4 4) '4)! ! 1& $ ,-(-4)4 4))4 3('))! AB !'&! $ -($)3 #1())!* 4(## ' '!& 1 +χ 1= x(t) r 3 f (r, t)d3 r, AFB 2& χ = (4ρπr04 )/(30 ) C ((4 #!# 47 &%*:9!/ -('!#!4# 6 )("(%6)2 #9)!* !7 # ' # '))7 #$# 6 0(-'2 1&( , 03)$+!! !# ")!$( ' '!& A>B ! &%)!! # ' # '3:9!4 ,(-4 ((4 ' #!# 4 -'%* %3"! 6 (-%!") 3 ! (-'! !* #!# 4 A (')'#)/ 03)$+!! & $(%#+)+!!B ! !##%&'( 6 $($ $'(-!(')'#) D (7 ($ ! '# 1&) +## 1()!-4 -(8&)!*7 # ( ! $(%#+)+!! ! 0(-'4 1& '2 &( (##4( ! '( #* # &!)1 -!+!/ 3 4 "!#%))2 .)!* $!) !"#$2 3('))!* AB # 3" 4 -($)( #1())!* 4(## AFB ('))!* .(%!#6 4 &4 $)")1 (-)# / )*')/ #147 $ (* *'%* #* 3# /"!'/ M(2 # $! ',() dr = 0, 01 7 .(2 '4)! dτ = 0, 02 (-)# )(* #14( &%* 3('))!* AB ! i = 1, 2, . . . , N n+1 n+1 − Ci fin+1 + Bi fi+1 = −Fi , AB Ai fi−1 2& fin C -)(")! 03)$+!! ' xi = i·dr ' 44) '4)! tn = n·dτ D00!+!) 2)$! Ai = Bi = dr(i − 0.5) 1 1 + − ; dr 2 (i − 0.5) 2rkr 2 AIB dr(i + 0.5) 1 1 − + ; + 0.5) 2rkr 2 AB dr 2 (i >>F Ci = J '()'(7 (#6$!) 1 dr dr 3 2 1 (3i + 0.25) + 2 + 2 + ; 3dτ dr (i + 0.5) dr (i − 0.5) 2rkr AB dr 3 (3i2 + 0.25) n fi . 3dτ AB Fi = (-)# )(* #14( &%* 3('))!* AB ! f0n+1 2 dr 1 dr 3 + 2+ − 24dτ dr 4rkr 2 G)(")! xn+1 = f1n+1 i = 0 dr 1 2 − + 2 dr 4rkr 2 + dr 3 n f0 + I0 exp −Ω20 xn . 24dτ A>B &%* #* !- -($)( #1())!* 4(##7 (-)# )(* #14( &%* $ 2 1= 1 xn+1 +χ A>>B (i · dr)3 fin+1 dr. N()!") 3#%'! )( ('/ 2()!+ fNn = 0 G)(")! ((4 ' I0 ! Ω0 ' A>B &%* (-) 3 ! (-'! !* #!# 4 $ %&"' "(& )" * ) &"+,- . #/+- &"0 ) & 0"& ( !# &# ('%) 2(0!$! 03)$+!! (#&%)!* (-4(47 # ' # '3:9! ((4 (4 I0 = 0, 1 ! Ω0 = 0, 4 ' 44) '4)! t1 = 5 7 t2 = 50 ! t3 = 200 (( 4 # ' # '3: '#$/ #$# ! ,(-'()!* -(&./ # !- (# '( ,%6.2 $%!"# '( "(# !+ !'&! $ ,# 43 3,'()!: #9)!* #!# 4 !7 #%&'( %6)7 3'%!")!: $! !"#$2 (&!3#( -(&.( A rkr1 = 1, 05 7 rkr2 = 4, 8 B -3%6 ( %!.6 4(% $%!"# ' "(# !+ (&!3#( *&$( $! !"#$2 A!2()!")(* ,%(# 6B &%8( (-'!'( 6#* ' )'3: 0(-3 I0 = 0, 1 & ' ($ % % & ) %* t1 = 5 t2 = 50 t3 = 200 & +$ * rkr1 = 1, 05 rkr2 = 4, 8 rkr3 = 6, 2 I0 = 0, 1 & $ t1 = 49 rkr1 = 4, 82 , - ($ - rkr2,3 = 1 & ) . / % t2 = 51, t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t1 = 49 7 $2&( (')'#) (#&%)! ($ !"#$! 3# ()'!%#6 A>7 !# FB7 ! t = 50 ''&! #* ').) &%)! %6) 1%( 8&)!7 $ !'&! $ # *))43 )!-$43 -)(")!: $! !"#$2 (&!3#( A rkr2,3 = 1 B A7 7 !# FB D 4 #%3"( ,%6. $%!"# ' -(&./ $(-'( #* ' -($! !"#$/ ,%( # ! ! &%8(: (# ! !- (# '( #6 +## -('.! #* $(%#+)+!/ # ,(-'()!4 4(%2 $%!"# '( $3)1 -) A4($#!434 03)$+!! (#&%)!* # *)) 3,'( B I0 = 0, 01 & ($ I0 = 0, 001 & ' ($ % - % & 0% & ) % t1 = 5 t2 = 50 t3 = 100 & +$ * rkr1 = 1 - ($ rkr2 = 1, 11 rkr3 = 1, 58 & ) %* t1 = 5 t2 = 50 t3 = 100 & +$ * rkr1 = 1 rkr2 = 2, 18 rkr3 = 5, 57 ( !# &# ('%) 2(0!$! 03)$+!! (#&%)!* (-4(47 # ' # '3:9! ((4 (4 I0 = 0, 01 ! Ω0 = 0, 4 ' 44) '4)! t1 = 5 7 t2 = 50 ! t3 = 100 %3 ")) -3%6 ( # ' # '3: #%&3:943 4)2 -(&./ 1&! "- $! !"#$!/ (&!3#7 #9)! (&( 7 2)(+!* $(9( #* ($#!434 03)$+!! (#&%)!* )( "!)( 3,'( 6 A7 7 !#B J%6. $%!"# ' -(&./ ,3&3 !4 6 (&!3# r > rkr ! &%8( (-'!'( 6#* ' )'3: 0(-3 7 $ $(83 #* ' !2()!")/ ,%(# !7 423 # -(4 )/ '* )# 6: (# '! 6#* 7 ()(%2!") '43 #%3"(:7 -(&.! # r < rkr ,3&3 (# '* 6#* ! D 4 #9)! (# '( ($8 )(") '-(# ( 67 ! 48 #)'( )("( 6#* 2)(+!* -(&./7 " ) !'"! (%6) )(,%:&(44 +##(4 -4))! 03)$+!! (#&%)!* # '4)47 %3"))/ ! I0 = 0, 001 ! Ω0 = 0, 4 7 " # ' # '3 4(%/ !) )#!')# ! ,(-'()!* -(&./7 &# ('%) )( !#I ! t1 = 5 7 t2 = 50 7 t3 = 100 J%6.!)# ' ,(-'('.!1#* -) 3# &(# ! & $! !"#$2 (-4( 8&7 "4 #9)! (# '( )(") #39# ')) 34)6.( 6#*7 ( $! !"#$!/ (&!3# (# ! A>7 !#IB 4 27 $! !"#$!/ (&!3# ,3& 3'%!"!'( 6#* 4&%)) #& )2 (-4( -)( A rkr1 = 1 7 rkr2 = 1, 11 7 rkr3 = 1, 58 B J%6.!)# ' ,(-3:9!1#* -( &./ ' +## # ( '/& ' -($! !"#$3: ,%(# 6 ! &%8! # !- (# '(A7 7 !#IB (%6)/. (-'! ! ,3& # ' # ''( 6 +##3 $(%#+)+!!7 ' -3%6 ( $ 2 ,(-3 #* 4(% $%!"# ' $3)1 -)7 -(%)*:9!1 '#6 ,E4 ! D 4 )3$%(+!* 48 )(,%:&( 6#* ! #% )("(%( +##( $(%#+)+!! 1 %&"' 2" #'' (##4 !4 "!#%)) .)! #!# 4 AB 7AFB &%* #%3"(* 42)'))2 -(8&)!* +) ' $!# (%%!-(+!! I0 = 0, 01 ! Ω0 = 0, 4 7 ) !# ")!$ (, ( ),%6. '4* ' )("(% +##( # ,%6./ !) )#!')# 6: A+## # ' # '3 , (54&%! %42'( =>?B J# $( !'.(*#* 2)(+!* -(&./ ) ,&)* (# '7 " -'%* ,(-'(' .!4#* -(&.(4 '/ ! ' -($! !"#$3: ,%(# 6 A>7 7 !# BA$! !"#$!/ (&!3# " ! ) >>I J '()'(7 (#6$!) '-(# ( rkr1,2 = 1 7 rkr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t1 = 1, 25 t2 = 10 t3 = 35 & +$ * rkr1,2 = 1 rkr3 = 1, 17 ) % % #$ % I0 = 0, 1 Ω0 = 3 3 %&".2" &, 1& "!#%))1 D$#!4) ' ,% '*'%) !&!"#$ .)! ! I0 = 0, 1 ! Ω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` >> 2$8 7 49=: 7 . 4;4 7 ) 4]: ; a / 1 5! # & 5"# ) - # # * >> ?@.$ 7 49=A 7 . =; ]B4<C 7K <46:%<44; ] bF\X\FHH c FHV EFGHIJ c _MdRI LJFHRPNJ\FSWNH WH \FSIJWFUR >> eFSIJWFUR R[WIH[I FHV SI[MHNUNGO 7 4994 7 ^NU ; 7 _ <=6 : cNfRNH g h eNVIUWHG SMI IiNUdSWNH NP XFJSW[UI RWjI VWRSJWfdSWNH VdJWHG Hd[UIFSWNH GJNTSM FHV [NFJRIHWHG >> eFSIJWFUR k[WIH[I FHV LI[MHNUNGO 7 <66D 7 ^NU <6 7 _ DD4%DD= = cNfRNH g h eNVIUUWHG XJI[WXWSFSWNH WH jWJ[NHWd\ HWNfWd\ FUUNOR >> gNdJHFU NP ld[UIFJ eFSIJWFUR <66= 7 ^NU A:: m A 7 _ D4;%D<< 9 hdSSF n eNVIUUWHG SMI oWHISW[R NP RSJFWH WHVd[IV XJI[WXWSFSWNH WH lf \W[JNFUUNOIV RSIIUR >> p[SF eFSIJWFUWF 7 <664 7 ^NU D9 m ; 7 _ :=;%:9D 46 a # ' q# # 7 5+ 8 # 49=6 # 6A6A69 LMI GIHIJFUWjIV \NVIU NP PNJ\FSWNH NP F HIT XMFRI WR [NHRWVIJIV LMI fFRW[ RSFGIR NP XJN[IRR NP GJNTSM FJI GFSMIJIV WH F \NVIU FS XMFRI SJFHRWSWNH NP SMI rJRS RNJS LMI Hd\IJW[FU RNUdSWNH NP SMI oWHISW[ IsdFSWNH NP tNooIJ%_UFH[o WR JI[IWiIV hIXIHVIH[I NP SMI RNUdSWNH NH XFJF\ISJIR NP RORSI\ WR WHiIRSWGFSIV pJIFR NP FXXUW[FfWUWSO NP FRRd\XSWNHR \FVI fO YIUVNiW[M uWPRMWSR FHV kUIjNi FJI JIiIFUIV pURN WS WR RMNTH SMFS VIXIHVWHG NH XFJF\ISJIR NP RORSI\ WS WR XNRRWfUI SN JIRIJiI fNSM IsdWUWfJWd\ VWRSJWfdSWNH FHV FdSN\NVIUUWHG VWRSJWfdSWNH NP uWPRMWSR%kUIjNi pS RN\I iFUdIR NP XFJF\ISJIR SMI IsdFSWNH MFR SMI NR[WUUFSNJO RNUdSWNH + XMFRI SJFHRWSWNH Hd[UIFSWNH [NFUIR[IH[I tNooIJ%_UFH[o oWHISW[ IsdFSWNH eFSMI\FSW[FU kdfvI[S KUFRRWr[FSWNHR+ =6p<< & . a # !# # 2 # D<]6AD 0 &*# 2 # 4 B# DC Z\FWU+ SfIRXwdV\Jd # ! # # !# # 2 # D<]6AD 0 &*# 2 # 4 B# DC Z\FWU+ iFRoWHwdVRdJd