oQ`H2bmM; *QKTmiiBQMH L2m`Qb+B2M+2
S`Q7X .`X :BHHBM Zm2Bbb2`
jyX CMm` kyR9
AM?Hibp2`x2B+?MBb
R 1BMH2BimM;
RXR qb Bbi *QKTmiiBQMH L2m`Qb+B2M+2\ X X X X X X X X X X X X X X X X X X X X
RXk m7;#2M /2` *QKTmiiBQMH L2m`Qb+B2M+2 X X X X X X X X X X X X X X X X X
RXj AM?Hi /2` oQ`H2bmM; X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X
9
9
9
9
k :`mM/H;2M /2` L2m`Q#BQHQ;B2
kXR oQM :2M2M xm o2`?Hi2M X X X X X X X X X X X X
kXk .B2 L2`p2Mx2HH2 X X X X X X X X X X X X X X X X X
kXkXR .B2 SHbKK2K#`M X X X X X X X X X X
kXkXk .2M/`Bi2M X X X X X X X X X X X X X X X X
kXkXj tQM X X X X X X X X X X X X X X X X X X X
kXkX9 aQK X X X X X X X X X X X X X X X X X X
kXkX8 w2HHF2`M X X X X X X X X X X X X X X X X X
kXkXe 1M/QTHbKiBb+?2b _2iBFmHmK U1_V X
kXkXd JBiQ+?QM/`B2M X X X X X X X X X X X X X
kXkX3 .b wviQbF2H2ii X X X X X X X X X X X X X
kXj o2`#BM/mM;2M xrBb+?2M L2`p2Mx2HH2M X X X X X
kXjXR :T CmM+iBQMb X X X X X X X X X X X X X
kXjXk *?2KBb+?2 avMTb2M X X X X X X X X X X
kX9 m7#m /2` :`QĽ?B`M`BM/2 X X X X X X X X X X X
kX8 1H2Fi`Bb+?2 1B;2Mb+?7i2M pQM L2m`QM2M X X X
kX8XR J2K#`MTQi2MiBH- J2K#`MFTxBi i
kX8Xk AQM2M;H2B+?;2rB+?i2 X X X X X X X X X X
kX8Xj :QH/KM@>Q/;FBM@Eix :H2B+?mM; X X
kX8X9 J2K#`Mbi`QK X X X X X X X X X X X X X
kX8X8 1tT2`BK2Mi2HH2` Mbix X X X X X X X X
kX8Xe J2bbmM;2M /2b _m?2TQi2MiBHb X X X X
kX8Xd 1BM [mBpH2Mi2` a+?HiF`2Bb X X X X X X
kX8X3 .b FiBQMbTQi2MiBH X X X X X X X X X X
kX8XN >Q/;FBM@>mtH2v@6Q`KHBbKmb X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
mM/ J2K#`MrB/2`biM/
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
e
e
d
d
d
3
3
3
3
N
N
N
Ry
Ry
RR
RR
Rk
Rj
R9
Rd
Rd
R3
RN
RN
ky
j JQ/2HHB2`mM; pQM L2m`QM2M A
jXR AMi2;`i2@M/@6B`2 JQ/2HH2 X X X X X X X
jXRXR G2Fv A6ĜJQ/2HH X X X X X X X X
jXRXk J2bb#`2 :`ƺĽ2M BK A6@JQ/2HH
jXRXkXR 62m2``i2 X X X X X X X X
X
X
X
X
X
X
X
X
kk
kk
kj
kj
kj
k
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
jXk
jXj
jXRXkXk aTBF2b T`Q w2Bi72Mbi2` X X X X X X X X X X
jXRXj aTMMmM;b#? M;B;2 G2Bi7 ?B;F2Bi2M X X X X X X X
jXRX9 J2K#`MFMH/vMKBF X X X X X X X X X X X X X X
.b >Q/;FBM@>mtH2v JQ/2HH X X X X X X X X X X X X X X X X
jXkXR JQ/2HH;H2B+?mM;2M X X X X X X X X X X X X X X X X X
jXkXk S`K2i2` X X X X X X X X X X X X X X X X X X X X X X
jXkXj wmbKK2M7bbmM; X X X X X X X X X X X X X X X X X
JQ/2HHB2`mM; pQM AQM2MFM H2M KBi /2K J`FQp@JQ/2HH
jXjXR HH;2K2BM2 6Q`KmHB2`mM; X X X X X X X X X X X X X
jXjXk aT2xB2HH2 J`FQp@JQ/2HH2 X X X X X X X X X X X X X
jXjXkXR Gƺb2M /2` J`FQp@:H2B+?mM;2M X X X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
k8
ke
kd
k3
k3
kN
kN
kN
jy
jR
jj
9 JQ/2HHB2`mM; pQM L2m`QM2M AA, P`i@ mM/ x2Bi#? M;B;2 JQ/2HH2
9XR .B2 E#2H;H2B+?mM; X X X X X X X X X X X X X X X X X X X X X X X X
9XRXR _M/@ mM/ o2`xr2B;mM;b#2/BM;mM;2M X X X X X X X X X
9XRXk w2BimM#? M;B;2 GƺbmM; /2` E#2H;H2B+?mM; X X X X X X
9Xk .b _HH@JQ/2HH X X X X X X X X X X X X X X X X X X X X X X X X X
9Xj JmHiB@*QKT`iK2Mi JQ/2HH2 X X X X X X X X X X X X X X X X X X
9X9 oQM JBF`QbFQTB2/i2M xm /BbF`2i2M :Bii2`M X X X X X X X X X X
9X9XR "BH/p2``#2BimM; X X X X X X X X X X X X X X X X X X X X X
9X9XRXR 1BM7+?2 :H iimM;b}Hi2` X X X X X X X X X X X X
9X9XRXk .BzmbBQMb}Hi2` X X X X X X X X X X X X X X X X X
9X8 :Bii2`;2M2`B2`mM; mb ;2}Hi2`i2M "BH//i2M X X X X X X X X X X
9X8XR AbQ~ +?2M#2`2+?MmM; X X X X X X X X X X X X X X X X X X X
9X8Xk oQM "BH/ xm :Bii2` X X X X X X X X X X X X X X X X X X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
j8
j8
jd
jN
9R
9R
9j
9j
9j
99
93
93
93
8 Gƺb2M /2` 6BHi2`@ mM/ JQ/2HH;H2B+?mM;2M
8XR .b JQ/2HHT`Q#H2K X X X X X X X X X X
8XRXR o2`7?`2M BM R. X X X X X X X
8XRXk .Bz2`2Mx2Mp2`7?`2M BM k. X
8Xk 6BMBi2 1H2K2Mi2 o2`7?`2M X X X X X X
8XkXR PT2`iQ`2M X X X X X X X X X X
8XkXk a+?r+?2 6Q`KmHB2`mM; X X X
8XkXj :H2`FBM@o2`7?`2M X X X X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
9N
9N
8y
89
88
8e
8d
8d
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
e JQ/2HHB2`mM; pQM L2m`QM2M AAA, j.@JQ/2HH2 pQM x2HHmH `2M mM/ bm#x2HHmH `2M
S`Qx2bb2M
ey
j
R 1BMH2BimM;
RXR qb Bbi *QKTmiiBQMH L2m`Qb+B2M+2\
RXk m7;#2M /2` *QKTmiiBQMH L2m`Qb+B2M+2
qb #2/2mi2i *QKTmiiBQM\
biM/ xm;`mM/2X
.2` FiBpBi i mMb2`2b :2?B`Mb HB2;i 2BM /BbF`2i2` wm@
qB2 bBM/ /B2 :`ƺĽ2M *Q/B2`mM; mM/ .vMKBF BM /2` K2bb#`2M M2m`QMH2M FiBpBi i
p2`b+?HɃbb2Hi\ MirQ`i ;2#2M FMM UKM+?KHV 2BM BMi2`/BbxBTHBM `2` Mbix U*QKTm@
iiBQMH L2m`Qb+B2M+2V,
o2`2BMB;mM; pQM .BbxBTHBM2M rB2 "BQHQ;B2- J2/BxBM- Ji?2KiBF- S?vbBF- AM7Q`KiBFX
RXj AM?Hi /2` oQ`H2bmM;
RX "BQHQ;Bb+?2 :`mM/H;2M
kX 1BM7Ƀ?`mM; BM /B2 i?2Q`2iBb+?2 L2m`QrBbb2Mb+?7i
9
V :`mM/H;2M
#V 1BM7+?2 JQ/2HH2 UF2BM2 ` mKHB+?2 m~ƺbmM;V
+V :`mM/H;2M xm ` mKHB+? m7;2Hƺbi2M JQ/2HH2M
jX .2iBHHB2`i2 JQ/2HHB2`mM;, 1`7bbmM; /2` ` mKHB+?2M mM/ x2BiHB+?2M .vMKBF M2m@
`QMH2` S`Qx2bb2
V 1BM7Ƀ?`mM; BM /B2 h?2Q`B2 /2` T`iB2HH2M .Bz2`2MiBH;H2B+?mM;2M
#V LmK2`Bb+?2 .BbF`2iBbB2`mM;bp2`7?`2M
+V GƺbmM;bp2`7?`2M 7Ƀ` ;`QĽ2 :H2B+?mM;bbvbi2K2 UpHHXV
/V "BH/p2``#2BimM; mM/ _2FQMbi`mFiBQMbp2`7?`2M 7Ƀ` x2HHmH `2 P`;M2HH2M
9X "2?M/HmM; pQM h?2K2M mb Fim2HH2` 6Q`b+?mM; UMr2M/mM; pQM kX mM/ jXV
LQiBx, "BQHQ;B2 mM/ LmK2`BF #H2B#2M ?B2` mMpQHHbi M/B;5 .B2 #BQHQ;Bb+?2M :`mM/H;2M
bBM/ D2/Q+? mb`2B+?2M/ 7Ƀ` /B2 ?B2` #2?M/2Hi2M h?2K2MX 6Ƀ` 2BM2 mb7Ƀ?`HB+?2 "2@
?M/HmM; /2` LmK2`BF bQrB2 JQ/2HHB2`mM; mM/ aBKmHiBQM b2B m7 M/2`2 oQ`H2bmM;2M
p2`rB2b2MX
8
k :`mM/H;2M /2` L2m`Q#BQHQ;B2
kXR oQM :2M2M xm o2`?Hi2M
Molekulare Ebene
Gene
(biochemische Messungen)
überlebenswichtige Proteine (Kanäle,
Membran, Messenger ...)
Zelluläre Ebene
Neuronen
uroonen
(Elektrophysiologie,
Imaging-Verfahren)
Kanalaktivität, Signalempfang, Signalweiterleitung
Synaptische
aptische
Kopplung
pplung
Synaptische Kopplung,
Neurotransmitter, Rezeptoren
Netzwerk Ebene
(Multielektrodensysteme)
kleine Ne
Netzwerke
etzwer
Zusammenfassung von funktionellen Einheiten
Areale
(MRT, EEG ...)
Gehirnregionen
regionen und
lokale Sch
Schaltkreise
altkre
Verbindung
ndung vo
von
Gehirnarealen
Gehirnareale
rnarealen
Makroskopische Informationsverarbeitung
Global
(Beobachtung)
Verhalten
1b 2tBbiB2`2M mMi2`b+?B2/HB+?2 aFH2M #2B /2` "2i`+?imM; pQM AM7Q`KiBQMbp2``#2BimM;
BK :2?B`M,
• KQH2FmH`2 aFH U#BQ+?2KBb+?2 lMi2`bm+?mM;2MV
• x2HHmH `2 aFH U1H2Fi`QT?vbBQHQ;B2- 1BMx2Hx2HH#H2BimM;2M- AK;BM;p2`7?`2MV
• FH2BM2 a+?HiF`2Bb2 UbXQXV
• L2ixr2`F2 UJmHiB2H2Fi`Q/2Mbvbi2K2V
• `2H2 UJ_h- 11:- XXXV
e
• o2`?Hi2M U"2Q#+?imM;V
6Q`b+?mM;b;2#B2i2 bBM/ m7 2BMx2HM2 aFH2M mb;2`B+?i2i- b2?` b+?rB2`B; xm` w2Bi Bbi /B2
1`7bbmM; /2` ɂ#2`; M;2 xrBb+?2M /2M aFH2M5
kXk .B2 L2`p2Mx2HH2
w2HHi?2Q`B2,
Abi /b :2?B`M 2BM iQTQHQ;Bb+? xmbKK2M? M;2M/2b L2ixr2`F Q/2` #2bi2?i 2b mb 2BM@
x2HM2M EQKTQM2Mi2M- /B2 KBi2BMM/2` FQKKmMBxB2`2M\
:QH;B@o2`7?`2M UR338V, EHBmK@/B+?`QKi Y aBH#2`MBi`i `2;B2`i BM /2` w2HH2 xm aBH#2`@
+?`QKi ⇒ 6 `#mM; 2BMx2HM2` w2HH2M BK L2ixr2`F- //m`+? Hbb2M bB+? /B2 2H2K2Mi`2M
1BM?2Bi2M Uw2HH2MV /`bi2HH2MX
kXkXR .B2 SHbKK2K#`M
.B2 SHbKK2K#`M Bbi /b /2}MB2`2M/2 1H2K2Mi 2BM2` w2HH2X aB2 i`2MMi /2M BMi`@ mM/
2ti`x2HHmH `2M _mK KBi >BH72 B?`2` "BHBTB/@ai`mFim`X
wmbKK2Mb2ixmM;,
88 W S`Qi2BM2
k8 W S?QbT?QHBTB/2
Rj W *?QH2bi2`BM
9 W GBTB/2
j W EQ?H2M?v/`i2
S`Qi2BM2 #BH/2M /2M "mbi2BM 7Ƀ` EM H2 UTbbBp Y FiBpV- /m`+? /B2 b2H2FiBp /2}MB2`i2
AQM2M xrBb+?2M AMi`@ mM/ 1ti`x2HHmH ``mK mb;2imb+?i r2`/2M FƺMM2M U/B2b Bbi 2BM
r2b2MiHB+?2` bT2Fi #2B /2` JQ/2HHB2`mM; x2HHmH `2` S`Qx2bb2VX
kXkXk .2M/`Bi2M
• p2`xr2B;i2 E#2Hbi`mFim`- KBi /ɃMM2` r2`/2M/2M Ÿbi2M- D2 r2Bi2` KM bB+? pQK
aQK r2;#2r2;i
• TbbBp2 J2K#`M2B;2Mb+?7i2M UaB;MH/ KT7mM; r ?`2M/ aB;MHmb#`2BimM;V
• 1KT7 M;2` 7Ƀ` aB;MH2 M/2`2` w2HH2M
• q2Bi2`H2BimM; 2H2Fi`Bb+?2` aB;MH2 xmK aQK mM/ tQM
d
kXkXj tQM
• ?MHB+?2 ai`mFim` rB2 .2M/`Bi2M- FH2BM2`2 E#2H/m`+?K2bb2`
• FiBp2 J2K#`M2B;2Mb+?7i UF2BM2 aB;MH/ KT7mM;V
• Jv2HBMmK?ɃHHmM; ⇒ b+?M2HH2`2 aB;MH;2b+?rBM/B;F2Bi
• aB;MHr2Bi2`H2BimM; M p2`#mM/2M2 w2HH2M
kXkX9 aQK
• #2?2`#2`;i /2M w2HHF2`M- 2M/QTHbKiBb+?2b _2iBFmHmK mM/ M/2`2 P`;M2HH2M UJB@
iQ+?QM/`B2M- :QH;B@TT`i- XXXV
• ;`ƺĽi2` h2BH /2b x2HHmH `2M oQHmK2Mb
• p2`#BM/2i .2M/`Bi2M mM/ tQM
• ;27ɃHHi KBi wviQbQH
kXkX8 w2HHF2`M
• #2}M/2i bB+? BK aQK /2` w2HH2
• AK w2HHF2`M Bbi /B2 .L m7;2rB+F2HiX
• .m`+? E2`MTQ`2M BM /2` "BHBTB/K2K#`M FƺMM2M AQM2M /BzmM/B2`2M- r2H+?2 7Ƀ` /B2
w2HH2 Ƀ#2`H2#2MbrB+?iB;2 #BQ+?2KBb+?2 S`Qx2bb2 FiBpB2`2M FƺMM2M UxX"X EHxBmKVX
• AK w2HHF2`M rB`/ xmbKK2M KBi /2K 2M/QTHbKiBb+?2M _2iBFmHmK /B2 .L BM
S`Qi2BM2 mK;2rM/2Hi UJ2K#`Mm7#m- EM H2- _2x2TiQ`2M 2i+XV → SHbiBxBi iX
kXkXe 1M/QTHbKiBb+?2b _2iBFmHmK U1_V
• mĽ2`2 J2K#`M /2b w2HHF2`Mb 2`bi`2+Fi bB+? BM /b 1_
• `m?2b 1_, avMi?2b2 pQM S`Qi2BM2M
• ;Hii2b 1_, S`Q/mFiBQM #2biBKKi2` GBTB/2- 62iib m`2M mM/ m7M?K2 pQM EHxBmK
• ;`QĽ2` EHxBmKbT2B+?2`, 1_@J2K#`M 2`Kƺ;HB+?i mbimb+? pQM EHxBmK xrB@
b+?2M wviQbQH mM/ 1_ Ua1_*- ASj _- _v_V → *A*_X
• 2`bi`2+Fi bB+? r2Bi BM /B2 .2M/`Bi2M
3
kXkXd JBiQ+?QM/`B2M
• 1M2`;B2p2`bQ`;mM;, ɂ#2` bi`F ;27Hi2i2 BMM2`2 J2K#`M rB`/ 2BM S`QiQM2M;`/B@
2Mi m7;2#miX .`Ƀ#2` rB`/ /B2 hS@S`Q/mFiBQM M;2i`B2#2M UxX"X wBi`QM2Mb m@
`2xvFHmbVX
• EHxBmKbT2B+?2`
kXkX3 .b wviQbF2H2ii
.b wviQbF2H2ii Bbi /b h`MbTQ`iM2ix /2` w2HH2X 1b #2bi2?i mb /vMKBb+?2M 6BHK2Mi2MX
1BM/BK2MbBQMH2 ai`mFim`2M, JBF`Qim#mHB
• kyMK .m`+?K2bb2`
• JQiQ`T`Qi2BM2 #2r2;2M JQH2FɃH2 mM/ P`;M2HH2M 2MiHM; pQM JBF`Qim#mHB
wr2B/BK2MbBQMH2 ai`mFim`2M, FiBM mM/ L2m`Q}HK2Mi2
SQHvK2`BbiBQM pQM FiBM 2`x2m;i 2BM2 k.@Ji`Bt- r2H+?2 BM /2` _2;2H KBi /2` w2HHK2K@
#`M p2`#mM/2M Bbi U6Q`Kbi#BHBi i /2` w2HH2VX
L2m`Q}HK2Mi2
• #BH/2M /B2 E2`MHKBM UBK w2HHF2`MV• #2ii2M /B2 .L 2BM-
• bi#BHBbB2`2M /B2 E2`MKQ`T?QHQ;B2X
.`2B/BK2MbBQMH2 ai`mFim`2M
o2`#BM/mM;2M xrBb+?2M R.@ mM/ k.@wviQbF2H2ii#mbi2BM2M HB272`i /2` w2HH2 2BM2 /`2B/B@
K2MbBQMH2 AM7`bi`mFim` 7Ƀ` /2M h`MbTQ`i pQM JQH2FɃH2M mM/ P`;M2HH2MX
kXj o2`#BM/mM;2M xrBb+?2M L2`p2Mx2HH2M
qB2 bBM/ L2`p2Mx2HH2M KBi2BMM/2` p2`#mM/2M\ P/2` xm2`bi, qMM FƺMM2M L2`p2Mx2HH2M
2BM2 o2`#BM/mM; 2BM;2?2M\
_Ƀ+F#HB+F, tQM H2Bi2i aB;MH r2Bi2`- .2M/`Bi2M 2KT7M;2M ⇒ tQM mM/ .2M/`Bi2M
KɃbb2M BM EQMiFi i`2i2M- ` mKHB+?2b E`Bi2`BmKX
N
kXjXR :T CmM+iBQMb
AM "2`2B+?2M- BM /2M2M L2`p2M#?M2M /B+?i mM/ T`HH2H p2`Hm72M- FƺMM2M bB+? #2B 2BM2K
#biM/ pQM k@9MK :T CmM+iBQMb mb#BH/2MX
:T CmM+iBQM,
• :T CmM+iBQMb bBM/ S`Qi2BMFQKTH2t2- r2H+?2 i`Mbx2HHmH `2M AQM2Mmbimb+? 2`@
Kƺ;HB+?2MX
• b2?` b+?M2HH2 aB;MHɃ#2`i`;mM;
• ;2`BM;2 a2H2FiBpBi i #2B /2` aB;MHɃ#2`i`;mM;
Z, q`mK MBKKi /B2 .B+?i2 /2` :T CmM+iBQMb #2B ?ƺ?2`2M G2#2r2b2M #\
, ;2`BM;2 a2H2FiBpBi i, :C@;2FQTT2Hi2 w2HH2M H2Bi2M aB;MH BKK2` r2Bi2` ⇒ EQKTH2tB@
i ib`2/mFiBQM5
1`?ƺ?mM; /2` EQKTH2tBi i → +?2KBb+?2 avMTb2MX
kXjXk *?2KBb+?2 avMTb2M
EQKTH2t2`2 aB;MHr2Bi2`H2BimM; Hb #2B :T CmM+iBQMbX
hvT2BMi2BHmM;, M+? `i /2b L2m`Qi`MbKBii2`b- ;`Q#, ?2KK2M/2 Q/2` 2``2;2M/2 avMT@
b2MX
S` bvMTb2
• L2m`Qi`MbKBii2` ;2HM;i Ƀ#2` /b tQM Q/2` /B2 T` bvMTiBb+?2 J2K#`M BM /2M
T` bvMTiBb+?2M "QmiQM• lKb2ixmM; M /2` o2bBF2HQ#2`~ +?2 xm bT2xB}b+?2K L2m`Qi`MbKBii2` mM/ "27ɃH@
HmM; /2` o2bBF2H• o2bBF2H rM/2`M xm T` bvMTiBb+?2` J2K#`MX
1tQxviQb2
• 6mbBQM /2` o2bBF2HK2K#`M KBi T` bvMTiBb+?2` J2K#`M,
FiBQMbTQi2MiBH → ŸM/2`mM; /2` J2K#`MbTMMmM; → ƹzMmM; pQM EHxBmKF@
M H2M → EHxBmK2BMbi`QK → 6mbBQM
1M/QxviQb2
• Jbb2M2`?HimM; #2B o2bBF2HM- /X?X M+? 1tQxviQb2 KɃbb2M M2m2 o2bBF2H 2Mibi2?2MX
.B2b2` S`Qx2bb rB`/ Hb 1M/QxviQb2 #2x2B+?M2i, mb /2` 7mbBQMB2`i2M J2K#`M UHH2
S`Qi2BMFQKTH2t2 pQ`?M/2MV rB`/ M2#2M /2` +iBp2 xQM2 J2K#`M #;2b+?MɃ`imK M2m2 UH22`2V o2bBF2H xm #m2MX .B2b2 r2`/2M Mb+?HB2Ľ2M/ #27ɃHHiX
Ry
aB;MHɃ#2`i`;mM; /m`+? /2M bvMTiBb+?2M aTHi
• 6`2B;2b2ixi2` L2m`Qi`MbKBii2` /BzmM/B2`i xm` TQbibvMTiBb+?2M J2K#`M• FiBpB2`mM; TQbibvMTiBb+?2` _2x2TiQ`2M• ƹzMmM; BQM2MbT2xB}b+?2` EM H2-
• ŸM/2`mM; /2` J2K#`MbTMMmM;-
• aB;MHr2Bi2`H2BimM; BM TQbibvMTiBb+?2` w2HH2X
kX9 m7#m /2` :`QĽ?B`M`BM/2
:`Q#2 1BMi2BHmM; BM
• T`BK `2 mM/ b2FmM/ `2 b2MbQ`Bb+?2 EQ`i2t`2H2• T`BK `2 mM/ b2FmM/ `2 KQiQ`Bb+?2 EQ`i2t`2H2-
• bbQxBiBQMbFQ`iBx2b UaT`+?2- AM7Q`KiBQMbp2``#2BimM;XXXVX
AM /B2 hB272, b+?B+?ir2Bb2 MQ`/MmM; /2` w2HH2MX a+?B+?i2M r2`/2M M+? /2K w2HHivT
#2MMMi- /2` bB+? /Q`i #2}M/2iX
> m};bi2` w2HHivT, Sv`KB/Hx2HH2M U3yWVX
> m}; mMi2`bm+?i2 :2?B`M`2;BQM,
>BTTQ+KTmb
• KĽ;2#HB+? #2i2BHB;i #2B G2`M2M mM/ :2/ +?iMBb-
• b2?` THM`2 ai`mFim`- /X?X 2tT2`BK2Mi2HH b2?` ;mi xm; M;HB+?
kX8 1H2Fi`Bb+?2 1B;2Mb+?7i2M pQM L2m`QM2M
AQM2MFM H2 BM /2` SHbKK2K#`M bQ`;2M 7Ƀ` /2M mbimb+? bT2xB}b+?2` AQM2MX .2`
mbimb+? 2`7QH;i 2MiHM; pQM EQMx2Mi`iBQMb@ mM/ aTMMmM;b;`/B2Mi2MX AQM2MTmKT2M
`#2Bi2M 2Mi;2;2M /2K :2bKi;`/B2Mi2M U1M2`;B2p2`#`m+?VX
EiBQM2M
E+
L+
*2+
J;2+
EiBQM2M iQiH
AMM2M (KJ)
Rk9
Ry
8
R9
R8j
mĽ2M (KJ)
k
Rk8
k
R
Rjy
RR
MBQM2M
*H−
>*P−
3
M/2`2
AMM2M (KJ)
k
Rk
RjN
mĽ2M (KJ)
dd
kd
ke
MBQM2M iQiH
R8j
Rjy
kX8XR J2K#`MTQi2MiBH- J2K#`MFTxBi i mM/ J2K#`MrB/2`biM/
J2K#`MTQi2MiBH M /2` SHbKK2K#`M #mi bB+? 2BM2 SQi2MiBH/Bz2`2Mx xrBb+?2M
BMi`@ mM/ 2ti`x2HHmH `2K _mK m7- /b J2K#`MTQi2MiBHX .b SQi2MiBH BK mĽ2M@
`mK rB`/ T2` EQMp2MiBQM m7 y ;2b2ixiX
.b J2K#`MTQi2MiBH p`BB2`i BK P`i ⇒ w2HH2M bBM/ MB+?i BbQTQi2MiBHX ⇒ AQM2M~mbb
/m`+? /B2 w2HH2 2MiHM; /2b SQi2MiBH;`/B2Mi2MX
aix RX a2B IL /2` tBH2 AQM2M~mbb- RL /2` tBH2 qB/2`biM/ mM/ rL /2` BMi`x2HHmH `2
bT2xB}b+?2 qB/2`biM/ 2BM2b wvHBM/2`b KBi /2` G M;2 L mM/ "bBb~ +?2 πa2 X "2B x = 0
HB2;2 /B2 aTMMmM; V1 M- #2B x = L /B2 aTMMmM; V2 X .MM ;BHi,
V2 − V1
RL
L
= rL 2
πa
IL =
RL
UkXRV
UkXkV
1H2Fi`QiQMBb+?2 1B;2Mb+?7i2M r2`/2M /m`+? /B2 6Q`K /2` w2HH2 #22BM~mbbiX
Z, qB2 rB`Fi bB+? /B2 ŸM/2`mM; /2b J2K#`MTQi2MiBHb BM w2HH2 mM/ " mb\
, JQ`T?QHQ;Bb+? FQKTFi2 w2HH2M bBM/ m+? 2H2Fi`QiQMBb+? FQKTFiX w2HH2 " FMM /?2`
M ?2`mM;br2Bb2 Hb BbQTQi2MiBH M;2MQKK2M r2`/2MX
⇒ 1BM7+?bi2b JQ/2HH, w2HH2 rB`/ Hb 2BM2 1BM?2Bi KQ/2HHB2`iX
J2K#`MFTxBi i .B2 J2K#`M FMM Hb EQM/2MbiQ` BMi2`T`2iB2`i r2`/2MX
aix kX a2B Cm /B2 J2K#`MFTxBi i- Q /B2 G/mM;b/Bz2`2Mx mM/ Vm /B2 J2K@
#`MbTMMmM;- /MM ;BHi
Q = C m Vm
Cm Bbi /#2B T`QTQ`iBQMH xm` w2HHQ#2`~ +?2 AX
Rk
UkXjV
.2}MBiBQM RX JBi cm rB`/ /B2 bT2xB}b+?2 J2K#`MFTxBi i #2x2B+?M2i,
cm :=
Cm
A
UkX9V
.B2b2 HB2;i ivTBb+?2`r2Bb2 #2B cm = 10nF /mm2 X
J2K#`MrB/2`biM/ q2MM 2BM2 w2HH2 FiBp Bbi- #2/2mi2i /B2b- B?` J2K#`MTQi2MiBH
#2}M/2i bB+? MB+?i BK _m?2TQi2MiBH- /X?X mK bB+? mb /2K _m?2xmbiM/ xm #2r2;2MKmbb /2` J2K#`MrB/2`biM/ Ƀ#2`rmM/2M r2`/2MX
P?KǶb+?2b :2b2ix,
Rm =
∆V
Ie
UkX8V
Ie b2B /#2B /2` ai`QK- /2` 2ir pQM 2BM2` 1H2Fi`Q/2 BMDBxB2`i rB`/- ∆V /B2 SQi2MiB@
H/Bz2`2MxX "2B L2`p2Mx2HH2M HB2;i /2` J2K#`MrB/2`biM/ ivTBb+?2`r2Bb2 BK "2`2B+?
Ry JǕ #Bb Ryy JǕX
.2}MBiBQM kX aT2xB}b+?2 :`ƺĽ2M,
rm , bT2xB}b+?2` qB/2`biM/
τm , J2K#`M@w2BiFQMbi2
.B2 J2K#`Mx2BiFQMbiMi2 τm Bbi /2}MB2`i Hb
τm := rm cm .
kX8Xk AQM2M;H2B+?;2rB+?i2
M /2` SHbKK2K#`M ?2``b+?i 2BM EQMx2Mi`iBQMb;`/B2Mi bQrB2 2BM 2H2Fi`Bb+?2` :`@
/B2MiX "2}M/2M bB+? /B2 ;`/B2Mi2M#? M;B;2M mbimb+?T`Qx2bb2 BK :H2B+?;2rB+?i- /MM
Bbi /2` L2iiQ@AQM2M~mbb MmHHX
"2`2+?MmM; /2b _m?2TQi2MiBHb, M /2` J2K#`M 2tBbiB2`i 2BM SQi2MiBHbT`mM; UVm =
Vi − Va V- /X?X 2BM2 1M2`;B2#``B2`2X
aix jX 1BM h2BH+?2M KBi G/mM; z · q #2MƺiB;i 2BM2 1M2`;B2 E pQM KBM/2bi2Mb
E ≥ −zqV
KBi
x, h2BH+?2MpH2Mx UMx?H TQbBiBp2` S`QiQM2MH/mM;2MV
[, S`QiQM2MH/mM;
xm` ɂ#2`[m2`mM; /2` J2K#`MX
Rj
UkXeV
.B2 q?`b+?2BMHB+?F2Bi /7Ƀ`- /bb 2BM h2BH+?2M /B2b2 FBM2iBb+?2 1M2`;B2 #2bBixi- #2`2+?@
M2i bB+? mb /2` "QHixKMM@o2`i2BHmM; /2` q?`b+?2BMHB+?F2Bib/B+?i2M,
aix 9X a2B T /B2 #bQHmi2 h2KT2`im` mM/ kb /B2 "QHixKMM@EQMbiMi2X .MM ;BHi,
P (E ≥ −zqV ) = 2tT(zqV /kb T )
UkXdV
.2}MBiBQM jX a2B R /B2 mMBp2`b2HH2 :bFQMbiMi2 mM/ F /B2 6`/v@EQMbiMi2X .B2
_272`2MxbTMMmM; VT b2B /2}MB2`i Hb /B2 aTMMmM;- /B2 #2MƺiB;i rB`/- mK 2BM 2BM7+?
TQbBiBp ;2H/2M2b h2BH+?2M UKBi /2` KBiiH2`2M 1M2`;B2 kb T V Ƀ#2` /B2 J2K#`M xm i`Mb@
TQ`iB2`2M,
!
"
R·T
kb T
VT =
=
UkX3V
F
q
AK :H2B+?;2rB+?i UV = Vgg V Kmbb /B2 Mx?H /2` h2BH+?2M- /B2 /B2 MQir2M/B;2 1M2`;B2
xm` ɂ#2`[m2`mM; /2` J2K#`M ?#2M- m7 #2B/2M a2Bi2M /2` J2K#`M ;H2B+? b2BMX m7
2BM2` a2Bi2 U?B2`, mĽ2MV #2bBix2M HH2 h2BH+?2M /B2 MQir2M/B;2 1M2`;B2 Uh`MbTQ`i KBi
/2K :`/B2Mi2MV- m7 /2` M/2`2M Mm` 2BM Mi2BH,
[mĽ2M] · 1 = [BMM2M] · P (E ≥ −zqV )
⇔ [mĽ2M] = [BMM2M] 2tT(zVgg /VT )
!
"
VT
[mĽ2M]
⇔ Vgg =
HM
(L2`Mbi@:H2B+?mM;)
z
[BMM2M]
kX8Xj :QH/KM@>Q/;FBM@Eix :H2B+?mM;
.B2 L2`Mbi@:H2B+?mM; #2`2+?M2i /b _m?2TQi2MiBH #2B "2i2BHB;mM; 2BM2b AQM2MivTbX
S`Q#H2K, mMi2`b+?B2/HB+?2 EM H2- p2`b+?B2/2M2 AQM2MivT2M
⇒ :>E@:H2B+?mM;
%
%

PK [K + ]a + PA [A− ]i
RT  K
A

%
lKF2?`TQi2MiBH Vm =
HM %
UkXNV
+
F
PK [K ]i + PA [A− ]a
K
A
>2`H2BimM; UM;2/2mi2iV,
• J2K#`M BQM2MT2`K2#2H
• AQM2M FƺMM2M T2` .BzmbBQM U6B+Fb+?2b :2b2ixV mM/ T2` 2H2Fi`Bb+?2M 6Hmbb U#2@
2BM~mbbi pQK 2H2Fi`Bb+?2K 62H/V /B2 J2K#`M 2MiHM; /2` EQQ`/BMi2 t TbbB2`2MX
6Ƀ` /B2 :2bKiBQM2M~mbb/B+?i2 jIon #2B MM?K2 2BM2b FQMbiMi2M 2H2Fi`Bb+?2M
62H/2b BMM2`?H# /2` J2K#`M ;BHi,
!
"
d [AQM] zAQM F Vm
(∗) jAQM = −DAQM
−
·
[AQM]
UkXRyV
dx
RT
L
z : oH2Mx
L : J2K#`M/B+F2
R9
>2`H2BimM; /2` :>E@:H2B+?mM; Umb7Ƀ?`HB+?V,
wr2B S`Qx2bb2 #2rB`F2M /B2 qM/2`mM; pQM AQM2M,
RX .BzmbBQM U6B+Fb+?2b :2b2ixVkX 2H2Fi`Bb+?2 q2+?b2HrB`FmM; U/2}MB2`i /m`+? 2H2Fi`Bb+?2b 62H/ #xrX SQi2MiBHV
.B2 :2bKi~mbb/B+?i2 2`;B#i bB+? Hb amKK2 #2B/2` h2BH~mbb/B+?i2M,


 d [AQM] zAQM F Vm


jAQM = −DAQM 
 dx − RT · L [AQM]
) *+ , )
*+
,
UkXRRV
2H2Fi`Bb+?
/B77mbBp
.B2b /2}MB2`i 2BM R@/BK2MbBQMH2b S`Q#H2K b2MF`2+?i xm` J2K#`MX
GƺbmM;,
(2.11)
⇔
AMi2;`iBQM Ƀ#2` J2K#`M,
⇒
⇔
⇔
⇔
ˆL
− DjII +
d[I]
dx
zI F
RT
·
Vm
L [I]
=1
ˆL
d[I]
dx
zI F
RT
dx =
1 dx
− DjII +
· VLm [I]
0
0
. !
"
!
"/
RT L
jI
zI F Vm
jI
zI F Vm
HM −
+
·
[I]a − HM −
+
·
[I]i
= L
zI F Vm
DI
RT L
DI
RT L
0 j
1
IF
− DII + zRT
· VLm [I]a
zI F Vm
= HM
IF
RT
− DjII + zRT
· VLm [I]i
e
zI F Vm
RT
=
− DjII +
− DjII +
zI F
RT
zI F
RT
·
·
Vm
L [I]a
Vm
L [I]i
F Vm
:
!RT
"
jI
zI F Vm
jI
zI F Vm
zI µ
e
−
+
·
[I]i = −
+
·
[I]a
DI
RT L
DI
RT L
DI zI µ [I]i · ezI µ − [I]a
jI = −
L
(1 − ezI µ )
a2B µ /2}MB2`i Hb µ :=
⇔
⇔
PI =
DI
⇔L
jI = PI · zI · µ ·
[I]a − [I]i · ezI µ
(1 − ezI µ )
UkXRkV
.2}MBiBQM 9X .B2 2H2Fi`Bb+?2 ai`QK/B+?i2 JI - /B2 pQK 6Hmbb /2` AQM2M I ?2`pQ`;2`m72M
rB`/- Bbi /2}MB2`i /m`+? JI := zI F · jI X
R8
JM #2i`+?i2i /2M :H2B+?;2rB+?ibxmbiM/ J;2b =
UJQMQpH2Mx- xX"X L+ - E+ - *H− V,
%
I
JI = 0 mM/ /2M 6HH nI = ±1
6Ƀ` EiBQM2M K UzK = +1V 2`;B#i bB+? mb kXRk
JK = PK µF
7Ƀ` MBQM2M A UzA = −1V ;BHi
[K]a − [K]i eµ
,
1 − eµ
[A]a − [A]i e−µ
[A]i − [A]a eµ
=
P
µF
.
A
1 − e−µ
1 − eµ
%
a2ixi KM #2B/2 mb/`Ƀ+F2 BM I JI = 0 2BM mM/ ;`mTTB2`i mK- 2`? Hi KM




JA = PA µF




2
2
2
 Fµ
2
 F µeµ



·
P
[K]
+
P
[A]
·
=
P
[K]
+
P
[A]
a
i
i
a
K
A
K
A

 1 − eµ

 1 − eµ
K



)
*+ A
,
)K
*+ A
,
=:u
⇔
⇔
⇔
⇔
u
e =
v
u
µ = HM
v
=:v
µ
%
%

PK [K]a + PA [A]i
F Vm
A

%
%
= HM  K
RT
( PK [K]i + PA [A]a
K
A
%
%

PK [K]a + PA [A]i
RT  K
A

%
Vm =
HM %
F
PK [K]i + PA [A]a
K
UkXRjV
A
"2BbTB2H2,
RX L+ - E+ - *H−
Vm
RT
=
HM
F
kX L+
Vm
!
PN a+ [N a+ ]a + PK + [K + ]a + PCl− [Cl− ]i
PN a+ [N a+ ]i + PK + [K + ]i + PCl− [Cl− ]a
RT
=
HM
F
⇒ L2`Mbi@:H2B+?mM;
!
PN a+ [N a+ ]a
PN a+ [N a+ ]i
Re
"
RT
=
HM
F
!
[N a+ ]a
[N a+ ]i
"
"
UkXR9V
UkXR8V
kX8X9 J2K#`Mbi`QK
.2}MBiBQM 8X Hb J2K#`Mbi`QK #2x2B+?M2i KM /2M :2bKiH/mM;b~mbb Ƀ#2` /B2
SHbKK2K#`MX
EQMp2MiBQM, SQbBiBp2` ai`QK- r2MM TQbBiBp2 G/mM;2M pQM 1ti`@ xm AMi2`x2HHmH ``mK
rM/2`Mc M2;iBp2` ai`QK- r2MM mK;2F2?`iX
.2}MBiBQM eX 6QH;2M/2 :`ƺĽ2M }M/2M ? m}; o2`r2M/mM;,
V − Ei
im =
2
i
gi
gi (V − Ei )
i`2B#2M/2 E`7i 7Ƀ` AQM i
G2Bi7 ?B;F2Bi UT`Q 6H +?2V
J2K#`Mbi`QK
kX8X8 1tT2`BK2Mi2HH2` Mbix
Rd
kX8Xe J2bbmM;2M /2b _m?2TQi2MiBHb
1tT2`BK2Mi2HH2 1`KBiiHmM; /2` lKF2?`TQi2MiBH2 BK o2`;H2B+? xm L2`Mbi@ mM/ :>E@
q2`i2MX
L2`Mbi;H2B+?mM; HB272`i 7Ƀ` FH2BM2 L+ b+?H2+?i2 q2`i2X
EN a = +55 Ko
EK = −75 Ko
R3
:>E 7Ƀ` L+ mM/ E+ , Vm ≈ −60Ko
PN a+ ≈
1
25
· PK +
kX8Xd 1BM [mBpH2Mi2` a+?HiF`2Bb
kX8X3 .b FiBQMbTQi2MiBH
"2Q#+?imM; U*QH2 *m`iBb RNjNV,
Z, qb #2rB`Fi 2BM2 .2TQH`BbB2`mM; #Bb 8y Ko \
• ŸM/2`mM; /2` :2bKiT2`K2#BHBi i\ L2BM- / #2B pƺHHB;2` .m`+?H bbB;F2Bi .2TQH@
`BbiBQM Mm` #Bb y KoX
• L@EM H2 UL2`Mbi, EN a+ ≈ 55 Ko V
RN
1tT2`BK2Mi2HH2 lMi2`bm+?mM;,
• ŸM/2`mM; /2` 2ti`x2HHmH `2M L+ @EQMx2Mi`iBQM ⇒ ŸM/2`mM; /2` KTHBim/2 UF2B@
M2 ŸM/2`mM; BK _m?2TQi2MiBHV
1`FH `mM; pQM *QH2 *m`iBb,
• ŸM/2`mM; BM /2` J2K#`MbTMMmM; #2rB`Fi ŸM/2`mM; BM /2` L+ @G2Bi7 ?B;F2BiX
1H2Fi`Q+?2KBb+?2` :`/B2Mi pQM L+ #2rB`Fi 1BMbi`QK pQM L+ @AQM2MX
• L+ @AQM2MFM H2 bBM/ bTMMmM;b#? M;B; UaTMMmM;bb2MbQ` BK S`Qi2BMFQKTH2tVX
kX8XN >Q/;FBM@>mtH2v@6Q`KHBbKmb
M;2bi`2#i rB`/ 2BM q2+?b2H pQM /2` "2b+?`2B#mM; /2` 1BMFMH/vMKBF xm` "2b+?`2B@
#mM; 2BM2` KF`QbFQTBb+?2M EMH/vMKBF mM/ /B2 "2Q#+?imM; /2` x2BiHB+?2M ŸM@
/2`mM; pQM G2Bi7 ?B;F2Bi mM/ G/mM;b~mbb /m`+? /B2 J2K#`MX
1tT2`BK2Mi2HH2` Mbix,
62bib2ix2M /2b J2K#`MTQi2MiBHb m7 mMi2`b+?B2/HB+?2M q2`i2MX
1tT R, .2TQH`BbB2`mM; mK ey Ko BM hBMi2M}b+?@tQM
UV mMi2` MQ`KH2M "2/BM;mM;2M
U#V Q?M2 L+ @AQM2M
U+V #2`2+?M2i2` L+ @"2Bi`;
1tT k, Vm = 60 − 150 Ko
UV mMi2` MQ`KH2M "2/BM;mM;2M
U#V Y hhs, mbb+?Hi2M /2` L+ @EM H2
U+V Y h1, mbb+?Hi2M /2` E+ @EM H2
AMi2`T`2iiBQM,
E+ ,
• JBi xmM2?K2M/2K Vm MBKKi /B2 G2Bi7 ?B;F2Bi /2` E+ @EM H2 xmX
• bT i2b 1BMb2ix2M
L+ ,
• Vm ↑ ⇒ L+ @G2Bi7 ?B;F2Bi ↑ ⇒ Vm ↑↑ ⇒ G2Bi7 ?B;F2Bi ↓ UL+ @EMH FiBpB2@
`mM; Y AMFiBpB2`mM;VX
• 7`Ƀ?2b 1BMb2ix2M
ky
.b FiBQMbTQi2MiBH
.b FiBQMbTQi2MiBH rB`/ BK q2b2MiHB+?2M #2biBKKi pQM /2` FiBpB2`mM; mM/ AMF@
iBpB2`mM; /2b L+ @EMHb bQrB2 pQM /2` E+ @EMHFiBpBi iX
⇒ >Q/;FBM@>mtH2v@JQ/2HH UbT i2`V
lMi2`b+?B2/2 K tQM,
RX mMKv2HBMB2`i2b tQM
kX Kv2HBMB2`i2b tQM
kR
j JQ/2HHB2`mM; pQM L2m`QM2M A
AK 2`bi2M ETBi2H Ƀ#2` /B2 JQ/2HHB2`mM; pQM L2m`QM2M #27bb2M rB` mMb KBi /2` 2BM@
7+?bi2M JQ/2HH/`bi2HHmM; pQM L2m`QM2M- BM /2` /b L2m`QM Hb 2BM2 1BM?2Bi KQ/2HHB2`i
rB`/ mM/ bQKBi Ƀ#2` F2BM2 ` mKHB+?2 m~ƺbmM; p2`7Ƀ;iX
JQiBpiBQM, JM+?2 w2HH2M bBM/ 2H2Fi`QiQMBb+? FQKTFiX
m7;#2, "2b+?`2B#mM; /2b wmbKK2M?M;b xrBb+?2M J2K#`MbTMMmM; mM/ G/mM;b@
p2`b+?B2#mM;2M Ƀ#2` /B2 J2K#`MX
1b ;BHi /B2 FTxBiBp2 "2b+?`2B#mM; /2` J2K#`M
UjXRV
Cm V = Q
mM/ /KBi 7Ƀ` /B2 x2BiHB+?2 o2` M/2`mM; pQM V
Cm
dV
dQ
=
.
dt
dt
UjXkV
.2` ai`QK dQ
dt rB`/ p2`m`b+?i /m`+? bvMTiBb+?2 ai`ƺK2- ai`ƺK2 Ƀ#2` J2K#`MFM H2
Q/2` m+? 1H2Fi`Q/2M UBK 1tT2`BK2MiVX qB` x2`H2;2M /2M ai`QK ?B2` BM 2BM2M /m`+? /b
SQi2MiBH ?2`pQ`;2`m72M2M 2H2Fi`Bb+?2M ai`QK Im U/2` Mim`;2K Ľ /2` 2B;2M2M l`b+?2
2Mi;2;2MrB`Fi- /?2` BM /2` .Bz2`2MiBH;H2B+?mM; KBi M2;iBp2K oQ`x2B+?2M p2`b2?2MV
mM/ 2BM2M 2ti2`M2M U1BM@Vai`QK Ie pQM TQbBiBp2M G/mM;2M,
dQ
= −Im + Ie
dt
dV
⇒ Cm
= −Im + Ie
dt
dV
Ie
⇔ cm
= −im +
dt
A
UjXjV
(1`BMM2`mM;, im =
2
i
gi (V − Ei ))
jXR AMi2;`i2@M/@6B`2 JQ/2HH2
o2`2BM7+?mM;, .B2 "BQT?vbBF /2` FiBQMbTQi2MiBH2 rB`/ MB+?i KQ/2HHB2`iX
⇒ .b JQ/2HH #2b+?`2B#i SQi2MiBHp2`H m72 #Bb xmK a+?r2HHr2`i UpQM 2ir −8y KoVX
"2B 1``2B+?2M /2b a+?r2HHr2`ib rB`/ /b SQi2MiBH m7 2BM2M Dz_m?2r2`idz xm`Ƀ+F;2b2ixiX
kk
jXRXR G2Fv A6ĜJQ/2HH
AK H2Fv AMi2;`i2@M/@6B`2@JQ/2HH r2`/2M Mm` TbbBp2 1B;2Mb+?7i2M /2` J2K#`M KQ@
/2HHB2`i,
dV
dt
dV
τm
)*+,
dt
cm
⇔
rm = g1
L
=cm rm
=
−gL (V − EL ) +
=
E L − V + Rm I e
Ie
A
UjX9V
mM/
V = Vth ! S mM/ wm`Ƀ+Fb2ix2M m7 V = Vreset
UjX8V
Gƺb2M /2` .Bz2`2MiBH;H2B+?mM;
RX Ie FQMbiMi, MHviBb+?2 GƺbmM;
V (t) = EL + Rm Ie + (V (to ) − (EL + Rm Ie )) e−
t−t0
τm
UjXeV
kX Ie x2Bi#? M;B;, .B2 GƺbmM; rB`/ M /BbF`2i2M w2BiTmMFi2M tk = t0 + k · ∆t, k ∈ N
#2`2+?M2i- BM/2K m7 D2/2K AMi2`pHH [tk , tk+1 ] /2` ai`QK Ie /m`+? 2BM2 EQMbiMi2
(k)
Ie M;2M ?2`i rB`/,
∆t
V (tk+1 ) = EL + Rm Ie + (V (tk ) − (EL + Rm Ie(k) )) e− τm .
6Ƀ` #2HB2#B;2b t ∈ [tk , tk+1 ] ?i KM /MM
V (t) = EL + Rm Ie + (V (tk ) − (EL + Rm Ie(k) )) e−
t−tk
τm
.
6Ƀ` ∆t → 0 FQMp2`;B2`i /B2 /BbF`2i2 GƺbmM; ;2;2M /B2 FQMiBMmB2`HB+?2X
jXRXk J2bb#`2 :`ƺĽ2M BK A6@JQ/2HH
jXRXkXR 62m2``i2
.2}MBiBQM dX .B2 62m2``i2 2BM2b L2m`QMb rB`/ /2}MB2`i Hb
r7B`2 :=
1
t7B`2
.
"2biBKKmM;,
• 2tT2`BK2Mi2HH,
IBMD FQMbiMi, J2bb2M /2` w2Bi#bi M/2 xrBb+?2M Sb (t7B`2 )
kj
UjXdV
• "2`2+?MmM; pQM r7B`2 mb A6@JQ/2HH,
JM 7Q`/2`i- /bb M+? 2BM2K S mM/ /2K //m`+? 2`7QH;i2M _2b2i m7 V`2b2i 2tFi
/B2 w2Bi t7B`2 p2`;2?i- #2pQ` rB2/2` /B2 J2K#`MbTMMmM; Vi? 2``2B+?i rB`/- 7Ƀ` /B2
2BM S mb;2Hƺbi rB`/,
!
V (t7B`2 ) = Vi? = EL + Rm IBMD + (V`2b2i − EL − Rm IBMD ) 2tT
!
"
t
Vi? − (EL + Rm IBMD )
− τ7B`2
m
⇔
= 2tT
V`2b2i − EL − Rm IBMD
!
"
V`2b2i − EL − Rm IBMD
⇔ t7B`2 = τm · HM
Vi? − EL − Rm IBMD
!
!
""−1
V`2b2i − EL − Rm IBMD
⇒ r7B`2 = τm HM
Vi? − EL − Rm IBMD
−
t7B`2
τm
"
UjX3V
DzS`Q#H2K2dz,
RX HM(1) = 0, V`2b2i += Vi?
kX HM(x), 1b Kmbb x > 0 ;2Hi2M,
⇒
V`2b2i − EL − Rm IBMD
>0!
Vi? − EL + Rm IBMD
L+?r2Bb, AK HH;2K2BM2M b2ixi KM V`2b2i < EL - / /m`+? V`2b2i /B2 >vT2`TQH`BbiB@
QM M+? 2BM2K S M+?;2bi2HHi rB`/X . IBMD > 0- Bbi HbQ /2` w ?H2` /2b GQ;`Bi?Kmb@
`;mK2Mib BKK2` M2;iBp- 2MibT`2+?2M/ Kmbb m+? /2` L2MM2` M2;iBp b2BM- /KBi /2`
mb/`m+F /2}MB2`i Bbi, Rm IBMD > Vi? − EL X .B2b2 6Q`/2`mM; Kmbb #2` Q?M2?BM 2`7ɃHHi
b2BM- /KBi /2` "2;`Bz /2` 62m2``i2 bBMMpQHH Bbi- /2MM Bbi bB2 MB+?i 2`7ɃHHi- `2B+?i /2`
1BMbi`QK MB+?i mb- mK /b J2K#`MTQi2MiBH Ƀ#2`?mTi m7 /2M MQir2M/B;2M a+?r2HH@
r2`i Vi? Mxm?2#2M- bQKBi rB`/ ;` MB+?i ;272m2`iX .?2` #2`2+?M2i KM /B2 62m2``i2
2MibT`2+?2M/ KBi
3 4
4
55−1
V
−EL −Rm IBMD
τm HM V`2b2i
, Rm IBMD > Vi? − E
−E
−R
I
m BMD
L
i?
r7B`2 =
.
0
bQMbi
6Ƀ` ;`QĽ2 q2`i2 pQM IBMD FMM KM /B2 62m``i2 HBM2` #b+? ix2M,
!
"
!
"
V`2b2i − EL − Rm IBMD
V`2b2i − Vi?
V`2b2i − Vi?
HM
= HM 1 +
≈
,
Vi? − EL − Rm IBMD
Vi? − EL − Rm IBMD
Vi? − EL − Rm IBMD
HbQ r7B`2 ≈
Vi? −EL −Rm IBMD
τm (V`2b2i −Vi? ) X
"2B /B2b2` #b+? ixmM; rB`/ /B2 L ?2`mM; HM(1 + x) ≈ x 7Ƀ` FH2BM2 x p2`r2M/2iX .B2b2
rB`/ mb /2` hvHQ`2MirB+FHmM; ;2rQMM2M,
k9
HH;2K2BM, "2B ?BM`2B+?2M/2` :Hii?2Bi /2` 6mMFiBQM f ;BHi BM 2BM2` mXlX b2?` FH2BM2M
lK;2#mM;R mK a /B2 1MirB+FHmM;
f " (a)
f "" (a)
f """ (a)
(x − a) +
(x − a)2 +
(x − a)3 + . . . .
1!
1!
1!
1MirB+F2Hi KM f (x) = HM(1 + x) mK 0- 2`? Hi KM
f (x) = f (a) +
ln" (1 + 0)
HM(1 + x) = HM(1 + 0) +
(x − 0) + · · · ≈ x.
) *+ , ) *+
1! ,
=0
=1
w2BiHB+?2 o2` M/2`mM; pQM 62m2``i2M
62m2``i2M bBM/ BM _2HBi i MB+?i FQMbiMiX
Z, qQ/m`+? p2` M/2`M bB+? 62m2``i2M\
, FiBp2 EM H2 /2TQH`BbB2`2M Q/2` ?vT2`TQH`BbB2`2M ⇒ K + p2`FH2BM2`i KBi FiBpB2@
`mM; /B2 62m2``i2 UK + ?vT2`TQH`BbB2`iV
1`r2Bi2`mM; /2b TbbBp2M A6@JQ/2HHb mK K + @EM H2,
JM 7Ƀ?`i 2BM2M EY @mbbi`QKi2`K BM /2` .Bz2`2MiBH;H2B+?mM; 2BM,
τm
dV
= EL − V − rm gb` (V − EK ) +Rm Ie
)
*+
,
dt
UjXNV
M2m5
MM?K2 7Ƀ` G2Bi7 ?B;F2Bi gb` , 2tTQM2MiB2HH2` #7HH, τb` dgdtb` = −gb` X
lT/i2 M+? D2/2K w2Bib+?`Bii, gb` −→ gb` + ∆gb`
jXRXkXk aTBF2b T`Q w2Bi72Mbi2`
JM KM #2i`+?i2i /B2 KBiiH2`2 62m2``i2 Si BMM2`?H# pQM w2Bi72Mbi2`M Ti = [ti , ti+1 ],
Si :=
O (aTBF2b BM [ti , ti+1 ])
ti+1 − ti
UjXRyV
o2`;H2B+? /2` Si (i = 1 . . . n) HB272`i pHHX AM7Q`KiBQM Ƀ#2` /B2 o2``#2BimM; pQM _2Bx2MX
"2B ŸM/2`mM; /2` w2Bi72Mbi2` Tik H bbi bB+? Kƺ;HB+?2 AMp`BMx /2` Sik mMi2` k mMi2`bm@
+?2MX
"2BbTB2H oBbm2HH2` AMTmi UxX"X 6BHKVX
lMi2`bm+?2
RX 62m2``i2M 2BMx2HM2` L2m`QM2M Uo2`i2BHmM;V
kX Si /2` L2m`QM2M Uo2`i2BHmM;V
jX Sik UAMp`BMx\V
⇒ wmQ`/MmM; 2BM2` ax2M2 U:2bB+?i- GM/b+?7iV xm Q#B;2M :`ƺĽ2MX
R
.B2 DzlK;2#mM;dz FMM BK mM;ɃMbiB;bi2M 6HH Mm` mb /2K SmMFi a b2H#bi #2bi2?2M UmM/ /KBi mb
Ki?2KiBb+?2` aB+?i F2BM2 2B;2MiHB+?2 lK;2#mM; b2BMV- #2BK GQ;`Bi?Kmb Bbi /b #2` MB+?i /2` 6HHX
k8
A6 Y _27`Fi `T?b2
RX 62biH2;mM; 2BM2b w2Bi72Mbi2`b- BM /2K F2BM aTBF2 Dz2`Hm#idz rB`/X
kX xmb ixHB+?2 h2`K2 KBi ;22B;M2i2K
dg
dt
wmbKK2M7bbmM;
RX A6 2BM7+? xm #2`2+?M2M
kX biiBbiBb+?2 AM7Q`KiBQM2M rB2 62m2``i2M- AMi2`bTBF2@AMi2`pHb- aTBF2@*QmMi .Bb@
i`B#miBQMb X X X
biiBbiBb+?2 .i2M ↔ AM7Q`KiBQMb+Q/B2`mM;
jX F2BM2 ` mKHB+?2 m~ƺbmM;
9X F2BM2 JQ/2HHB2`mM; /2` #BQT?vbBFHBb+?2M 1B;2Mb+?7i2M pQM L2m`QM2M UrB2 p2``@
#2Bi2i /B2 w2HH2\V
jX Y 9X 7Ƀ?`2M xm 2BM2` ;2Mm2`2M "2b+?`2B#mM; /2` J2K#`M/vMKBFX
jXRXj aTMMmM;b#? M;B;2 G2Bi7 ?B;F2Bi2M
.B2 ŸM/2`mM; /2b J2K#`MTQi2MiBHb 7Ƀ?`i xm biQ+?biBb+?2K ƹzMmM;b@ mM/ a+?HB2Ľ@
p2`?Hi2M /2` EM H2X
JQ/2HHB2`mM;, qB2 ? M;i /B2b2b o2`?Hi2M pQM 6FiQ`2M rB2 J2K#`MbTMMmM;- L2m@
`Qi`MbKBii2`M UavMTb2V- *2+ 2i+X #\
1`BMM2`mM;, >>- "2b+?`2B#mM; /2b o2`?Hi2Mb pB2H2` EM H2X
⇒ /2i2`KBMBbiBb+?2` UFQMiBMmB2`HB+?2`V Mbix 7Ƀ` EMH/B+?i2M U;2`2+?i72`iB;i /m`+? ?Q@
?2 w?H /2` EM H2V
mbM?K2M, SQbi@bvMTiBb+?2 _2x2TiQ`2M, KM+?KH b2?` r2MB;2X
MM?K2, ƹzMmM;br?`b+?2BMHB+?F2Bi 2BM2b EMHb ≈ Mi2BH Qz2M2` EM H2 `2HiBp xm
/2` :2bKix?H HH2` EM H2X
ƹzMmM;br?`b+?2BMHB+?F2Bi mM/ G2Bi7 ?B;F2Bi
.2}MBiBQM 3X 6QH;2M/2 :`ƺĽ2M r2`/2M p2`r2M/2i,
EM H2 hvT i)
Pi := O(Qz2M2
O(EM H2 hvT i)
gi : KtBKH2 G2Bi7 ?B;F2Bi /2` EM H2 /2b hvTb igi := gi · Pi : 2z2FiBp2 G2Bi7 ?B;F2Bi /2` EM H2 /2b hvTb BX
ke
jXRX9 J2K#`MFMH/vMKBF
qB` mMi2`b+?2B/2M BK HH;2K2BM2M xrBb+?2M T2`bBbi2Mi2M mM/ i`MbB2Mi2M EM H2MX
UV S2`bBbi2Mi2 EM H2 PK (Vm ) MBKKi KBi .2TQH`BbB2`mM; xm- KBi >vT2`TQH`BbB2@
`mM; #X
.B2 6mMFiBQMbr2Bb2 /2b EMHb Ua+?rBM;iɃ`V rB`/ KBi >BH72 pQM k FɃMbiHB+?2M EMH@
mMi2`2BM?2Bi2M KQ/2HHB2`iX wm D2/2K k 2tBbiB2`i 2BM2 ƹzMmM;br?`b+?2BMHB+?F2Bi nX .B2
q?`b+?2BMHB+?F2Bi- /bb 2BM EMH Qz2M Bbi- Bbi /2KM+?
PK = nk
UjXRRV
n ∈ [0, 1]
k Kmbb mb 2tT2`BK2Mi2HH2M .i2M #2biBKKi r2`/2M- xX"X k = 4 7Ƀ` E+ @EMHX
.2}MBiBQM NX FiBpB2`mM;b@ mM/ .2FiBpB2`mM;b`i2 rB`/ KBi αn (V ) mM/ βn (V ) MQiB2`iX
qB2 M/2`i bB+? n\
⇔
KBi
"2`2+?MmM; pQM αn mM/ βn
dn
= αn (V )(1 − n) − βn (V )n
dt
dn
τn (V )
= n∞ (V ) − n
dt
1
τn (V ) =
αn (V ) + βn (V )
αn (V )
n∞ =
αn (V ) + βn (V )
UjXRkV
UjXRjV
UjXR9V
UjXR8V
FiBpB2`mM;, ɂ#2`rBM/mM; 2BM2` 1M2`;B2#``B2`2 qBα V
Bα Bbi #? M;B; pQM /2` G/mM; mM/ /2` h`MbTQ`ibi`2+F2 UJ2K#`M/B+F2VX
1M2`;B2M 2tT (−qBα /kB T ) `2B+?2M xm` ɂ#2`rBM/mM; /2` 1M2`;B2#``B2`2 mbX
⇒ αn (V ) := Aα 2tT(−qBα /kB T )(≡ Aα 2tT(−Bα V /VT )
UjXReV
Aα rB`/ /#2B mb 2BM2K 2tT2`BK2Mi2HH2M 6BiiBM; ;2rQMM2MX
MHQ;,
βn (V ) := Aβ 2tT(−qBβ V /kB T )
⇒ n∞ (V ) =
1
1+
βn
αn (V
)
=
1+
kd
!
Aβ
Aα
UjXRdV
· 2tT
1
!
(Bα −Bβ )·V
VT
""
U"V h`MbB2Mi2 EM H2
• k S`Qx2bb2 bi2m2`M /B2 ƹzMmM; Ua+?HB2ĽmM;V /2` EM H2X
• .B2b2 bBM/ 2Mi;2;2M;2b2ixi bTMMmM;b#? M;B;X
ƹzMmM;br?`b+?2BMHB+?F2Bi U"bTX L+ @EMHV,
RX a+?rBM;iɃ`, MHQ; xm T2`bBbi2Mi2K EMH, mk (xX"X k = 3) → FiBpB2`mM;
kX Em;2H, AMFiBpB2`mM; KBi q?`b+?2BMHB+?F2Bi hi (xX"X i = 1)
→ PN a+ = m3 h
ŸM/2`mM; pQM m mM/ h,
dm
= αm (V )(1 − m) − βm (V ) · m
dt
dh
= αh (V )(1 − h) − βh (V ) · h
dt
UjXR3V
UjXRNV
αm , αh ; βm , βh bBM/ MHQ; xm αn mM/ βm /2}MB2`iX
τm (V )
dm
dt
UjXkyV
= m∞ (V ) − m
τm (V ) =
m∞ (V ) =
1
αm (V ) + βm (V ))
αm (V )
αm (V ) + βm (V )
UjXkRV
UjXkkV
jXk .b >Q/;FBM@>mtH2v JQ/2HH
.b >Q/;FBM@>mtH2v JQ/2HH #2b+?`2B#i /B2 L+ @ mM/ E+ @G2Bi7 ?B;F2Bi2M mM/ /2M /KBi
p2`#mM/2M2M J2K#`Mbi`QKX
jXkXR JQ/2HH;H2B+?mM;2M
RX
im = g i (V − EL ) + gK n4 (V − EK ) + g N a m3 h(V − EN a )
)
*+
, )
*+
, )
*+
,
UjXkjV
g K = y.je Ka KK
UjXk8V
G2+Fbi`QK
EHBmKbi`QK
g i = y.yyj Ka KK−k
Li`BmKbi`QK
EL = −89.j3d Ko
EK = −dd Ko
−k
g N a = R.k Ka KK−k
EN a = +8y Ko
k3
UjXk9V
UjXkeV
kX
Cm
dV
Ie
= −im +
dt
A
UjXkdV
jX
dm
dt
dn
τn (V )
dt
dh
τh (V )
dt
τm (V )
= m∞ (V ) − m
UjXk3V
= n∞ (V ) − n
UjXkNV
= h∞ (V ) − h
UjXjyV
jXkXk S`K2i2`
0.01(V + 55)
1 − 2tT(−0.1 (V + 55))
βn (V ) = 0.125 2tT(−0.0125 (V + 65))
0.1(V + 40)
αm (V ) =
1 − 2tT(−0.1 (V + 40))
βm (V ) = 4 2tT(−0.0556 (V + 65))
UjXjRV
αh (V ) = 0.07 2tT(−0.05 (V + 65))
1
βh (V ) =
1 + 2tT(−0.1 (V + 35))
UjXj8V
αn (V ) =
UjXjkV
UjXjjV
UjXj9V
UjXjeV
wm #2+?i2M Bbi- /bb /B2 _2FiBQMb`i2M ?B2` BM 1BM?2Bi2M pQM Kb−R mM/ /b J2K#`M@
TQi2MiBH BM 1BM?2Bi2M pQM Ko xm #2?M/2HM bBM/X
jXkXj wmbKK2M7bbmM;
• ;mi2 "2b+?`2B#mM; /2` FiBp2M 1B;2Mb+?7i2M pQM L+ @ mM/ E+ @EM H2M
• JQ/2HHB2`mM; /2b ;2bKi2M SQi2MiBHp2`Hm7b UMB+?i Mm` bm#@i?`2b?QH/V
• #Bb?2` MQ+? F2BM2 ` mKHB+?2 mb/2?MmM; US@mb#`2BimM; m7 ;2bKi2K tQMV
• F2BM2 /2iBHHB2`i2 JQ/2HHB2`mM; pQM 1ti`@ mM/ AMi`x2HHmH ``mK US`Q#H2K, EQTT@
HmM; 2H2Fi`Bb+?2b aB;MH KBi bm#x2HHmH `2M S`Qx2bb2MV
jXj JQ/2HHB2`mM; pQM AQM2MFM H2M KBi /2K J`FQp@JQ/2HH
"2BbTB2H, _2Bxr2Bi2`H2BimM; M /2` avMTb2
RX FiBQMbTQi2MiBH 2``2B+?i avMTb2 UL+ - E+ V
kX o2` M/2`mM; /2` J2K#`MbTMMmM;
→ *2+ @ EM H2 ƺzM2M
kN
jX 6mbBQM pQM o2bBF2HM- mbb+?ɃiimM; pQM L2m`Qi`MbKBii2`M U:"- :HmiKi- X X X V
BM bvMTiBb+?2M aTHi
9X FiBpB2`mM; TQbibvMTiBb+?2` _2x2TiQ`2M
→ ƹzMmM; pQM L+ - E+ @ EM H2M
→ TQbibvMTiBb+?2` ai`QK
.#2B bBM/ UR@jV T` bvMTiBb+?2 S`Qx2bb2 UbT i2`V mM/ U9V /B2 TQbibvMTiBb+?2 _2Bxr2B@
i2`H2BimM;X
P7i bBM/ M /2` AQM2M/vMKBF M 2BM2` J2K#`M EM H2 #2i2BHB;i- /B2 B?`2 EQM7Q`KiBQM
BM #? M;B;F2Bi pQK J2K#`MTQi2MiBH Q/2` pQK "BM/2M 2BM2b #2biBKKi2M "Qi2MbiQzb
M BM bB2 BMi2;`B2`i2 _2x2TiQ`2M M/2`M UBK 2BM7+?bi2M 6HH ;B#i 2b /B2 xr2B EQM7Q`K@
iBQM2M Qz2M mM/ ;2b+?HQbb2MVX
jXjXR HH;2K2BM2 6Q`KmHB2`mM;
A/22, "2b+?`2B#mM; /2` EQM7Q`KiBQMbɃ#2`; M;2 MHQ; xm +?2KBb+?2M _2FiBQM2MX
JBi Si (i = 1 . . . n) b2B2M /B2 /BbF`2i2M S`Qi2BMFQM7Q`KiBQM2M #2x2B+?M2i- xrBb+?2M /2@
M2M +?2KBb+?2 _2FiBQM2M #2biBKKi2 ɂ#2`; M;2 /2}MB2`2M, Si ! Sj , (i, j) ∈ {1, . . . , n}2 X
.B2 wmbiM/br?`b+?2BMHB+?F2Bi P (Si , t) #2b+?`2B#2 /B2 q?`b+?2BMHB+?F2Bi- /bb 2BM S`Q@
i2BM xmK w2BiTmMFi t /B2 EQM7Q`KiBQM Si 2BMMBKKiX .B2 x2BiHB+?2 1MirB+FHmM; /B2b2`
:`ƺĽ2 FMM Hb :H2B+?;2rB+?ibT`Qx2bb #2b+?`B2#2M r2`/2M,
n
n
j=1
j=1
2
dP (Si , t) 2
=
P (Sj , t)P (Sj → Si ) −
P (Si , t)P (Si → Sj )
dt
UjXjdV
oQM 2BM2K EMH xm pB2H2M
a2B si /2` Mi2BH /2` EM H2 BK wmbiM/ Si M /2` :2bKiK2M;2X
6Ƀ` (i, j) ∈ {1 . . . n}2 ;2#2M /B2 _2FiBQMb`i2M rij #xrX rji /2` _2FiBQM2M xrBb+?2M Si
mM/ Sj ;2K Ľ /2` _2FiBQM Si "
rij
rji
!S
MX L+? /2K Jbb2MrB`FmM;b;2b2ix ;BHi /MM
j
7Ƀ` /B2 Mi2BH2 /2` EM H2 BK wmbiM/ i /B2 .Bz2`2MiBH;H2B+?mM;
n
n
j=1
j=1
2
dsi 2
=
sj rji −
si rij
dt
UJ`FQp@EBM2iBFV.
UjXj3V
aTMMmM;b#? M;B;2 EM H2
lMi2` bTMMmM;b#? M;B;2M EM H2M p2`bi2?i KM bQH+?2- /2`2M _i2M rij pQM /2` J2K@
#`MbTMMmM; V #? M;2MX 1b ;BHi,
Si "
rij (V )
!S .
j
rji (V )
jy
UjXjNV
.2` ɂ#2`;M; pQM Si M+? Sj 2`7Q`/2`2 /B2 1M2`;B2 Uij X .2` Mi2BH /2` EM H2 BK wm@
biM/ Si - /B2 Ƀ#2` KBM/2bi2Mb /B2b2 1M2`;B2 p2`7Ƀ;2M- Bbi ;2;2#2M /m`+? 2tT(−Uij /kb T )
U"QHixKMM@6FiQ`V- /KBi ;BHi 7Ƀ` /B2 _2FiBQMb`i2
rij (V ) = Rij 2tT(−Uij (V )/kb T )
UjX9yV
KBi /2` "QHixKMM@EQMbiMi2 kb mM/ /2` h2KT2`im` h mM/ 2BM2` KtBKH2M lKbix@
`i2 Rij X
.B2 "2biBKKmM; pQM Uij (V ) Bbi BK HH;2K2BM2M b+?rB2`B;- KM #2?BH7i bB+? KBi 2BM2`
HBM2`2M TT`QtBKiBQM
Uij (V ) ≈ c0 + c1 V
UjX9RV
⇒ rij (V ) = Rij 2tT(−Uij (V )/kb T )
UjX9kV
= Rij 2tT(−(c0 + c1 V )/kb T ) = Rij 2tT
a2ix2
−c0
aij := Rij 2tT kb T ,
bij :=
!
V
⇒ rij (V ) = aij · 2tT −
bij
KBi EQMbiMi2M aij mM/ bij X
−c0
kb T
· 2tT
−c1 V
kb T
kb T
.
c1
"
UjX9jV
jXjXk aT2xB2HH2 J`FQp@JQ/2HH2
RX wr2B EMHxmbi M/2
Hb 2BM7+?bi2b J`FQp@JQ/2HH H bbi bB+? 2BM wr2B@wmbiM/b@JQ/2HH 7Q`KmHB2`2MC"
r1 (V )
r2 (V )
!O
UjX99V
BM r2H+?2K /2` EMH xrBb+?2M ;2b+?HQbb2M mM/ Qz2M r2+?b2HM FMMX .b >Q/;FBM@
>mtH2v@JQ/2HH H bbi bB+? m7 /B2b2 q2Bb2 Hb aT2xBH7HH pQM J`FQp@JQ/2HH2M M2m
BMi2`T`2iB2`2M,
m"
h"
αm (V )
! m∗ ,
βm (V )
αh (V )
! h∗ .
βh (V )
/B2 EMHH2Bi7 ?B;F2Bi #2`2+?M2i bB+? /MM Hb o = m3 hX
jR
UjX98V
UjX9eV
kX .`2B EMHxmbi M/2
Hb M +?bi2b H bbi bB+? MM2?K2M- /bb 2BM EMH /`2B /BbF`2i2 wmbi M/2 2BMM2?K2M
FMM, ;2b+?HQbb2M U*V- BMFiBp UAV mM/ Qz2M UPV,
r1
!O
%
!!!! r6 r2 r4 """""
!!!!
"
"
""""
r5 !!!
!!# "$"""" r3
C &!!" !
UjX9dV
I
.B2 _i2M r1 , . . . , r6 FƺMM2M bTMMmM;b#? M;B; b2BM Q/2` FƺMM2M LmHH b2BM- D2
M+?/2K- Q# 2BM EMH mMB/B`2FiBQMH wmbiM/b M/2`mM; pQ`M2?K2M FMMX 1t@
T2`BK2Mi2HH2 .i2M xm p2`b+?B2/2M2M EMHivT2M 2tBbiB2`2M mM/ FQMMi2M ;2Mmixi
r2`/2M- mK bT2xB}b+?2 J`FQp@JQ/2HH2 xm 2MirB+F2HMX
"2BbTB2H2,
V LJ.
C &!!" !
%O
!!!! r6 r2 r4 """
!!!!
"
""
r5 !!!
!!# ""
UjX93V
I
r1 = 0
r2 = e.N b−R
r3 = 0
r4 = Rey b−R
r5 = 9.d b−R
r6 = RNy b−R KJ−R
#V :"
r1
!O
%
!!!! r6 r2 r4 """""
!!!!
"
"
""""
r5 !!!
!!# "$"""" r3
C &!!" !
UjX9NV
I
r1 = R8y b−R KJ−R
r2 = kyy b−R
r3 = kk b−R
r4 = RR b−R
r5 = j9 b−R
r6 = RNy b−R KJ−R
jX JmHiBxmbiM/bKQ/2HH2
Hb "2BbTB2H 2BM2b JmHiBxmbiM/bKQ/2HH b2B /b JQ/2HH pQM oM/2M#2`; mM/ "2x@
MBHH URNNRV /2b Li`BmKFMHb ;2MMMi,
C"
r5
r6
!C "
1
r5
r6
!C
(2
r6
O
( "
r7
r4
r3
!C "
4
r2
r1
r9
'
I"
r5
'
!C
(3
r8
r1
r2
!I "
4
jk
r3
r4
r10
'
!I
3
UjX8yV
jXjXkXR Gƺb2M /2` J`FQp@:H2B+?mM;2M
AM J`FQp@JQ/2HH2M H bbi bB+? 7Ƀ` D2/2M wmbiM/ UD2/2 lM#2FMMi2V 2BM2 .Bz2`2MiBH@
;H2B+?mM; m7bi2HH2MX
EBM2iBF2M 2`bi2` P`/MmM;
AK 6HH pQM EBM2iBF2M 2`bi2` P`/MmM;
C"
r1
r2
!O
UjX8RV
2`;B#i bB+? 7QH;2M/2b avbi2K,
dC
= r2 · O − r1 · C
UjX8kV
dt
dO
= r1 · C − r2 · O
UjX8jV
dt
lK /2M Mi2BH Qz2M2` EM H2 xm #2biBKK2M- Bbi /B2 xr2Bi2 :H2B+?mM; xm Hƺb2MX C H bbi
bB+? b+?`2B#2M Hb (1 − O)- bQKBi 2`;B#i bB+? 2BM2 .Bz2`2MiBH;H2B+?mM; /2` 6Q`K
dO
= r1 · (1 − O) − r2 · O
UjX89V
dt
JBi /2` M7M;b#2/BM;mM; O(t0 ) = O∗ H bbi bB+? /B2 GƺbmM; MHviBb+? M;2#2M,
KBi
O(t) = O∞ + K1 2tT (−(t − t0 )/τ1 )
UjX88V
K1 = O ∗ − O ∞
r1
O∞ =
r 1 + r2
1
τ1 =
r 1 + r2
UjX8eV
EQMi`QHH@_2+?MmM;,
dO
dt
!
"
1
1
1
= K1 −
2tT(−t/τ1 ) − O∞ + O∞
τ1
τ1
τ1
!
"
1
1
=
−
O(t) + O∞
τ1
τ1
= (r1 + r2 ) · O∞ − (r1 + r2 )O(t)
= r1 − (r1 + r2 )O(t).
O(t0 ) = O∞ + K1 · 1
= O∞ + O∗ − O∞
= O∗ .
jj
UjX8dV
UjX83V
EBM2iBF2M xr2Bi2` P`/MmM;
AK 6HH pQM EBM2iBF2M xr2Bi2` P`/MmM;
r1
!O
%
!!!! r6 r2 r4 """""
!!!!
"
"
""""
r5 !!!
!!# "$"""" r3
C &!!" !
UjX8NV
I
2`;2#2M bB+? 7Ƀ` /B2 lM#2FMMi2M O mM/ I 7QH;2M/2 :H2B+?mM;2M,
dO
dt
dI
dt
= r1 (1 − O − I) − (r2 + r3 ) O + r4 I
UjXeyV
= r6 (1 − O − I) − (r4 + r5 ) I + r3 O
UjXeRV
JBi /2M M7M;b#2/BM;mM;2M O(t0 ) = O∗ mM/ I(t0 ) = I ∗ 2`;2#2M bB+? /B2 HH;2K2BM2M
GƺbmM;2M
O(t − t0 ) = O∞ + K1 2tT(−(t − t0 )/τ1 ) + K2 2tT(−(t − t0 )/τ2 )
I(t − t0 ) = I
∞
+ K3 2tT(−(t − t0 )/τ1 ) + K4 2tT(−(t − t0 )/τ2
UjXekV
UjXejV
KBi
K1 =
(O∗ − O∞ )(a + 1/τ2 ) + b(I ∗ − I ∞ )
1
1
τ2 − τ1
K2 = (O∗ − O∞ ) − K1
−a − 1/τ1
K3 = K1
b
−a − 1/τ1
K4 = K2
b
br
−
dr
6
1
O∞ =
ad − bc
cr1 − ar6
I∞ =
ad − bc
a = −(r1 + r2 + r3 ) , b = −r1 + r4 ,
c = r3 − r6 , d = −(r4 + r5 + r6 )
a + d 16
τ1/2 = −
±
(a − b)2 + 4bc.
2
2
EBM2iBF2M HH;2K2BM2` P`/MmM;
HH;2K2BM Hbb2M bB+? GƺbmM;2M /2` `2bmHiB2`2M/2M avbi2K2 ;2rƺ?MHB+?2` .Bz2`2MiBH@
;H2B+?mM;2M mb J`FQp@EBM2iBF2M MB+?i MHviBb+? /`bi2HH2MX ./m`+? Bbi KM /`m7
M;2rB2b2M- MmK2`Bb+?2 GƺbmM;bp2`7?`2M xm p2`r2M/2M UbB2?2 ETBi2H :`mM/H;2M Mm@
K2`Bb+?2` Ji?2KiBFVX
j9
9 JQ/2HHB2`mM; pQM L2m`QM2M AA, P`i@ mM/
x2Bi#? M;B;2 JQ/2HH2
9XR .B2 E#2H;H2B+?mM;
"Bb?2` rm`/2M L2m`QM2M Mm` Hb mb/2?MmM;bHQb2 SmMFiQ#D2Fi2 #2?M/2Hi- m7 /2M2M
aB;MH2 2BM2M x2BiHB+?2M o2`Hm7 ?#2MX 1b Bbi FH`- /bb /B2b 2BM2 b2?` HBKBiB2`i2 "2@
b+?`2B#mM; /2` 2H2Fi`Q+?2KBb+?2M S`Qx2bb2 BM 2BM2K L2m`QM /`bi2HHic iib +?HB+? r2`@
/2M 2H2Fi`Bb+?2 aB;MH2 2MiHM; pQM .2M/`Bi2M KBi KTHBim/2Mp2`Hmbi mM/ x2Bi#? M;B;
i`MbTQ`iB2`iX
.B2b2b o2`?Hi2M H bbi bB+? /m`+? /B2 E#2Hi?2Q`B2 #2b+?`2B#2MX
MM?K2M,
• .2M/`Bi2M bBM/ DzE#2Hdz, `QiiBQMbbvKK2i`Bb+?2 Eƺ`T2` KBi p`B#H2K _/Bmb aX
• G/mM;2M #2r2;2M bB+? BM 2BM2 _B+?imM; x 2MiHM; /2b E#2Hb UiM;2MiBH xm`
J2K#`M- Zm2`#2r2;mM;2M r2`/2M p2`M+?H bbB;i- /X?X /b E#2H rB`/ /2 7+iQ
Mm` KBi 2BM/BK2MbBQMH2` mb/2?MmM; KQ/2HHB2`iVX
→ V (x, t) Bbi #? M;B; pQM _mK mM/ w2Bi- x mM/ tX
• .b mĽ2M`mKTQi2MiBH Bbi FQMbiMiX
##BH/mM; 9XR, aFBxx2 xm` >2`H2BimM; /2` E#2H;H2B+?mM;X
wm` >2`H2BimM; /2` E#2H;H2B+?mM; #2i`+?i2M rB` 2BM UBM}MBi2bBKHV Fm`x2b aiɃ+F /2b
E#2Hb rB2 BM ##BH/mM; 9XRX
j8
1b ;BHi 7Ƀ` /B2 BM /2K U/ ∆x Hb FH2BM M;2MQKK2M rB`/, xvHBM/`Bb+?2MV oQHmK2M 2BM@
;2b+?HQbb2M2 G/mM; Q m7;`mM/ /2` G/mM;b2`?HimM;
Cm
∂V
∂Q
=
= IL (x) − IL (x + ∆x) − Im + Ie ,
∂t
∂t
U9XRV
rQ#2B /B2 HBMF2 :H2B+?mM; 2BM2 FTxBiBp2 "2b+?`2B#mM; mM/ /B2 `2+?i2 :H2B+?mM; 2BM2
"2b+?`2B#mM; Ƀ#2` /B2 1BM@ mM/ mbbi`ƺK2 /`bi2HH2MX
Cm Bbi ?B2` /B2 ETxBi i /2b mMi2`bm+?i2M E#2H@oQHmK2MbV /B2 J2K#`MbTMMmM; K E#2HbiɃ+FIL /B2 ai`QKbi `F2 H M;b /2b E#2HbIm /2` mbbi`QK Ƀ#2` /B2 J2K#`M Up2`m`b+?i /m`+? G2+Fbi`QK mM/ EMHFiBpBi i2MV
Ie 2BM 2ti2`M2` 1BMbi`QK U2ir /m`+? bvMTiBb+?2 FiBpBi i Q/2` 2BM2 1H2Fi`Q/2VX
.B2 ai`QKbi `F2 IL H bbi bB+? m+? Ƀ#2` /2M qB/2`biM/ RL mM/ /b SQi2MiBH Φ BK
E#2H /`bi2HH2M, 1b ;BHi KBi /2K P?Kb+?2M :2b2ix 7Ƀ` 2BM BM}MBi2bBKH FH2BM2b dx,
Φ(x + dx) − Φ(x) = −RL (x) · IL (x),
rQ#2B /2` qB/2`biM/ RL pQM /2` :2QK2i`B2 /2b E#2Hb #? M;i- M KHB+? ;2`/2 BM /2`
6Q`K RL (x) = rL πadx
2 (x) UD2 H M;2` /b #2i`+?i2i2 aiɃ+F- /2biQ ;`ƺĽ2` /2` qB/2`biM/- D2
#`2Bi2` /B2 Zm2`b+?MBiib~ +?2- /2biQ ;2`BM;2`V- rQ#2B /2` bT2xB}b+?2 qB/2`biM/ rL 2BM2
Ji2`BHFQMbiMi2 U/2b wviQbQHbV BbiX JM 2`? Hi HbQ mb /2` pQ`M;2;M;2M2M :H2B+?mM;
Φ(x + dx) − Φ(x) = −rL
dx
· IL (x).
πa2 (x)
.BpB/B2`i KM #2B/2 a2Bi2M /m`+? dx mM/ 7Ƀ?`i /2M :`2MxɃ#2`;M; dx → 0 mb- 2`? Hi
KM
∂Φ
rL
= − 2 · IL .
∂x
πa
:2?i KM
4 pQM 2BM2K5FQMbiMi2M mĽ2M`mKTQi2MiBH mb U2ir Φa ≡ 0V- FMM KM
∂V
i −Φa )
KBi ∂x = ∂(Φ∂x
B/2MiB}xB2`2M mM/ 2`? Hi 7Ƀ` /B2 ai`QKbi `F2 /2M mb/`m+F
IL = −
πa2 ∂V
.
rL ∂x
∂Φ
∂x
U9XkV
"2pQ` /B2b2b 1`;2#MBb BM :H2B+?mM; U9XRV 2BM;2b2ixi r2`/2M rB`/- bQHH MQ+? 2BM2 r2Bi2`2
:H2B+??2Bi #2;`ɃM/2i r2`/2M, qB` rQHH2M x2B;2M- /bb UmM/ rB2V m+? /B2 ETxBi i Cm
/2b mMi2`bm+?i2M E#2HbiɃ+Fb pQM /2bb2M ;2QK2i`Bb+?2` 6Q`K #? M;iX
.xm 7bb2M rB` /B2 J2K#`M Hb B/2H2M EQM/2MbiQ` m7- BM /2K /B2 "2xB2?mM;
V =E·d
je
U9XjV
;BHi- rQ#2B d 7Ƀ` /B2 .B+F2 /2` J2K#`M bi2?i mM/ E = −∇Φ 7Ƀ` /B2 U`/BH2V ai `F2
/2b 2H2Fi`Bb+?2M 62H/2b BK AMM2`M /2` J2K#`MR X .B2 2`bi2 Jtr2HH;H2B+?mM; bi2HHi /2M
wmbKK2M?M; xrBb+?2M /2` G/mM;b/B+?i2 ρi BK E#2HbiɃ+F U/2bb2M oQHmK2M KBi Ω
#2x2B+?M2i b2BV mM/ /2K SQi2MiBH ?2`,
−∆Φ =
ρi
.
'0
>B2` Bbi '0 2BM2 Lim`FQMbiMi2- /B2 oFmmK@S2`K2iiBpBi iX AMi2;`B2`i KM #2B/2 a2Bi2M
mM/ r2M/2i HBMFb /2M aix pQM :mĽ M- 2`? Hi KM
ˆ
ˆ
ρi
Q
Q
−∇Φ · n dν =
dµ ⇔ 2πa∆xE =
⇔ E=
,
'0
'0
2πa∆x'0
∂Ω
Ω
rQ#2B KM ?B2`#2B pQM /2` MM?K2 mb;2?i- /bb /b 2H2Fi`Bb+?2 62H/ BKK2` b2MF`2+?i
m7 /2` J2K#`M bi2?iX a2ixi KM /2M bQ ;27mM/2M2M mb/`m+F 7Ƀ` E MmM BM U9XjV 2BM2`;B#i bB+?
d
1
'0
V =
Q ⇒ Cm =
2πa∆x.
U9X9V
' 2πa∆x
d
)*+,
) 0 *+ ,
=:cm
−1
=Cm
T2` /27X
LmM Bbi HH2b #2`2Bi2i, JM b2ixi U9XkV mM/ U9X9V BM U9XRV 2BM,
∂V
πa2 (x) ∂V
πa2 (x + ∆x) ∂V
cm 2πa∆x
= −
(x) − (−1) ·
(x + ∆x) − Im + Ie .
) *+ , ∂t
rL ∂x
rL
∂x
)
*+
,
)
*+
,
Cm
IL (x)
IL (x+∆x)
.B2b2 :H2B+?mM; i2BHi KM m7 #2B/2M a2Bi2M /m`+? 2πa(x)∆x mM/ #2+?i2i /2M ɂ#2`@
;M; pQM /2M ai`ƺK2M Im mM/ Ie xm /2M ai`QK/B+?i2M im #xrX ie - mK b+?HB2ĽHB+? /2M
:`2MxɃ#2`;M; ∆x → 0 /m`+?xm7Ƀ?`2M mM/ 2M/HB+? /B2 E#2H;H2B+?mM; xm 2`?Hi2M,
cm
∂V
1 ∂
=
∂t
2arL ∂x
!
a2
∂V
∂x
"
− im + ie
U9X8V
Pz2MbB+?iHB+? ?M/2Hi 2b bB+? KBi #H2BimM;2M bQrQ?H M+? /2` w2Bi Hb m+? BK P`i MB+?i
K2?` Ĝ rB2 #Bb?2` Ĝ mK 2BM2 ;2rƺ?MHB+?2- bQM/2`M mK 2BM2 T`iB2HH2 .Bz2`2MiBH;H2B+?mM;X
9XRXR _M/@ mM/ o2`xr2B;mM;b#2/BM;mM;2M
.KBi /B2 E#2H;H2B+?mM; 2BM/2miB; Hƺb#` Bbi- bBM/ M7M;b@ mM/ _M/#2/BM;mM; MQi@
r2M/B;X lK /B2 E#2H;H2B+?mM; mĽ2`/2K m+? 7Ƀ` p2`xr2B;i2 .2M/`Bi2M Mr2M/2M xm
R
.B2b2 A/2HBbB2`mM; Bbi MiɃ`HB+? Mm` xmH bbB;- bQHM;2 d ! a- / bQMbi BK J2K#`MxrBb+?2M`mK F2BM
MM ?2`M/ FQMbiMi2b 62H/ xm 2`r`i2M BbiX
jd
FƺMM2M- #`m+?i KM xmb ixHB+?2 "2/BM;mM;2M M /2M o2`xr2B;mM;bTmMFi2MX
_M/#2/BM;mM;
RX a2H2/@2M/@"2/BM;mM; Ĝ LBKKi KM M- /bb K 1M/2 /2b .2M/`Bi2M /2` tBH2 qB@
/2`biM/ b2?` ;`QĽ Bbi- FƺMM2M F2BM2 AQM2M /`Ƀ#2` 2BM@ Q/2` mb~B2Ľ2M UL2mKMM@
y@_M/#2/BM;mM;V,
dV
= 0.
U9XeV
dX 2M/
kX EBHH2/@2M/@"2/BM;mM; Ĝ ai2HHi KM bB+? pQ`- /bb /b .2M/`Bi2M2M/2 Qz2M Bbi U"2@
b+? /B;mM;- /m`+? S` T`ipQ`#2`2BimM; BK 1tT2`BK2Mi 2i+XV- bQ FƺMM2M AQM2M 7`2B
xrBb+?2M AMi`@ mM/ 1ti`x2HHmH ``mK mb;2imb+?i r2`/2MX aQKBi ;Bi 2b F2BM2M
SQi2MiBHbT`mM; K E#2H2M/2 U.B`B+?H2i@y@_M/#2/BM;mM;V,
V
2M/
= 0.
U9XdV
jX SQi2MiBH#2/BM;mM; Ĝ Abi /b SQi2MiBH VL K 1M/2 /2b E#2Hb #2FMMi UrB`/
2b xX"X /m`+? 2BM2 1H2Fi`Q/2 m7 2BM2K 72bi2M q2`i ;2?Hi2MV- ;BHi U.B`B+?H2i@
_M/#2/BM;mM;V,
V 2M/ = V2M/ .
U9X3V
M7M;b#2/BM;mM;
JM MBKKi xmK "2BbTB2H M- /bb /B2 w2HH2 xm "2;BMM BbQTQi2MiBH KBi J2K#`M`m?2@
TQi2MiBH Bbi,
V (·, t0 ) ≡ V`2bi .
o2`xr2B;mM;bTmMFi2
RX .b J2K#`MTQi2MiBH M o2`xr2B;mM;bTmMFi2M x∗ Kmbb BM HH2M EQKT`iBK2Mi2M
1 . . . n- /B2 M x∗ M;`2Mx2M- ;H2B+? b2BM- /X?X
V1 (x∗ ) = V2 (x∗ ) = · · · = Vn (x∗ ).
U9XNV
kX K o2`xr2B;mM;bTmMFi /`7 F2BM2 G/mM; 2Mibi2?2M Q/2` p2`MB+?i2i r2`/2M- /@
?2` Kmbb /B2 amKK2 /2` ai`ƺK2 2MiHM; /2b .2M/`Bi2M M /B2b2M SmMFi2M LmHH
2`;2#2M,
n
n
2
2
πa2 ∂Vi
Ii (x∗ ) =
= 0.
U9XRyV
rL ∂x x∗
i=1
i=1
j3
9XRXk w2BimM#? M;B;2 GƺbmM; /2` E#2H;H2B+?mM;
MM?K2M,
RX "2i`+?i2 .2M/`Bi2M U/X?X F2BM2 MB+?i@HBM2`2M "2Bi` ;2 rB2 >>@ai`ƺK2VkX Mm` 1H2Fi`Q/2MbiBKmHiBQMjX FQMbiMi2 _/B2M- /X?X a mM#? M;B; pQM xX
.MM Bbi im HBM2` UJ2K#`M Hb P?Kb+?2` qB/2`biM/V mM/ 2b ;BHi
im =
V − V`2bi
.
rm
a+?`2B#2 v := V − V`2bi mM/ b2ix2 2BM BM U9X8V,
cm
∂v
a ∂2v
v
=
−
+ ie .
2
∂t
2r ∂x
r
) m*+ ,
) L*+ ,
MMX j
MMX R mM/ k
.2}MB2`i KM τm := rm cm UrB2 ;2?#iV mM/ λ :=
KM /B2 HBM2`BbB2`i2 E#2H;H2B+?mM;
τm
7
arm
2rL
U2H2Fi`QiQMBb+?2 G M;2V- 2`? Hi
∂v
∂2v
= λ 2 2 − v + rm i e .
∂t
∂x
U9XRRV
lM2M/HB+?2b E#2H
qB` rQHH2M /B2b2 :H2B+?mM; 7Ƀ` 2BM2M aT2xBH7HH- M KHB+? 2BM mM2M/HB+? HM;2b E#2H KBi
2BM2` TmMFi7ƺ`KB;2M 1H2Fi`Q/2- BK biiBQM `2M wmbiM/ U/X?X ∂v
∂t = 0V MHviBb+? Hƺb2MX
• _M/#2/BM;mM;2M, v → 0 7Ƀ` |x| → ∞-
• 1BMbi`QK#2/BM;mM;, FQMbiX 1H2Fi`Q/2Mbi`QK Ie BM x = 0 UTmMFi7ƺ`KB;2 Zm2HH2V
lK /B2 TmMFi7ƺ`KB;2 1H2Fi`Q/2 xm KQ/2HHB2`2M- r2`/2M rB` xmM +?bi pQM 2BM2` 1H2Fi`Q/2
KBi /2` mb/2?MmM; 2ε mb;2?2M- bQ/bb BK "2`2B+? |x| < ε 2BM2 FQMbiMi2 1H2Fi`Q/2M@
Ie
bi`QK/B+?i2 ie = 2πa·2ε
MHB2;iX aT i2` r2`/2M rB` /MM /2M :`2MxɃ#2`;M; ε → 0
pQHHxB2?2MX
mb /2` HBM2`BbB2`i2M E#2H;H2B+?mM; U9XRRV 2`;B#i bB+? /B2 biiBQM `2 :H2B+?mM;
λ2
d2 v
= v − rm i e .
dx2
U9XRkV
6Ƀ` ie ≡ 0 U/X?X BM /2M "2`2B+?2M x < −ε mM/ x > ε V p2`2BM7+?i bB+? /B2b2 MQ+? r2Bi2`
xm` ?QKQ;2M2M :H2B+?mM;
d2 v
λ2 2 = v,
dx
jN
7Ƀ` r2H+?2 /B2 HH;2K2BM2 GƺbmM; /m`+?
x
x
v(x) = B1 2tT(− ) + B2 2tT( )
λ
λ
;2;2#2M BbiX
m7 /2` HBMF2M a2Bi2 Ux < εV ;BHi /B2 _M/#2/BM;mM; v(x) → 0 (x → −∞)- HbQ ?i KM
4 x5
v(x) → B1HBMFb 2tT −
= 0 (x → −∞) ⇒ B1HBMFb = 0,
λ
2MibT`2+?2M/ m7 /2` `2+?i2M a2Bi2 Ux > εV, JBi v(x) → 0 (x → ∞) ;BHi
4x5
v(x) → B2`2+?ib 2tT
= 0 (x → ∞) ⇒ B2`2+?ib = 0.
λ
mb avKK2i`B2;`ɃM/2M Kmbb #2` MiɃ`HB+? /MM B1`2+?ib = B2HBMFb =: B ;2Hi2M mM/ 2b
Bbi /?2` 7Ƀ` x ∈
/ [−ε, ε]
!
"
|x|
v(x) = B 2tT −
.
λ
LmM Kmbb Mm` MQ+? /B2 EQMbiMi2 " #2biBKKi r2`/2M- /B2 bB+? BM B`;2M/2BM2` 6Q`K mb
/2K 1BMbi`QK 2`;2#2M Kmbb,
AM /2` lK;2#mM; [−ε, ε] mK LmHH ;BHi /B2 BM?QKQ;2M2 UmM/ /?2` b+?rB2`B;2` xm #2?M@
/2HM/2V :H2B+?mM;
d2 v
λ2 2 = v − rm i e .
U9XRjV
dx
JM FMM /B2b2 :H2B+?mM; m7 #2B/2M a2Bi2M BMi2;`B2`2M,
ˆε
−ε
⇔ λ
2
λ
2d
!
2v
dx2
dx =
ˆε
−ε
(v − rm ie ) dx
dv
dv
(ε) −
(−ε)
dx
dx
"
=
ˆε
−ε
v dx − rm ie · 2ε =
ˆε
−ε
v dx − rm
Ie
2πa
U9XR9V
dv
dv
.B2 r2b2MiHB+?2 ɂ#2`H2;mM; Bbi MmM 7QH;2M/2, .B2 #H2BimM;2M dx
(−ε) mM/ dx
(ε) KɃbb2M
dv
dv
;2`/2 KBi /2M #H2BimM;b;`2Mxr2`i2M HBK dx (t) #xrX HBK dx (t) /2` U#Bb m7 "V b+?QM #2@
t↑−ε
t↓ε
FMMi2M HBMF2M mM/ `2+?i2M GƺbmM;2M Ƀ#2`2BMbiBKK2M- mK /B2 bi2iB;2 .Bz2`2MxB2`#`F2Bi
/2` GƺbmM; xm ;`MiB2`2MX .B2b2 #H2BimM;2M bBM/ #2` ;2`/2
9 :
8 B
dv
2tT λt9 , : t < −ε
λ
(t) =
t
−B
dx
λ 2tT − λ , t > ε
mM/ bQKBi 2`;B#i bB+? 2BM;2b2ixi BM U9XR9V
!
! ""
ˆε
4 ε5
4 ε5 B
B
−ε
rm I e
2
−2λB 2tT −
= λ − 2tT −
− 2tT
=
v dx −
.
λ
λ
λ
λ
λ
2πa
−ε
9y
1b Bbi MB+?i MQir2M/B;- /b p2`#H2B#2M/2 AMi2;`H Ƀ#2` v xm Hƺb2M- /2MM r2MM KM MmM
/2M :`2MxɃ#2`;M; ε → 0 mb7Ƀ?`i- FQMp2`;B2`i /B2 :H2B+?mM; Ur2;2M ai2iB;F2Bi pQM vV
;2;2M
rm I e
−2λB · 1 = 0 −
2πa
mM/ KM FMM M+? " m~ƺb2M,
rm I e
B=
.
4πaλ
rm
.B2 /MM BM HH2M SmMFi2M x ∈ R ;ɃHiB;2 GƺbmM; Bbi /?2` UKBi Rλ := 2πaλ
V
!
"
Rλ I e
|x|
v(x) =
2tT −
.
2
λ
9Xk .b _HH@JQ/2HH
TT`QtBKiBQM 2BM2b p2`xr2B;i2M .2M/`Bi2M#mKb /m`+? 2BM [mBpH2Mi2b E#2HX
q?H /2b E#2Hb,
.2}MBiBQM
RyX
4
5 JBi
7
arm
λ := 2rL .
L
λ
rB`/ /B2 2H2Fi`QiQMBb+?2 G M;2 2BM2b E#2Hb U.2M/`Bi2MV #2x2B+?M2i
P#2`~ +?2 2πaL := .2M/`Bi2MQ#2`~ +?2 SD
SD
lK λ xm #2`2+?M2M- #2`2+?M2 a /m`+? a = 2πL
X
a2ix2
L
λ
mM/ SD ;H2B+?5
9Xj JmHiB@*QKT`iK2Mi JQ/2HH2
EQKTH2t2 M2m`QMH2 ai`mFim`2M bBM/ MHviBb+? MB+?i xm 2`7bb2MX
→ ##BH/2M M2m`QMH2` ai`mFim`2M m7 /BbF`2i2 wmbi M/2X
Gƺb2M, MmK2`Bb+?
• 1BMi2BHmM; /2b L2m`QMb BM 2M/HB+? pB2H2 xvHBM/`Bb+?2 EQKT`iBK2Mi2
• C2/2b EQKT`iBK2Mi rB`/ Hb BbQTQi2MiBH M;2MQKK2M U/X?X 2BM2 lM#2FMMi2 T`Q
EQKT`iBK2MiVX
• Gƺb2 V 7Ƀ` D2/2b EQKT`iBK2Mi µX
Gƺb2M pQM oµ
qB2 #2B /2` >2`H2BimM; /2` E#2H;H2B+?mM; Ĝ :H2B+?mM; U9XRV Ĝ b2ixi KM M #2B /2`
ai`QK#BHMx Ƀ#2` /b µ@i2 EQKT`iBK2Mi Up;HX ##X 9XkV,
!
"
!
"
∂Vµ
1
1
Cm
= IL xµ − L µ − IL xµ + L µ − I m + I e
U9XR8V
∂t
2
2
9R
##BH/mM; 9Xk, a+?2K, .`2B EQKT`iBK2Mi2 mb /2K JmHiB@*QKT`iK2Mi@JQ/2HHX
mM/ ;2Mm rB2 /Q`i /`Ƀ+Fi KM /MM /B2 tBH2M ai`ƺK2 IL M /2M 1M/2M /2b EQKT`iB@
K2Mib Ƀ#2` /b SQi2MiBH mM/ /2M P?KKb+?2M qB/2`biM/ mbX wrBb+?2M /2M SmMFi2M7Ƀ` /B2 Vµ #xrX Vµ+1 ;2;2#2M Bbi- HB2;i 2BM qB/2`biM/- /2` bB+? mb xr2B Dz?H#2Mdz xv@
HBM/2`7ƺ`KB;2M EQKT`iBK2Mi2M xmbKK2Mb2ixiX "2B 2BM2` bQH+?2M _2B?2Mb+?HimM; pQM
qB/2`bi M/2M r2`/2M /B2 2BMx2HM2M qB/2`bi M/2 2BM7+? //B2`i- bQ/bb ;BHi,
!
"
!
"
Φµ − Φµ−1
Φµ+1 − Φµ
1
1
I L x µ − Lµ =
,
I L x µ + Lµ =
.
1
1
L
L
L
L
2
2
2 µ−1
2 µ
rL 2πaµ2 + rL 2πaµ+1
2
rL πa
+ rL πa
2
2
µ
µ+1
µ−1
µ
LBKKi KM rB2/2` 2BM FQMbiMi2b mĽ2M`mKTQi2MiBH M- FMM KM Φ /m`+? V bm#@
biBimB2`2MX JBi /2` BM :H2B+?mM; U9X9V ?2`;2H2Bi2i2M A/2MiBi i 7Ƀ` /B2 ETxBi i- /B2 ?B2`
'0
Cm =
2πaµ Lµ Hmi2i- #2FQKKi KM /KBi mb U9XR8V /B2 :H2B+?mM;
d
)*+,
=:cm
∂Vµ
= −iµm + iµe + gµ−1,µ (Vµ − Vµ−1 ) − gµ,µ+1 (Vµ+1 − Vµ )
∂t
KBi
0
1−1
aµ a2µ−1
Lµ−1
Lµ
−1
4
5,
gµ−1,µ = rL
+
r
(2πa
L
)
=
µ µ
L
2πa2µ
2πa2µ−1
rL Lµ Lµ−1 a2µ + Lµ a2µ−1
0
1−1
aµ a2µ+1
Lµ
Lµ+1
−1
4
5
gµ,µ+1 = rL
+
r
(2πa
L
)
=
µ
µ
L
2πa2µ
2πa2µ+1
r L L a2 + L
a2
cm
L µ
µ µ+1
U9XReV
U9XRdV
U9XR3V
U9XRNV
µ+1 µ
7Ƀ` 2BM mMp2`xr2B;i2b E#2HX
.B2 :`ƺĽ2 gµ,µ+1 #2b+?`2B#i /2M tBH2M AQM2M~mbb xrBb+?2M ;2FQTT2Hi2M EQKT`iBK2M@
i2M µ mM/ µ +1 BM #? M;B;F2Bi pQM /2M /Q`i ?2``b+?2M/2M SQi2MiBH2MX .B2b2 :H2B+?mM;
FMM 7Ƀ` D2/2b EQKT`iBK2Mi m7;2bi2HHi r2`/2M- bQ/bb KM 2BM avbi2K pQM ;2rƺ?M@
HB+?2M .Bz2`2MiBH;H2B+?mM;2M U#H2BimM;2M Mm` MQ+? BM /2` w2BiV 2`? Hi- /b KBi /2M
; M;B;2M MmK2`Bb+?2M J2i?Q/2M ;2Hƺbi r2`/2M FMMX
.B2 /BbF`2i2 GƺbmM; /2b J2K#`MTQi2MiBHb Bbi BM B?`2` m~ƺbmM; /m`+? /B2 Mx?H
U#xrX :`ƺĽ2V /2` EQKT`iBK2Mi2 #2biBKKiX
hQQH, xX"X L2m`QMX
9k
9X9 oQM JBF`QbFQTB2/i2M xm /BbF`2i2M :Bii2`M
.b :2M2`B2`2M pQM _2+?2M;Bii2`M- /B2 /B2 #BQHQ;Bb+?2 _2HBi i `2T` b2MiB2`2M- Bbi 2BM2
/2` b+?rB2`B;bi2M m7;#2M BM /2` aBKmHiBQMX .B2 _2FQMbi`mFiBQM #2;BMMi KBi /2` 1`@
x2m;mM; pQM JBF`QbFQTB2/i2MX .B2b2 bBM/ BK _2;2H7HH HH2`/BM;b MB+?i mMKBii2H#` p2`@
r2M/#`- mK KBi B?M2M miQKiBbB2`i :Bii2` xm 2`x2m;2MX wmK2Bbi K+?i /B2 o2`r2M/mM;
2BM2` JBF`QbFQTB2i2+?MBF bT2xB}b+?2 L+?#2`2BimM;bb+?`Bii2 MQir2M/B;- mK /B2 ZmHBi i
/2` .i2M m7 2BM #2MƺiB;i2b LBp2m xm ?2#2MX AM .B2b2K #b+?MBii r2`/2M a+?`Bii2 /2`
.i2M@L+?#2`#2BimM; mM/ /B2 H2ixi2M/HB+?2 :Bii2`2`x2m;mM; #2?M/2HiX
9X9XR "BH/p2``#2BimM;
JBF`QbFQTB2/i2M 2Mi?Hi2M Dz62?H2`[m2HH2Mdz- xX"X,
• Gƺ+?2` BM ai`mFim`2M• _mb+?2MX
.B2 :Bii2`H;Q`Bi?K2M KɃbb2M mb /2M "BH//i2M P#2`~ +?2Mbi`mFim`2M 2ti`?B2`2MX
.KBi /B2b ;2HBM;2M FMM- KɃbb2M x2HHmH `2 ai`mFim`2M BM /2M .i2M DzFQMiBMmB2`HB+?dz
b2BM- HbQ F2BM2 xm ;`QĽ2M aT`ɃM;2 BM /2M :`mr2`i2M m7r2Bb2MX .xm Kmbb bB+? /b
2B;2MiB+?2 "BH/ pQK _mb+?2M #?2#2MX lK _mb+?2M mb /2M .i2M xm 2Mi72`M2M mM/
/MM P#2`~ +?2M;2QK2i`B2M xm `2FQMbi`mB2`2M- #2Mmixi KM FHbbBb+?2 J2i?Q/2M /2`
"BH/p2``#2BimM; rB2 6BHi2` mM/ a2;K2MiB2`2`X
:H iimM;b}Hi2` Dz;H ii2Mdz :`mr2`i#BH/2`- /X?X p2``BM;2`M /B2 #bi M/2 xrBb+?2M :`m@
r2`i2M #2M+?#`i2` oQt2H- BM/2K Ƀ#HB+?2`r2Bb2 D2/2` oQt2H@:`mr2`i Ƀ#2` 2BM2 ;2rBbb2
lK;2#mM; BM ;22B;M2i2` q2Bb2 ;2KBii2Hi rB`/X
9X9XRXR 1BM7+?2 :H iimM;b}Hi2`
RX JBii2Hr2`i}Hi2` U"Qt}Hi2`V Ĝ AK ;2}Hi2`i2M "BH/ ?i oQt2H j /2M JBii2Hr2`i HH2`
SBt2H BM b2BM2` lK;2#mM;,
k
12
FB (Gj ) =
Gi .
k
i=1
.B2b2bk
6BHi2` Bbi 2ti`2K H2B+?i xm BKTH2K2MiB2`2M mM/ KBi ;2`BM;bi2K _2+?2M@
m7rM/ p2`#mM/2MX a2BM2 :H iimM;b2B;2Mb+?7i2M bBM/ D2/Q+? b+?H2+?i, >Q?2 6`2@
[m2Mx2M r2`/2M Mm` mMxm`2B+?2M/ ;2/ KT7i mM/ bQ;` T?b2Mp2`b+?Q#2MX
k
kX J2/BM}Hi2` Ĝ AK m`bT`ɃM;HB+?2M _Q?#BH/ G rB`/ BM 2BM2` lK;2#mM; pQM k oQt2HM
UɃ#HB+?2`r2Bb2 2BM2` bvKK2i`Bb+?2M lK;2#mM;- HbQ 2ir /2` kd2`@lK;2#mM;- /B2
HH2 oQt2H 2Mi? Hi- /B2 2BM2 a2Bi2- EMi2 Q/2` 1+F2 KBi /2K oQt2H j i2BH2MV /2`
J2/BM ;2#BH/2i,
FM (Gj ) = K2/BM {Gj1 , . . . , Gjk } .
U9XkyV
AK "BH/p2``#2BimM;b@6+?D`;QM Bbi 6BHi2` Ƀ#HB+?2`r2Bb2 2BM L2mi`mKX
9j
.B2b Bbi 2BM MB+?i@HBM2`2b 6BHi2`X
jX :mĽ}Hi2`, "2BK :mĽ}Hi2` rB`/ /2` :`mr2`i 2BM2b oQt2Hb j Hb /2` ;2rB+?i2i2
JBii2Hr2`i Ƀ#2` 2BM2 lK;2#mM; 2`KBii2HiX .B2 :2rB+?imM;b7FiQ`2M 2`;2#2M bB+?
UM ?2`mM;br2Bb2V mb 2BM2` :mĽ7mMFiBQM U2tTQM2MiB2HH #7HH2M/- D2 r2Bi2` pQM
oQt2H j 2Mi72`MiV,
FG (Gj ) =
k
2
i=1
KBi g(j, i) =
U9XkRV
g(j, i) · Gi
1
d
(2π) 2 σ
k
% 1
d
4
5
2
· 2tT − 12 'i−j'
σ2
l=1 (2π) 2 σ
· 2tT
4
2
− 12 'i−l'
σ2
5.
.#2B Bbi d /B2 .BK2MbBQM /2b "BH/2b- σ bi2?i 7Ƀ` /B2 Dz"`2Bi2dz /2` :mĽ7mMFiBQM
mM/ 0i − j0 7Ƀ` /B2 U2mFHB/Bb+?2V .BbiMx xrBb+?2M i@i2K mM/ j@i2K SBt2HX .2` L2M@
M2` BM /2M EQ2{xB2Mi2M g(j, i) /B2Mi H2/B;HB+? /2` LQ`KHBbB2`mM; UBM amKKiBQM
KɃbb2M /B2 EQ2{xB2Mi2M R 2`;2#2MVX
.B2 :H iimM;b2B;2Mb+?7i2M /2b :mĽ}Hi2`b bBM/ b2?` ;mi- /B2 "2`2+?MmM; Bbi 2BM@
7+?- #2` 2irb m7r M/B;2` Hb 2ir #2BK "Qt}Hi2`X
S`Q#H2K
.B2b2 6BHi2` M2?K2M F2BM2 _Ƀ+FbB+?i m7 x2HHmH `2 ai`mFim`2M- /B2 bB+? B/2H2`r2Bb2 /@
/m`+? mbx2B+?M2M- DzEMi2Mdz M /2M P#2`~ +?2M xm ?#2M- /X?X ?Q?2 :`mr2`i@:`/B@
2Mi2M #2BK ɂ#2`;M; 2ir pQM J2K#`M xm lK;2#mM; Q/2` *viQTHbKX .B2b2 EMi2M
bBM/ 2`rɃMb+?i- / bB2 /B2 ai`mFim` /2}MB2`2M- /B2 #2B /2` :Bii2`2`x2m;mM; M+?;2#BH/2i
r2`/2M bQHH2MX .B2 :H iimM;b}Hi2` ;H ii2M /B2b2 EMi2M #2` UbB2 p2`b+?rBKK2MV- bQ/bb
bB2 b+?rB2`B;2` xm /2i2FiB2`2M r2`/2MX
qɃMb+?2Mbr2`i Bbi 2BM 6BHi2`- /b bi`mFim`2`?Hi2M/ ;H ii2iX
9X9XRXk .BzmbBQMb}Hi2`
A/22, .b :`mr2`i#BH/ rB`/ Hb o2`i2BHmM; T?vbBFHBb+?2` Jbb2 BMi2`T`2iB2`i, D2 /mMF@
H2` 2BM oQt2H- /2biQ ?ƺ?2` /B2 EQMx2Mi`iBQM M /B2b2` ai2HH2X wmK :H ii2M H bbi KM
/MM /B2 :`mr2`i2 /BzmM/B2`2MX
Z, qb TbbB2`i- r2MM /B2 .BzmbBQM MB+?i #22M/2i rB`/\
, >QKQ;2M2b "BH/, C2/2b oQt2H ?i /2Mb2H#2M :`mr2`i- M KHB+? /2M JBii2Hr2`i HH2`
m`bT`ɃM;HB+?2M :`mr2`i2X
.BzmbBQM
wm` >2`H2BimM; 2BM2` :H2B+?mM; 7Ƀ` /B2 "2b+?`2B#mM; /2` .BzmbBQM- r2`/2M /`2B ɂ#2`H2@
;mM;2M xmbKK2M;27Ƀ?`i,
99
RX 6B+Fb+?2b :2b2ix Ĝ EQMx2Mi`iBQMb;27 HH2 r2`/2M mb;2;HB+?2M, .2` /BzmbBp2 6Hmbb
2`7QH;i BKK2` KBi /2K :27 HH2 /2` EQMx2Mi`iBQM u U/B2b Bbi 2BM2 2KTB`Bb+?2 hi@
b+?2- /B2 HH2`/BM;b m+? mb 7mM/K2MiH2`2M S`BMxBTB2M /2` h?2`KQ/vMKBF
?2`;2H2Bi2i r2`/2M FMMV,
,
6Hmbb/B+?i272H/ ,j = −D∇u,


U9XkkV
∂/∂x
, =  ∂/∂y  UBM j.V,
KBi /2K .Bz2`2MiBHQT2`iQ` :`/B2Mi, ;`/ = ∇
∂/∂z
/2` BKK2` BM /B2 _B+?imM; /2b bi `Fbi2M MbiB2;bX D Bbi BK HH;2K2BM2M 2BM h2MbQ`
xr2Bi2` aim72- /2` .BzmbBQMbi2MbQ`- Bbi /B2b2` BbQi`QT- FMM 2` /m`+? 2BM2M aFH`
2`b2ixi r2`/2MX
kX Jbb2M2`?HimM;, a2B V 2BM #2HB2#B;2b oQHmK2M2H2K2Mi mM/ u /B2 EQMx2Mi`iBQM
BM V X .MM H bbi bB+? /B2 ŸM/2`mM; /2` Jbb2MK2M;2 BM V b+?`2B#2M Hb,
ˆ
∂u
(,x) d,x.
∂t
V
"2}M/2M bB+? F2BM2 Zm2HH2M UmM/ a2MF2MV BM V - bQ M/2`i bB+? u mbb+?HB2ĽHB+?
/m`+? /2M 6Hmbb Ƀ#2` /B2 _ M/2` pQM V ,
ˆ
ˆ
∂u
− ,j · ,n ds =
(,x) d,x.
U9XkjV
∂t
V
∂V
>B2`#2B Bbi ,j · ,n /b aFH`T`Q/mFi /2b 6Hmbb/B+?i272H/b KBi /2` mĽ2`2M P#2`~ @
+?2MMQ`KH2MX
jX :mĽb+?2` AMi2;`Hbix Ĝ .2` ;mĽb+?2 AMi2;`Hbix HB272`i 2BM2M Ki?2KiBb+?2M
wmbKK2M?M; xrBb+?2M /2K AMi2;`H Ƀ#2` 2BM o2FiQ`72H/ m7 2BM2` oQHmK2M@
Q#2`~ +?2 KBi /2K AMi2;`H Ƀ#2` /B2 .Bp2`;2Mx /2b o2FiQ`72H/2b BM /2K oQHmK2MX
HH;2K2BM ;BHi 7Ƀ` ?BM`2B+?2M/ `2;mH `2 oQHmKBM V mM/ o2FiQ`72H/2` F, ,
ˆ
ˆ
,
/Bp F (,x) d,x =
F, (,x) · ,n ds
U9Xk9V
V
∂V
, · u /2}@
.2` .Bz2`2MiBHQT2`iQ` /Bp Bbi /#2B /m`+? /b aFH`T`Q/mFi /Bp u = ∇
MB2`iX
q2M/2i KM MmM 9Xk9 m7 9Xkj M- 2`? Hi KM
ˆ
ˆ
ˆ
∂u
(,x) d,x = − ,j · ,n ds = − /Bp ,j d,x,
∂t
V
V
∂V
98
mM/ / /B2b 7Ƀ` #2HB2#B;2 oQHmKBM V ;BHi- FMM KM /B2 AMi2;`H2 r2;Hbb2M mM/ ,j
KBi?BH72 pQM 9Xkk 2`b2ix2M- mK /B2 .BzmbBQMb;H2B+?mM; xm 2`?Hi2M,
4
5
∂u
,
= /Bp D∇u
.
∂t
AK 6HH- rQ D 2BM aFH` Bbi- Mmixi KM /B2 GBM2`Bi i /2` .Bp2`;2Mx- mK
∂u
, = ∆u
= D /Bp ∇u
∂t
xm b+?`2B#2M KBi /2K GTH+2@PT2`iQ` ∆ := /Bp ;`/ =
n
%
i=1
∂2
∂x2i
7Ƀ` n .BK2MbBQM2MX
Z, qB2 Hbb2M bB+? bT2xB2HH2 6BHi2` 2MirB+F2HM\
, :22B;M2i2 q?H pQM DX
"2BbTB2H2 

1 0 0
UV D =  0 1 0  ⇒ BbQi`QT2 .BzmbBQM UxX"X :mĽ@6BHi2`V
0 0 1


5 0 0
U#V D =  0 1 0  ⇒ MBbQi`QT2 .BzmbBQM
0 0 1
q?H /2b .BzmbBQMbi2MbQ`b .
1`BMM2`mM;, .b _Q?#BH/ rB`/ Hb /BbF`2i2 Jbb2Mp2`i2BHmM; BMi2`T`2iB2`iX
1BM B/2H2b D 2`F2MMi /B2 Dzai`mFim`2Mdz mM/ 2`Hm#i .BzmbBQM 2MiHM; /2` ai`mFim` #2`
MB+?i [m2` /xm- mK bB2 bQ xm 2`?Hi2MX
qb rB` Ƀ#2` /b DzT?vbBFHBb+?2dz "BH/ rBbb2M,
2
Jbb2 :
M :=
mi
i
a+?r2`TmMFi :
1 2
R :=
mi,ri
M
,ri : P`ibp2FiQ` /2b oQt2Hb xi
h` ;?2Bibi2MbQ`
1BM KbbBp2` Eƺ`T2` #2bBixi _QiiBQMb2M2`;B2,
TR =
3
1 2
Jlm ωl ωm
2
l,m=1
J : h` ;?2Bibi2MbQ`
ω : qBMF2H;2b+?rBM/B;F2Bibp2FiQ`
9e
U9Xk8V
.#2B 2`;2#2M bB+? /B2 EQKTQM2Mi2M /2b h` ;?2Bibi2MbQ`b M+? /2` oQ`b+?`B7i
2
9
:
Jlm =
mi ri2 · δlm − ril rim ,
U9XkeV
i
8
1, l = m
/b E`QM2+F2`@.2Hi Bbi mM/ ril #xrX rim /B2 l@i2- #xrX m@i2
0, bQMbi
EQQ`/BMi2 /2b i@i2M "BH/TmMFi2b M;B#iX
rQ δlm =
J Bbi bvKK2i`Bb+? TQbBiBp /2}MBi mM/ ?i 7Ƀ` P#D2Fi2 KBi mb/2?MmM; BM HH2 .BK2MbBQ@
M2M pQHH2M _M;X L+? /2K aT2Fi`Hbix Hbb2M bB+? /?2` Q`i?QMQ`KH2 1B;2Mp2FiQ`2M
v1 , v2 , v3 KBi /2M xm;2?ƺ`B;2M 1B;2Mr2`i2M 0 < λ1 ≤ λ2 ≤ λ3 }M/2M- bQ/bb ;BHi


9
: λ1
9
:
 v1 v 2 v 3 T .
λ2
J = v1 v2 v3 
λ3
.B2b2 lK7Q`KmM; M2MMi KM >mTi+?b2Mi`Mb7Q`KiBQMj X .B2 1B;2Mr2`i2 λ1 #Bb λ3
;2#2M mbFmM7i Ƀ#2` /B2 "2b+?z2M?2Bi /2b #2i`+?i2i2M Eƺ`T2`b- BMb#2bQM/2`2,
λ1 FH2BM BK o2`;H2B+? xm λ2 , λ3 → HBM2` ;27Q`Ki2` Eƺ`T2`λ1 , λ2 FH2BM BK o2`;H2B+? xm λ3 → THM` ;27Q`Ki2` Eƺ`T2`λ1 ≈ λ2 ≈ λ3 → BbQi`QT2` Eƺ`T2`X
EQMbi`mFiBQM /2b .BzmbBQMbi2MbQ`b mb /2K h` ;?2Bibi2MbQ`
AM D2/2K oQt2H /2b _Q?#BH/2b FMM KM MmM Ƀ#2` 2BM2 ;2rBbb2 lK;2#mM; /2M h` ;@
?2Bibi2MbQ` #2`2+?M2MX
6BM/2i KM 2BM2 HBM2`2 lK;2#mM; U2ir- r2MM /b oQt2H h2BH 2BM2b .2M/`Bi2M Q/2`
tQMb BbiV- b2ixi KM /2M .BzmbBQMbi2MbQ` 7Ƀ` /B2b2b oQt2H 2MibT`2+?2M/ bQ- /bb Mm`
2MiHM; /2` _B+?imM; /2` lK;2#mM; ;2;H ii2i rB`/- MB+?i #2` Q`i?Q;QMH /xm U/B2
#2B/2M M/2`2M _B+?imM;2MV,


9
: 1
9
:
λ1
λ2
 v1 v2 v3 T . U9XkdV
ε
2 1,
≈ 1 : D = DL := v1 v2 v3 
λ2
λ3
ε
6BM/2i KM 2BM2 THM`2 lK;2#mM; U2ir- r2MM /b oQt2H h2BH /2` w2HHF2`M@J2K#`M
BbiV- b2ixi KM /2M .BzmbBQMbi2MbQ` 7Ƀ` /B2b2b oQt2H bQ- /bb Mm` T`HH2H xm` pQM /2`
lK;2#mM; /2}MB2`i2M 1#2M2 ;2;H ii2i rB`/- MB+?i #2` b2MF`2+?i xm` 1#2M2,


1
9
:
9
:
λ1
λ1
 v1 v2 v3 T . U9Xk3V
1
2 1,
≈ 1 : D = DP := v1 v2 v3 
λ3
λ2
ε
.B2b2b 6BHi2` rB`/ Hb i` ;?2Bib#bB2`i2b- MBbQi`QT2b .BzmbBQMb}Hi2` #2x2B+?M2iX
j
hib +?HB+? bBM/ /B2 1B;2Mp2FiQ`2M ;2`/2 /B2 Mb+?mHB+?2M >mTi+?b2M /2b KbbBp2M Eƺ`T2`b- r2MM
/B2b2` Ƀ#2` bQH+?2 p2`7Ƀ;iX
9d
9X8 :Bii2`;2M2`B2`mM; mb ;2}Hi2`i2M "BH//i2M
9X8XR AbQ~ +?2M#2`2+?MmM;
• HH2 SmMFi2- /B2 xm 2BM2` ;2;H ii2i2M J2K#`M ;2?ƺ`2M- ?#2M ?MHB+?2 :`mr2`i2X
• 1BM2 AbQ~ +?2 p2`H m7i /m`+? SmMFi2 KBi B/2MiBb+?2M :`mr2`i2MX
9X8Xk oQM "BH/ xm :Bii2`
RX .2}MBiBQM 2BM2b :`mr2`ib G∗
kX J`+?BM; h2i`?2/` UQ/2` J`+?BM; *m#2bV
3 oQt2H
w2`H2;2 BM 8 h2i`2/2`X
AM D2/2K h2i`2/2`@EMQi2M HB2;i 2BM :`mr2`iX
V AMi2`TQHB2`2 UHBM2`V 2MiHM; HH2` h2i`2/2`@EMi2M mM/ #2biBKK2 /B2 SQbBiB@
QM pQM G∗ X
xX"X G∗ = 90
#V o2`#BM/2 HH2 AbQ~ +?2MTmMFi2 xm 2BM2K ;2b+?HQbb2M2M P#2`~ +?2M;Bii2` Uj@
1+Fb;Bii2`VX
oQHmK2M;Bii2`, b2?` T`Q#H2K#? M;B;
⇒ pB2H2 :Bii2`;2M2`iQ`2MX
"bTX, oQM _Q?@ → P#2`~ +?2MfoQHmK2M;Bii2` U6QHB2MV
93
8 Gƺb2M /2` 6BHi2`@ mM/ JQ/2HH;H2B+?mM;2M
AM /B2b2K ETBi2H r2`/2M BM ;`Q#2M wɃ;2M h2+?MBF2M pQ`;2bi2HHi- KBi /2`2M >BH72 T`@
iB2HH2 .Bz2`2MiBH;H2B+?mM;2M MmK2`Bb+? /BbF`2iBbB2`i mM/ ;2Hƺbi r2`/2M FƺMM2MX .B2b2
o2`7?`2M r2`/2M 2bb2MiB2HH- bQ#H/ KM bB+? m7 /2K h2``BM /2` /`2B/BK2MbBQMH2M
.`bi2HHmM; pQM /2iBHHB2`i2M w2HH;2QK2i`B2M mM/ /`m7 xm KQ/2HHB2`2M/2M #BQHQ;Bb+?2M
S`Qx2bb2M #2}M/2iX .B2b2b ETBi2H /B2Mi mbb+?HB2ĽHB+? Hb 1BM#HB+F BM /B2 J2i?Q/2M /2`
JQ/2HHB2`mM; mM/ aBKmHiBQMX .2` BMi2`2bbB2`i2 G2b2` bQHHi2 bB+? /2b?H# /KBi MB+?i xm@
7`B2/2M ;2#2M mM/ #2;H2Bi2M/2 oQ`H2bmM;2M /2` JQ/2HHB2`mM; mM/ aBKmHiBQM #2bm+?2MX
8XR .b JQ/2HHT`Q#H2K
.B2 BK pQ`?2`B;2M ETBi2H ?2`;2H2Bi2i2 .BzmbBQMb;H2B+?mM;- r2H+?2 /2M .BzmbBQMb}Hi2`
/2}MB2`i2- /B2Mi mMb ?B2` Hb JQ/2HH;H2B+?mM;X
AM 2BM2K #2b+?` MFi2M :2#B2i Ω ⊂ Rd ;2Hi2,
∂u
= D∆u.
∂t
U8XRV
. /b Gƺb2M 2BM2` bQH+?2M FQMiBMmB2`HB+?2M .Bz2`2MiBH;H2B+?mM; 2BM mM2M/HB+?@/BK2M@
bBQMH2b S`Q#H2K Bbi U/B2 GƺbmM; Kmbb xm mM2M/HB+? pB2H2M w2Bi@ mM/ _mKTmMFi2M #2@
`2+?M2i r2`/2MV- #2bi2?i /2` 2`bi2 a+?`Bii #2B /2` MmK2`Bb+?2M TT`QtBKiBQM /`BM- /b
S`Q#H2K m7 2BM 2M/HB+?@/BK2MbBQMH2b xm `2/mxB2`2MX
.xm /BbF`2iBbB2`i KM Ƀ#HB+?2`r2Bb2 xmM +?bi /b :2#B2i- m7 /2K /B2 .Bz2`2MiBH;H2B@
+?mM; /2}MB2`i Bbi- rBHH ?2BĽ2M, JM M ?2`i /b :2#B2i /m`+? 2BM UQ7iV TQHv;QMH #2`M@
/2i2b :2#B2i M mM/ i`BM;mHB2`i /B2b2b /MM- /X?X KM x2`H2;i 2b BM FH2BM2 h2BHK2M;2M/B2 2BM7+?2 :2biHi ?#2M UxmK "2BbTB2H BM .`2B2+F2VX 6Ƀ` /B2 wr2+F2 /B2b2b #b+?MBiib
rB` /b i`BM;mHB2`i2 /BbF`2i2 :2#B2i BKK2` 2BM mMB7Q`K2b :Bii2` mb d@/BK2MbBQMH2M
Zm/2`M /2` a2Bi2MH M;2 h b2BMX
.MM Dz/ɃMMidz KM /2M GƺbmM;b`mK mb- BM/2K KM Mm` MQ+? BM /2M 2M/HB+? pB2H2M
:Bii2`TmMFi2M U/2M 1+FTmMFi2M /2` Zm/2`V 2BM2 GƺbmM; #2`2+?M2iX .B2 GƺbmM; /2b
/BbF`2i2M S`Q#H2Kb Bbi HbQ 2BM o2FiQ` KBi 6mMFiBQMbr2`i2M M /2M :Bii2`TmMFi2MX
. /2` FQMiBMmB2`HB+?2 "2;`Bz /2` #H2BimM; 7Ƀ` 2BM2 6mMFiBQM- /B2 Mm` MQ+? BM BbQHB2`i2M
1BMx2HTmMFi2M /2}MB2`i Bbi- b2BM2M aBMM p2`HB2`i- Kmbb KM m+? /b /BbF`2i2 S`Q#H2K
MTbb2M Ĝ mM/ xr` /2`;2biHi- /bb /B2 GƺbmM; /B2b2b S`Q#H2Kb m7 bBMMpQHH2 q2Bb2
DzM?2dz M /2` GƺbmM; /2b FQMiBMmB2`HB+?2M S`Q#H2Kb HB2;i- HbQ #2BbTB2Hbr2Bb2 bQ- /bb
/B2 q2`i2 /2` /BbF`2i2M mM/ /2` FQMiBMmB2`HB+?2M GƺbmM; BM /2M /2}MB2`i2M :Bii2`TmMF@
i2M Kƺ;HB+?bi M? #2BbKK2M HB2;2MX
9N
.xm TT`QtBKB2`i KM m+? /B2 .Bz2`2MiBHQT2`iQ`2M m7 2BM2 /BbF`2i2 q2Bb2- bQ/bb
bB2 2BM2M o2FiQ` m7 2BM2M o2FiQ` ##BH/2M Ubiii 2BM2 6mMFiBQM m7 2BM2 6mMFiBQMVX
HH2 /BbF`2iBbB2`i2M P#D2Fi2 2`?Hi2M /2M AM/2t h - bQ/bb /b /BbF`2i2 S`Q#H2K bB+? BM
/2` 7QH;2M/2M 6Q`K /`bi2HHi,
∂ h uh
= −D∆h uh
∂t
m7 Ωh .
U8XkV
1BM2 b2?` M?2HB2;2M/2 `i- /B2 .Bz2`2MiBHQT2`iQ`2M xm /BbF`2iBbB2`2M- Bbi /B2 J2i?Q@
/2 /2` 6BMBi2M .Bz2`2Mx2M U6.V, . #H2BimM;2M MB+?ib M/2`2b bBM/ Hb .Bz2`2MiBH@
[mQiB2Mi2M UHbQ :`2Mxr2`i2 pQM .Bz2`2Mx2M[mQiB2Mi2MV- TT`QtBKB2`i KM bB2 /m`+?
.Bz2`2Mx2M[mQiB2Mi2M5
8XRXR o2`7?`2M BM R.
a+?`Bii R, TT`QtBKB2`2
∂u
∂t X
"2BbTB2H, 1tTHBxBi2b 1mH2`p2`7?`2MX
JM #2i`+?i2 /b S`Q#H2K u" (t) = f (t, u(t))- #2B /2K /B2 #H2BimM; 2BM2` mM#2FMi2M
6mMFiBQM /m`+? 2BM2M mb/`m+F f ;2;2#2M Bbi- /2` pQM /2` 6mMFiBQM b2H#bi mM/ pQM /2`
w2Bi #? M;iX JBi /2` L ?2`mM; /m`+? .Bz2`2Mx2M[mQiB2Mi2M 2`;B#i bB+?
u" (t) ≈
u(t + ht ) − u(t)
ht
mM/ /`mb 2`? Hi KM Hb Ai2`iBQMb7Q`K2H
uht (t + ht ) − uht (t)
= f (t, uht (t))
ht
⇔
uht (t + ht ) = uht + ht · f (t, uht (t))
U8XjV
U8X9V
JM #2`2+?M2i HbQ uht (t + ht ) Hb 2BM2 TT`QtBKiBQM pQM u(t + ht )X
LmK2`Bb+? Bbi bB+?2`xmbi2HH2M- /bb /b o2`7?`2M FQMp2`;B2`i- /X?X
uh t → u
7Ƀ` ht → 0.
U8X8V
HH;2K2BM, .b 1mH2`p2`7?`2M ;2?ƺ`i xm /2M 1BMb+?`Biip2`7?`2MX .B2 :H2B+?mM;
u" (t) = f (t, u(t))
rB`/ BM /BbF`2i2M w2BiTmMFi2M t + k · ht , (k = 1 . . . n) TT`QtBKB2`i /m`+?
uh (t + h) = uh (t) + h · Φh (t, uh (t), uh (t + h))
U8XeV
.#2B /2}MB2`i /B2 bQ;X o2`7?`2Mb7mMFiBQM Φ 2BM2 TT`QtBKiBQM M f X
qB+?iB;, "2B /2` EQMbi`mFiBQM pQM TT`QtBKiBQMbp2`7?`2M Bbi m7 /2`2M EQMbBbi2Mx
mM/ EQMp2`;2Mx xm +?i2M,
8y
RX EQMbBbi2Mx
Φh → f
7Ƀ` h → 0
U8XdV
kX EQMp2`;2Mx
uh → u
7Ƀ` h → 0
U8X3V
o2`7?`2M ?ƺ?2`2` P`/MmM;
"2i`+?i2 /B2 hvHQ`2MirB+FHmM;
h2
hp
u(t + h) = u(t) + h · u" (t) + u"" (t) + . . . + u(p) (t) + _2bi
)
*+
, 2
p!
)
2`bi2 P`/MmM; U1mH2`V
*+
U8XNV
,
kX P`/MmM;
#? M;B; /pQM- rQ /B2 hvHQ`2MirB+FHmM; #;2#`Q+?2M rB`/- Bbi /B2 ZmHBi i mM/ /B2
P`/MmM; /2b o2`7?`2Mb #2biBKKiX
a+?`Bii k, TT`QtBKB2`2 ∆u X
AK 2BM/BK2MbBQMH2M 6HH ;BHi, ∆u = u"" X
A/22, MHQ; xm` w2Bii2`[email protected]`2iBbB2`mM; FMM ?B2` 2#2M7HHb 2BM .Bz2`2Mx2Mp2`7?`2M
M;2rM/i r2`/2MX
.Bz2`2Mx2Mp2`7?`2M 7Ƀ` −∆u = f
RX TT`QtBKiBQM pQM m" X
1b ;BHi
u" (x) = HBK
h→0
)
u(x + h) − u(x)
*+ h
,
U8XRyV
.Bz2`2MiBH[mQiB2Mi
TT`QtBKB2`2 KBi .Bz2`2Mx2M[mQiB2Mi
u(x + h) − u(x)
h
u(x)
−
u(x − h)
u" (x) ≈
h
u(x
+
h)
− u(x − h)
u" (x) ≈
2h
u" (x) ≈
Q/2`
Q/2`
8R
`2+?ibb2BiB;
U8XRRV
HBMFbb2BiB;
U8XRkV
x2Mi`H
U8XRjV
DzZmHBi idz /B2b2` TT`QtBKiBQM2M,
hvHQ`, JBi 2BM2K ξ1 ∈ (x − h, x), #xrX ξ2 ∈ (x, x + h) ;BHi 7Ƀ` `2+?ib@ mM/ HBMFbb2BiB;2
.Bz2`2Mx2M[mQiB2Mi2M
h2 ""
u (ξ2/1 )
2
u(x + h) − u(x)
h
= u" (x) + u"" (ξ2 )
h
2
u(x) − u(x − h)
h ""
"
= u (x) − u (ξ1 )
h
2
u(x ± h) = u(x) ± hu" (x) +
⇒
mM/
U8XR9V
U8XR8V
U8XReV
U8XRdV
mM/ 72`M2` 7Ƀ` /2M x2Mi`H2M .Bz2`2Mx2M[mQiB2Mi2M
h2 ""
h3
u (x) ± u""" (ξ2/1 )
2
6
2
9
:
u(x + h) − u(x − h)
h
= u" (x) +
u""" (ξ1 ) + u""" (ξ2 )
2h
6
u(x ± h) = u(x) ± hu" (x) +
⇒
U8XR3V
U8XRNV
.B2 P`/MmM; /2b `2+?ib@ U∂ + V #xrX HBMFbb2BiB;2M U∂ − V .Bz2`2Mx2M[mQiB2Mi2M Bbi HbQ Rr ?`2M/ /2` x2Mi`H2 .Bz2`2Mx2M[mQiB2Mi P`/MmM; k 2`xB2HiX
kX TT`QtBKiBQM pQM m"" (t)
A/22, HBMFb@ mM/ `2+?ibb2BiB;2 2`bi2 #H2BimM;2M FQK#BMB2`2M,
− +
(∂ ∂ u)(x) :=
=
u(x+h)−u(x)
h
− u(x)−u(x−h)
h
h
u(x + h) − 2u(x) + u(x − h)
h2
U8XkyV
hvHQ`2MirB+FHmM;,
h2 ""
h3
h4
u (x) ± u""" (x) + u(4) (ξ2/1 )
U8XkRV
2
6
4!
5
h4 4 (4)
⇒ u(x + h) + u(x − h) = 2u(x) + h2 u"" (x) +
u (ξ1 ) + u(4) (ξ2 )
4!
5
2 4
h
⇒ (∂ − ∂ + u)(x) = u"" (x) +
u(4) (ξ1 ) + u(4) (ξ2 )
24
u(x ± h) = u(x) ± hu" (x) +
.B2b2b .Bz2`2Mx2Mp2`7?`2M 7Ƀ` u"" ?i HbQ /B2 P`/MmM; kX
wm #2+?i2M Bbi- /bb /B2b2 P`/MmM; UmM/ 2MibT`2+?2M/ ;BHi /B2b m+? 7Ƀ` /B2 M/2`2M
o2`7?`2MV Mm` mMi2` /2` MM?K2 ;2MɃ;2M/2` _2;mH`Bi i /2` GƺbmM; ;BHi- /KBi /B2
#H2BimM;2M /2` #2MƺiB;i2M P`/MmM; U?B2`, KBM/2bi2Mb 95V Ƀ#2`?mTi 2tBbiB2`2MX P# /B2@
b2 6Q`/2`mM; 2`7ɃHHi Bbi- ? M;i bQrQ?H pQM /2M :H2B+?mM;2M Hb m+? pQK :2#B2i #- m7
8k
/2K bB2 ;2Hi2MX 6Ƀ` /B2 K2Bbi2M S`Q#H2K2 /2` Mr2M/mM; FMM pB2`7+?2 .Bz2`2MxB2`@
#`F2Bi /m`+?mb MB+?i ;2x2B;i r2`/2M5
JBi u ∈ C 4 (Ω̄) #2` FMM HbQ 7Ƀ` /b FQMiBMmB2`HB+?2 S`Q#H2K
− ∆u = f (BM Ω)
U8XkkV
− ∂ − ∂ + uh (x) = f (x) (BM Ωh )
U8XkjV
2BM /BbF`2i2b S`Q#H2K /2` 6Q`K
7Q`KmHB2`i r2`/2M- #2B /2K KM 2BM2M HQFH2M 62?H2` pQM O(h2 ) K+?iX
.B2b2 /BbF`2i2 :H2B+?mM; Kmbb BM D2/2K /2` BMM2`2M SmMFi2 pQM Ωh ;2Hi2M- rb BK
2BM/BK2MbBQMH2M 6HH 7Ƀ` /B2 n + 1 lM#2FMMi2M UBMM2`2 SmMFi2 mM/ #2B/2 _M/TmMF@
i2V n − 1 :H2B+?mM;2M ;2M2`B2`iX .B2 Ƀ#`B;2M #2B/2M :H2B+?mM;2M 2`;2#2M bB+? mb /2M
_M/#2/BM;mM;2MX AK 6HH pQM .B`B+?H2i@_M/#2/BM;mM;2M UoQ`;#2 /2` GƺbmM; BM /2M
_M/TmMFi2MV `2/mxB2`i bB+? /B2 Mx?H /2` lM#2FMMi2M mK /B2 _M/FMQi2MX
lK /b /BbF`2i2 S`Q#H2K Hb :H2B+?mM;bbvbi2K 7Q`KmHB2`2M xm FƺMM2M- 7bbi KM /B2
6mMFiBQMbr2`i2 BM /2M :Bii2`TmMFi2M BM 2BM2K o2FiQ` xmbKK2M- 2BM2` bQ;X :Bii2`@
7mMFiBQMX q2MM /b :2#B2i Ωh ;2`/2 /b AMi2`pHH (0, 1) Bbi mM/ h = n−1 - ?i /B2b2
:Bii2`7mMFiBQM /B2 :2biHi


uh (h)
 uh (2h) 


uh = 
U8Xk9V
.
XX


X
uh (1 − h)
.MM FMM KM /b /BbF`2i2 S`Q#H2K FQKTFi Hb :H2B+?mM;bbvbi2K
Lh u h = q h
b+?`2B#2M KBi

2 −1
 −1
2 −1

1 
−1
2
Lh = 2 
h 

X
XX





XX

X


XX
X −1 
−1
2



mM/ qh = 

f (h) + h−2 ϕ0
f (2h)
XX
X
f (1 − h) + h−2 ϕ1
rQ ϕ0 mM/ ϕ1 /B2 HBMF2M #xrX `2+?i2M .B`B+?H2i@_M/r2`i2 bBM/X
8j



,

wmbKK2M7bbmM;
∂u
= D · (∆u)
) *+ ,
∂t
)*+,
TT`QtBKiBQM /m`+? 1aoxX"X 1mH2`
u(t + ht ) − u(t)
u" (t) ≈
ht
U8Xk8V
TT`QtBKiBQM /m`+? .Bz2`2Mx2Mp2`7?`2Mc
Gƺb2 :H2B+?mM;bbvbi2K Lh uh = qh
1BM2 Jƺ;HB+?F2Bi- w2Bi@ mM/ P`ib/BbF`2iBbB2`mM; xmbKK2Mxm7Ƀ?`2M- Bbi /B2 bQ;X _Qi?2@
J2i?Q/2X .#2B `Ƀ+Fi KM b+?`Biir2Bb2 BM /2` w2Bi pQ` mM/ Hƺbi /#2B BM D2/2K /2`
a+?`Bii2 2BM /MM Mm` MQ+? Q`ib#? M;B;2b :H2B+?mM;bbvbi2KX
6Ƀ` /B2 .BzmbBQMb;H2B+?mM; BM R. 2`;B#i bB+? BM D2/2K BMM2`2M SmMFi x #2B o2`r2M/mM;
/2b BKTHBxBi2M 1mH2`@o2`7?`2Mb mM/ /2` ∂ − ∂ + @.BbF`2iBbB2`mM; /2` xr2Bi2M #H2BimM;,
uh (t + ht , x) = uh (t, x) +
ht D
(uh (t + ht , x − h) − 2uh (t + ht , x) + uh (t + ht , x + h)) ,
h2
rb UrB2/2` 7Ƀ` Ωh = (0, 1) mM/ h = n−1 V BM D2/2K w2Bib+?`Bii xm 2BM2K :H2B+?mM;bbvbi2K
/2` 7QH;2M/2M 6Q`K 7Ƀ?`i,


2

 
2 + hht D
−1
uh (t + ht , h)
uh (t, h) + hht 2D ϕ0


2
X
  uh (t + ht , 2h)  
XX
uh (t, 2h)
ht D 
−1 2 + hht D


 
=





X
XX
XX
XX
XX

h2 
 
X
X
X
−1  

2
uh (t + ht , 1 − h)
uh (t, 1 − h) + hht 2D ϕ1
−1 2 + hht D
8XRXk .Bz2`2Mx2Mp2`7?`2M BM k.
:2;2#2M b2B /b `2;mH `2 [m/`iBb+?2 :2#B2i
Ω = (0, 1) × (0, 1) = {(x, y) : 0 < x < 1, 0 < y < 1} .
U8XkeV
.BbF`2iBbB2`mM;
.b :2#B2i Ω rB`/ xm 2BM2` J2M;2 Ωh pQM (n − 1) × (n − 1) :Bii2`TmMFi2M mM/ /2` _M/
∂Ω xm 2BM2` J2M;2 Γh pQM 4n :Bii2`TmMFi2M xm` :Bii2`r2Bi2 h /BbF`2iBbB2`i,
KBi h =
Ωh = {(x, y) ∈ Ω : x/h, y/h ∈ Z}
Γh = {(x, y) ∈ ∂Ω : x/h, y/h ∈ Z} .
qB` #2i`+?i2M rB2/2` /b SQBbbQM@S`Q#H2K
−∆u = −uxx − uyy = f
u=ϕ
89
BM Ω,
m7 Γ = ∂Ω.
1
,
n



.

TT`QtBKiBQM /m`+? HBMFb@ mM/ `2+?ibb2BiB;2 .Bz2`2Mx2M BM #2B/2 _mK/BK2MbBQM2M,
9
:
(−∆h u)(x, y) := −∂x− ∂x+ − ∂y− ∂y+ u(x, y)
U8XkdV
=
−h−2 (u(x − h, y) + u(x + h, y)
+u(x, y − h) + u(x, y + h) − 4u(x, y))
.B2b2 .Bz2`2Mx2M7Q`K2H rB`/ m+? 6ɃM7@SmMFi@ai2`M ;2MMMi- KM MQiB2`i bB2 m+? Fm`x


−1
− ∆h = h−2 −1 4 −1 .
U8Xk3V
−1
LmM bQHH mb /B2b2` .BbF`2iBbB2`mM; 2BM HBM2`2b :H2B+?mM;bbvbi2K ?2`pQ`;2?2MX q ?`2M/
/B2 MQ`/MmM; /2` :H2B+?mM;2M BK 2BM/BK2MbBQMH2M 6HH m7 MiɃ`HB+?2 q2Bb2 UpQM HBMFb
M+? `2+?ibV 2`7QH;i- ?i KM BM K2?` _mK/BK2MbBQM2M p2`b+?B2/2M2 Jƺ;HB+?F2Bi2M- /B2
:Bii2`FMQi2M xm MmKK2`B2`2M- mb /2M2M bB+? mMi2`b+?B2/HB+?2 :2biHi2M 7Ƀ` /B2 Ji`Bt
/2b :H2B+?mM;bbvbi2Kb 2`;2#2MX
"2B /2` H2tBFQ;`T?Bb+?2M MQ`/MmM; MmKK2`B2`i KM /B2 :Bii2`TmMFi2 xm2`bi x2BH2M@
r2Bb2 mM/ /B2 w2BH2M /MM bTHi2Mr2Bb2 UQ/2` mK;2F2?`iV- HbQ
(h, h), (2h, h), . . . , (1 − h, h); (h, 2h), . . . , (1 − h, 2h); . . . ; (h, 1 − h), . . . , (1 − h, 1 − h).
1b 2Mibi2?i 2BM :H2B+?mM;bbvbi2K

T −I

−I T
Lh = h−2 

XX

X
Lh uh = qh KBi

XX


,

XX
X −I 
−I T
X

4

−1
T =


−1
4
XX
X
XX



.

XX
X −1
−1 4
X
T mM/ I bBM/ (n − 1) × (n − 1)@Ji`Bx2M- I Bbi /B2 1BM?2BibKi`BtX
AMb;2bKi Bbi /B2 avbi2KKi`Bt BK 6HH2 H2tBFQ;`T?Bb+?2` P`/MmM; HbQ 2BM2 /ɃMM #2@
b2ixi2 U"M/@VJi`Bt KBi 1BMi` ;2M BM /2` .B;QMH2M mM/ /2M 2`bi2M mM/ (n − 1)@i2M
L2#2M/B;QMH2MX
8Xk 6BMBi2 1H2K2Mi2 o2`7?`2M
"2/2mi2M/2 L+?i2BH2 /2` 6BMBi2M .Bz2`2Mx2M,
• ai`mFim`B2`i?2Bi /2b :Bii2`b Bbi oQ`mbb2ixmM; 7Ƀ` /B2 bBMMpQHH2 Mr2M/mM; /2`
6BMBi2M .Bz2`2Mx2MX
• LmK2`Bb+?2 MHvb2 /2` 6BMBi2 .Bz2`2Mx2M@o2`7?`2M p2`HM;i ?Q?2 _2;mH`Bi i
/2` GƺbmM; Uu ∈ C 4 (Ω)VX
AK 6QH;2M/2M rB`/ 2BM o2`7?`2M pQ`;2bi2HHi- /b BM #2B/2M SmMFi2M pQ`i2BH?7i2`2 1B@
;2Mb+?7i2M ?i mM/ /2` MHvb2 KBi Ki?2KiBb+?2M J2i?Q/2M ;mi xm; M;HB+? BbiX
88
8XkXR PT2`iQ`2M
qB` r2`/2M BM HH2` EɃ`x2 /b EQMx2Ti pQM 6mMFiBQM2M` mK2M Hb mM2M/HB+?@/BK2MbBQ@
MH2M o2FiQ`` mK2M M?M/ 2BM2b 2BM7+?2M 2BM/BK2MbBQMH2M "2BbTB2Hb KBi /2M r2b2Mi@
HB+?2M "2;`Bz2M bFBxxB2`2MX
.2}MBiBQM RRX a2B A ∈ Rn×n 2BM2 Ji`BtX .MM Bbi /B2 i`MbTQMB2`i2 Ji`Bt AT ∈ Rn×n
/2}MB2`i Hb /B2D2MB;2 Ji`Bt- r2H+?2 7Ƀ` #2HB2#B;2 o2FiQ`2M u, v ∈ Rn /B2 "2/BM;mM;
U8XkNV
(Au, v) = (u, AT v)
2`7ɃHHi- rQ#2B (·, ·) /b aiM/`/bFH`T`Q/mFi /2b Rn /`bi2HHiX
AK 2M/HB+?@/BK2MbBQMH2M Rn Bbi /b aFH`T`Q/mFi 2BM2 FQKTQM2Mi2Mr2Bb2 amKK2X qB2
Ƀ#2`i` ;i bB+? /B2b2b EQMx2Ti m7 2BM2M mM2M/HB+?@/BK2MbBQMH2M 6mMFiBQM2M`mK\
.2}MBiBQM RkX a2B2M u, v 6mMFiBQM2M mb 2BM2K 6mMFiBQM2M`mKR X .MM rB`/ /B2 JmH@
iBTHBFiBQM (u, v) /2}MB2`i /m`+?
(u, v) :=
ˆ1
U8XjyV
u(x)v(x) dx.
0
mb /2` amKK2 BK 2M/HB+?@/BK2MbBQMH2M o2FiQ``mK rB`/ HbQ 2BM AMi2;`H BK mM2M/HB+?@
/BK2MbBQMH2M 6mMFiBQM2M`mKX
qb #2/2mi2M MmM A mM/ AT BK 6mMFiBQM2M`mK\ mb /2` Ji`Bt A rB`/ BK 6mMFiBQ@
M2M`mK 2BM PT2`iQ` mM/ mb AT rB`/ 2BM PT2`iQ` A∗ - /2` Hb /DmM;B2`i2` PT2`iQ`
#2x2B+?M2i rB`/ mM/ ;2MmbQ /2}MB2`i Bbi rB2 BK 2M/HB+?@/BK2MbBQMH2M 6HH- HbQ 7Ƀ` HH2
6mMFiBQM2M u, v mb /2K xm;`mM/2HB2;2M/2M _mK /B2 7QH;2M/2 1B;2Mb+?7i 2`7ɃHHi,
U8XjRV
(Au, v) = (u, A∗ v).
d
"2BbTB2H, a2B A = dx
/2` #H2BimM;bQT2`iQ`- /2` 2BM2 6mMFiBQM m7 /2`2M #H2BimM;
##BH/2iX .MM ;BHi 7Ƀ` HH2 u, v KBi LmHHr2`i2M K _M/ /2b AMi2`pHHb [0, 1],
(Au, v) =
⇒
R
!
d
dx
"∗
!
d
u, v
dx
"
=
ˆ1
d
u(x)v(x) dx =
dx
0
ˆ1
0
d
=− .
dx
!
"
A1
d
A
u(x) · − v(x) dx + u v A
dx
) *+ ,0
=0
:2K2BMi Bbi T` xBb2 /2` _mK L2 (Ω) U?B2` KBi Ω = [0, 1]V- rB2 2` BM D2/2K G2?`#m+? Ƀ#2` JĽ@ mM/
AMi2;`iBQMbi?2Q`B2 xm }M/2M b2BM bQHHi2X
8e
8XkXk a+?r+?2 6Q`KmHB2`mM;
qB2 Q#2M 2`r ?Mi- Bbi #2B [email protected]`2Mx2M@o2`7?`2M xX"X /B2 6Q`/2`mM; pQM ?Q?2`
_2;mH`Bi i M /B2 GƺbmM; u 2BM L+?i2BHX .m`+? /2M ɂ#2`;M; xm 2BM2` bQ;2MMMi2M
b+?r+?2M 6Q`KmHB2`mM; /2` T`iB2HH2M .Bz2`2MiBH;H2B+?mM; Hbb2M bB+? /B2 _2;mH`B@
i ib7Q`/2`mM;2M #b+?r +?2MX qB2 #2B 6BMBi2M .Bz2`2Mx2M r2`/2M rB` m+? ?B2` 2BM xm
Hƺb2M/2b HBM2`2b :H2B+?mM;bbvbi2K Hb MmK2`Bb+?2 TT`QtBKiBQM /2` GƺbmM; ?2`H2B@
i2MX aiii D2/Q+? xm #2`2+?M2M/2 #H2BimM;2M xm TT`QtBKB2`2M mM/ /`mb 2BM HBM2`2b
:H2B+?mM;bbvbi2K m7xmbi2HH2M- r2`/2M rB` ?B2` BK 6mMFiBQM2M`mK ;22B;M2i2 6mMFiBQ@
M2M bm+?2M- /B2 m7 2BM HBM2`2b :H2B+?mM;bbvbi2K 7Ƀ?`2MX
"2i`+?i2M rB` mMb2`2 JQ/2HH;H2B+?mM;- /B2 2BM/BK2MbBQMH2 SQBbbQM@:H2B+?mM;,
!
"
d
d
−
D(x) u = f.
U8XjkV
dx
dx
qB` #BH/2M /b aFH`T`Q/mFi KBi 2BM2` bQ;X h2bi7mMFiBQM v(x),
!
!
" "
d
d
−
D(x) u , v = (f, v)
dx
dx
!
"
ˆ1
ˆ1
d
d
⇔ −
D(x) u v(x) dx =
f (x)v(x) dx
dx
dx
0
U8XjjV
U8Xj9V
0
qB` b2ix2M /2` ɂ#2`bB+?i ?H#2` BK 6QH;2M/2M D = 1 mM/ 2`?Hi2M KBi?BH72 T`iB2HH2`
AMi2;`iBQM
ˆ1
0
d
−
dx
!
"
du
(x) · v(x) dx
dx
=
ˆ1
0
A1
du
dv
du
A
(x) (x) dx −
(x)v(x)A
dx
dx
dx
0
U8Xj8V
U6Ƀ` /2M 2BM7+?2`2M 6HH KBi u(x) = v(x) = 0 m7 Γ - /2M rB` BK 6QH;2M/2M #2i`+?i2MA1
A
;BHi − du
(x)v(x)
A = 0XV
dx
0
q`mK K+?2M rB` /b\
RX q2MM Q#B;2 b+?r+?2 6Q`KmHB2`mM; 7Ƀ` HH2 v Umb 2BM2K #2biBKKi2M 6mMFiBQM2M@
`mKV ;BHi- /MM ;BHi UBM ;2rBbb2K aBMM2V m+? /B2 bi`F2 6Q`KmHB2`mM;X
kX JBi q?H ;22B;M2i2` 6mMFiBQM2M rB`/ mb /2K FQMiBMmB2`HB+?2M 2BM /BbF`2i2b S`Q@
#H2K UbB2?2 mMi2MVX
8XkXj :H2`FBM@o2`7?`2M
L2#2M /2` q?H pQM h2bi7mMFiBQM2M v- r2H+?2 2BM2 b+?r+?2 6Q`KmHB2`mM; UAMi2;`H/`@
bi2HHmM;V /2` SQBbbQM@:H2B+?mM; 2`x2m;2M- #2MƺiB;2M rB` r2Bi2` 2BM2 TT`QtBKiBQM U
8d
mMb2`2` GƺbmM; uX .b ?B2` M;2/2mi2i2 :H2`FBM@o2`7?`2M r ?Hi bQ;2MMMi2 Mbix@
7mMFiBQM2M- bQ /bb bB+? U Hb GBM2`FQK#BMiBQM /B2b2` Mbix7mMFiBQM2M /`bi2HH2M
H bbiX .KBi 2`x2m;2M rB` b+?HB2ĽHB+? mMb2` TT`QtBKB2`2M/2b :H2B+?mM;bbvbi2KX
a+?`Bii R, q ?H2 Φ1 (x), . . . Φn (x) Mbix7mMFiBQM2MX TT`QtBKB2`2 /B2 GƺbmM; u KBi
u(x) ≈ U (x) = U1 Φ1 (x) + . . . + Un Φn (x).
U8XjeV
.B2 UMm` MQ+? 2M/HB+? pB2H2MV EQ2{xB2Mi2M U1 . . . Un KɃbb2M #2biBKKi r2`/2M mM/ /7Ƀ`
r2`/2M 2#2MbQ pB2H2 :H2B+?mM;2M #2MƺiB;iX
a+?`Bii k, q ?H2 h2bi7mMFiBQM2M V1 . . . Vn xm` "2biBKKmM; pQM U1 . . . Un X
:H2`FBM UQ`B;BMHV, q ?H2 R Q/2` k b2?` ;mi2 Vi X
6BMBi2 1H2K2Mi2 URyy C?`2 bT i2`V, q ?H2 pB2H2 b2?` 2BM7+?2 Vi - Ƀ#HB+?2`r2Bb2,
Vi = Φi i = 1 . . . nX
a+?`Bii j, "BH/2 /b :H2B+?mM;bbvbi2K KU = F X
.xm r ?Hi KM ;2rƺ?MHB+? Mbix@ mM/ h2bi7mMFiBQM2M /2``i- /bb bB2 Kƺ;HB+?bi Mm`
b2?` HQFH pQM LmHH p2`b+?B2/2M bBM/ UKM bT`B+?i pQM HQFH2K h` ;2`V- xX"X /bb D2/2
pQM B?M2M m7 ;2Mm 2BM2K /2` /BbF`2i2M :Bii2`TmMFi2 /2M q2`i R MMBKKi- m7 HH2M
M/2`2M /2M q2`i y mM/ /xrBb+?2M D2r2BHb HBM2` p2`H m7iX . /B2b xm 2BM2K ?Ƀi+?2M@
`iB;2M o2`Hm7 /2` 6mMFiBQM2M 7Ƀ?`i- M2MMi KM bB2 m+? Q7i Dz>mi7mMFiBQM2MdzX
qB2 #2`2+?M2i KM MmM /B2 1BMi` ;2 pQM K mM/ F \
C2/2 w2BH2 /2b :H2B+?mM;bbvbi2Kb `2T` b2MiB2`i /b Mr2M/2M 2BM2` h2bi7mMFiBQM mM/
D2/2 aTHi2 /2` Ji`Bt /B2 MirQ`i 2BM2` /2` Mbix7mMFiBQM2M m7 /2M aix /2` h2bi@
7mMFiBQM2M- HbQ 7Ƀ` /B2 i@i2 w2BH2,
ˆ1
ˆ1
dU
dVi
(x)
(x) dx =
f (x)Vi (x) dx
dx
dx
0
0


1
ˆ 2
ˆ1
n
dU
dΦ
j
j
⇔ 
(x) Vi (x) dx =
f (x)Vi (x) dx.
dx dx
0
j=1
0
>B2`#2B bi2?i HBMFb /b S`Q/mFi /2` i@i2M w2BH2 pQM K KBi /2K o2FiQ` /2` mM#2FMMi2M
EQ2{xB2Mi2M (Ui )i=1...n mM/ `2+?ib /2` i@i2 1BMi`; /2b o2FiQ`b F X 6Ƀ` /2M (i, j)@i2M
1BMi`; pQM E ?i KM /2KM+?
Kij =
ˆ1
dVj
dΦi
(x)
(x) dx.
dx
dx
0
.B2 q?H pQM Mbix@ mM/ h2bi7mMFiBQM2M KBi HQFH2K h` ;2` bB+?2`i ?B2`- /bb /B2
K2Bbi2M /B2b2` 1BMi` ;2 LmHH bBM/- M KHB+? /MM- r2MM bB+? /B2 h` ;2` pQM Mbix@ mM/
83
h2bi7mMFiBQM MB+?i b+?M2B/2M- bB2 HbQ BM F2BM2K oQHmK2M ;H2B+?x2Bi; mM;H2B+? MmHH bBM/X
.?2` Bbi K 2BM2 /ɃMM #2b2ixi2 Ji`Bt- rQ/m`+? /b :H2B+?mM;bbvbi2K bB+? KBi FHbbB@
b+?2M MmK2`Bb+?2M o2`7?`2M H2B+?iU2`V Hƺb2M H bbiX
.b ;2Mm2 "2b2ixmM;bKmbi2` /2` Ji`Bt ;2Mm rB2 B?`2 1BMi` ;2 ? M;2M #2` BK HH@
;2K2BM2M BMb#2bQM/2`2 pQM /2` q?H /2b :Bii2`b #X 6Ƀ` 2BM 2BM/BK2MbBQMH2b :Bii2` KBi
[mB/BbiMi2M EMQi2M 2`;B#i bB+? D2/Q+? KBi >mi@Mbix@ mM/ @h2bi7mMFiBQM2M U7biV /B2
;H2B+?2 Ji`Bt rB2 /m`+? /B2 Mr2M/mM; 6BMBi2` .Bz2`2Mx2M UrB2 KM H2B+?i M+?T`Ƀ7iVM KHB+?


2 −1


XX


X
2
−1 −1
.
K=h 

XX
XX

X
X −1
−1 2
8N
e JQ/2HHB2`mM; pQM L2m`QM2M AAA,
j.@JQ/2HH2 pQM x2HHmH `2M mM/
bm#x2HHmH `2M S`Qx2bb2M
ey