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Gene
SynaptischeKopplung
kleine Netzwerke
Neuronen
Gehirnregionen undlokale Schaltkreise
Verbindung vonGehirnarealen
Verhalten
Zelluläre Ebene
Netzwerk Ebene
Areale
Global
überlebenswichtige Proteine (Kanäle,Membran, Messenger ...)
Kanalaktivität, Signalempfang, Signalweiterleitung
Synaptische Kopplung, Neurotransmitter, Rezeptoren
Zusammenfassung von funktionellen Einheiten
Makroskopische Informationsverarbeitung
Molekulare Ebene
(biochemische Messungen)
(Elektrophysiologie, Imaging-Verfahren)
(Multielektrodensysteme)
(MRT, EEG ...)
(Beobachtung)
tischeung
etzwer
onen
ionenaltkre
ung vorealen
apppl
Ne
uro
regSch
ndrna
SynaptischeKopplung
kleine Netzwerke
Neuronen
Gehirnregionen und Schaltkreise
Verbindung vonGehirnareale
Gene
•
•
•
•
•
•
⇒
•
•
•
•
•
•
• ⇒
•
•
•
•
•
•
•
•
•→
•
•
•
•→
•
•
•
•
•
•
•
•
⇒
•
•
•
⇒
→
•
•
•
•→ →
→ →
•
•
•
•
•
•
•
•
•
•
•
+
+
2+
2+
−−3
⇒ ⇒
IL RL rLL πa2 x = 0
V1 x = L V2
IL =V2 − V1
RL
RL = rLL
πa2
⇒
Cm Q Vm
Q = CmVm
Cm A
cm
cm :=Cm
A
cm = 10nF/mm2
Rm =∆V
Ie
Ie ∆V
rmτm
τm
τm := rmcm.
Vm =Vi − Va
z · q E
E ≥ −zqV
T kb
P (E ≥ −zqV ) = (zqV /kbT )
R FVT
kbT
VT =R · TF
(=
kbT
q
)
V = Vgg
[ ] · 1 = [ ] · P (E ≥ −zqV )
⇔ [ ] = [ ] (zVgg/VT )
⇔ Vgg =VT
z
([ ]
[ ]
)( )
⇒
Vm =RT
F
∑K
PK [K+]a +∑APA[A−]i
∑K
PK [K+]i +∑APA[A−]a
•
•
jIon
(∗) j = −D
(d [ ]
dx− z F
RT· Vm
L[ ]
)
z :L :
j = −D
d [ ]
dx︸ ︷︷ ︸− z F
RT· Vm
L[ ]
︸ ︷︷ ︸
(2.11) ⇔d[I]dx
− jIDI
+ zIFRT · Vm
L [I]= 1
⇒L
0
d[I]dx
− jIDI
+ zIFRT · Vm
L [I]dx =
L
0
1 dx
⇔ RTL
zIFVm
[ (− jIDI
+zIF
RT· Vm
L[I]a
)−
(− jIDI
+zIF
RT· Vm
L[I]i
)]= L
⇔ zIFVm
RT=
(− jI
DI+ zIF
RT · VmL [I]a
− jIDI
+ zIFRT · Vm
L [I]i
)
⇔ ezIFVm
RT =− jI
DI+ zIF
RT · VmL [I]a
− jIDI
+ zIFRT · Vm
L [I]i
µ µ :=FVm
RT:
⇔ ezIµ(− jIDI
+zIF
RT· Vm
L[I]i
)= − jI
DI+
zIF
RT· Vm
L[I]a
⇔ jI = −DIzIµ
L
[I]i · ezIµ − [I]a(1− ezIµ)
PI=DIL⇔ jI = PI · zI · µ · [I]a − [I]i · ezIµ
(1− ezIµ)
JI IJI := zIF · jI
J =∑
I JI = 0 nI = ±1+ + −
K zK = +1
JK = PKµF[K]a − [K]i e
µ
1− eµ,
A zA = −1
JA = PAµF[A]a − [A]i e
−µ
1− e−µ= PAµF
[A]i − [A]a eµ
1− eµ.
∑I JI = 0
∑
K
PK [K]a +∑
A
PA[A]i
︸ ︷︷ ︸=:u
· Fµ
1− eµ=
∑
K
PK [K]i +∑
A
PA[A]a
︸ ︷︷ ︸=:v
· Fµeµ
1− eµ
⇔ eµ =u
v
⇔ µ =u
v
⇔ FVm
RT=
∑K
PK [K]a +∑APA[A]i
(∑K
PK [K]i +∑APA[A]a
⇔ Vm =RT
F
∑K
PK [K]a +∑APA[A]i
∑K
PK [K]i +∑APA[A]a
+ + −
Vm =RT
F
(PNa+ [Na+]a + PK+ [K+]a + PCl− [Cl−]iPNa+ [Na+]i + PK+ [K+]i + PCl− [Cl−]a
)
+
Vm =RT
F
(PNa+ [Na+]aPNa+ [Na+]i
)=
RT
F
([Na+]a[Na+]i
)
⇒
V − Ei i
gi
im =∑
i
gi (V − Ei)
+
ENa = +55EK = −75
+ + Vm ≈ −60
PNa+ ≈ 125 · PK+
•
• ENa+ ≈ 55
• + ⇒
• +
+ +
• +
+
+
Vm = 60− 150
+
+
+
• Vm+
•+
• Vm ↑ ⇒ + ↑ ⇒ Vm ↑↑ ⇒ ↓ +
•
+ +
⇒
CmV = Q
V
CmdV
dt=
dQ
dt.
dQdt
Im
Ie
dQ
dt= −Im + Ie
⇒ CmdV
dt= −Im + Ie
⇔ cmdV
dt= −im +
IeA
( im =∑
i
gi (V − Ei))
⇒ −
cmdV
dt= −gL (V − EL) +
IeA
⇔rm= 1
gL
τm︸︷︷︸=cmrm
dV
dt= EL − V +RmIe
V = Vth ! V = Vreset
Ie
V (t) = EL +RmIe + (V (to)− (EL +RmIe)) e− t−t0
τm
Ie tk = t0 + k ·∆t, k ∈ N[tk, tk+1] Ie
I(k)e
V (tk+1) = EL +RmIe + (V (tk)− (EL +RmI(k)e )) e−∆tτm .
t ∈ [tk, tk+1]
V (t) = EL +RmIe + (V (tk)− (EL +RmI(k)e )) e−t−tkτm .
∆t → 0
r :=1
t.
•I (t )
• rV
t V
V (t ) = V = EL +RmI + (V − EL −RmI )
(− t
τm
)
⇔V − (EL +RmI )
V − EL −RmI=
(− t
τm
)
⇔ t = τm ·(V − EL −RmI
V − EL −RmI
)
⇒ r =
(τm
(V − EL −RmI
V − EL −RmI
))−1
(1) = 0 V += V
(x) x > 0
⇒V − EL −RmI
V − EL +RmI> 0 !
V < EL VI > 0
RmI > V − EL
V
r =
(τm
(V −EL−RmIV −EL−RmI
))−1, RmI > V − E
0.
I
(V − EL −RmI
V − EL −RmI
)=
(1 +
V − V
V − EL −RmI
)≈ V − V
V − EL −RmI,
r ≈V −EL−RmIτm(V −V )
(1 + x) ≈ x x
fa
f(x) = f(a) +f ′(a)
1!(x− a) +
f ′′(a)
1!(x− a)2 +
f ′′′(a)
1!(x− a)3 + . . . .
f(x) = (1 + x) 0
(1 + x) = (1 + 0)︸ ︷︷ ︸=0
+ln′(1 + 0)
1!︸ ︷︷ ︸=1
(x− 0) + · · · ≈ x.
⇒ K+
K+
K+
τmdV
dt= EL − V − rmg (V − EK)︸ ︷︷ ︸+RmIe
g τ dgdt = −g
g −→ g +∆g
Si Ti = [ti, ti+1]
Si :=( [ti, ti+1])
ti+1 − ti
Si (i = 1 . . . n)T ki Sk
i k
Si
Ski
⇒
a
dgdt
↔
2+
⇒
≈
Pi :=( i)
( i)
gi : igi := gi · Pi :
PK(Vm)
kk n
PK = nk n ∈ [0, 1]
k k = 4 +
αn(V ) βn(V )
n
dn
dt= αn(V )(1− n)− βn(V )n
⇔ τn(V )dn
dt= n∞(V )− n
τn(V ) =1
αn(V ) + βn(V )
n∞ =αn(V )
αn(V ) + βn(V )
αn βn qBαV
Bα
(−qBα/kBT )
⇒ αn(V ) := Aα (−qBα/kBT )(≡ Aα (−BαV /VT )
Aα
βn(V ) := Aβ (−qBβV /kBT )
⇒ n∞(V ) =1
1 + βnαn
(V )=
1
1 +
(Aβ
Aα·
((Bα−Bβ)·V
VT
))
•
•+
mk ( k = 3) →hi ( i = 1)
→ PNa+ = m3h
m h
dm
dt= αm(V )(1−m)− βm(V ) ·m
dh
dt= αh(V )(1− h)− βh(V ) · h
αm,αh; βm,βh αn βm
τm(V )dm
dt= m∞(V )−m
τm(V ) =1
αm(V ) + βm(V ))
m∞(V ) =αm(V )
αm(V ) + βm(V )
+ +
im = gi(V − EL)︸ ︷︷ ︸+ gKn4(V − EK)︸ ︷︷ ︸+ gNam3h(V − ENa)︸ ︷︷ ︸
gi = . − EL = − .
gK = . − EK = −gNa = . − ENa = +
CmdV
dt= −im +
IeA
τm(V )dm
dt= m∞(V )−m
τn(V )dn
dt= n∞(V )− n
τh(V )dh
dt= h∞(V )− h
αn(V ) =0.01(V + 55)
1− (−0.1 (V + 55))
βn(V ) = 0.125 (−0.0125 (V + 65))
αm(V ) =0.1(V + 40)
1− (−0.1 (V + 40))
βm(V ) = 4 (−0.0556 (V + 65))
αh(V ) = 0.07 (−0.05 (V + 65))
βh(V ) =1
1 + (−0.1 (V + 35))
−
• + +
•
•
•
+ +
→ 2+
→ + +
→
Si (i = 1 . . . n)Si ! Sj , (i, j) ∈ 1, . . . , n2
P (Si, t)t Si
dP (Si, t)
dt=
n∑
j=1
P (Sj , t)P (Sj → Si)−n∑
j=1
P (Si, t)P (Si → Sj)
si Si
(i, j) ∈ 1 . . . n2 rij rji Si
Sj Si
rij !! Sjrji""
i
dsidt
=n∑
j=1
sjrji −n∑
j=1
sirij .
rijV
Si
rij(V )!! Sj .
rji(V )""
Si Sj Uij
Si (−Uij/kbT )
rij(V ) = Rij (−Uij(V )/kbT )
kbRij
Uij(V )
Uij(V ) ≈ c0 + c1V
⇒ rij(V ) = Rij (−Uij(V )/kbT )
= Rij (−(c0 + c1V )/kbT ) = Rij
−c0kbT ·
−c1VkbT
aij := Rij
−c0kbT , bij :=
kbT
c1.
⇒ rij(V ) = aij ·(− V
bij
)
aij bij
Cr1(V )
!! Or2(V )""
mαm(V )
!! m∗,βm(V )""
hαh(V )
!! h∗.βh(V )""
o = m3h
Cr1 !!
r6
##!!!
!!!!
! Or2
""
r3$$""""""""
I
r4%%""""""""r5
&&!!!!!!!!
r1, . . . , r6
Cr6
##!!!
!!!!
! Or2
""
I
r4%%""""""""r5
&&!!!!!!!!
r1 = 0 r3 = 0 r5 = . −
r2 = . − r4 = − r6 = − −
Cr1 !!
r6
##!!!
!!!!
! Or2
""
r3$$""""""""
I
r4%%""""""""r5
&&!!!!!!!!
r1 = − − r3 = − r5 = −
r2 = − r4 = − r6 = − −
Cr5 !! C1r6
""r5 !! C2r6
""
r5''
Or4 !!
r9''
C4r3""
r2 !! C3r1""
r6
((
r10''
I
r7
((
r1 !! I4r2""
r3 !! I3r4""
r8
((
Cr1 !! Or2
""
dC
dt= r2 ·O − r1 · C
dO
dt= r1 · C − r2 ·O
C(1−O)
dO
dt= r1 · (1−O)− r2 ·O
O(t0) = O∗
O(t) = O∞ +K1 (−(t− t0)/τ1)
K1 = O∗ −O∞
O∞ =r1
r1 + r2
τ1 =1
r1 + r2
dO
dt= K1
(− 1
τ1
)(−t/τ1)−
1
τ1O∞ +
1
τ1O∞
=
(− 1
τ1
)O(t) +
1
τ1O∞
= (r1 + r2) ·O∞ − (r1 + r2)O(t)
= r1 − (r1 + r2)O(t).
O(t0) = O∞ +K1 · 1= O∞ +O∗ −O∞
= O∗.
Cr1 !!
r6
##!!!
!!!!
! Or2
""
r3$$""""""""
I
r4%%""""""""r5
&&!!!!!!!!
O I
dO
dt= r1 (1−O − I)− (r2 + r3)O + r4 I
dI
dt= r6 (1−O − I)− (r4 + r5) I + r3O
O(t0) = O∗ I(t0) = I∗
O(t− t0) = O∞ +K1 (−(t− t0)/τ1) +K2 (−(t− t0)/τ2)
I(t− t0) = I∞ +K3 (−(t− t0)/τ1) +K4 (−(t− t0)/τ2
K1 =(O∗ −O∞)(a+ 1/τ2) + b(I∗ − I∞)
1τ2
− 1τ1
K2 = (O∗ −O∞)−K1
K3 = K1−a− 1/τ1
b
K4 = K2−a− 1/τ1
b
O∞ =br6 − dr1ad− bc
I∞ =cr1 − ar6ad− bc
a = −(r1 + r2 + r3) , b = −r1 + r4,
c = r3 − r6 , d = −(r4 + r5 + r6)
τ1/2 = −a+ d
2± 1
2
√(a− b)2 + 4bc.
• a
• x
→ V (x, t) x t
•
∆xQ
Cm∂V
∂t=∂Q
∂t= IL(x)− IL(x+∆x)− Im + Ie,
Cm
VILImIe
IL RL Φdx
Φ(x+ dx)− Φ(x) = −RL(x) · IL(x),
RL
RL(x) = rLdx
πa2(x)
rL
Φ(x+ dx)− Φ(x) = −rLdx
πa2(x)· IL(x).
dx dx → 0
∂Φ
∂x= − rL
πa2· IL.
Φa ≡ 0 ∂Φ∂x
∂V∂x
(= ∂(Φi−Φa)
∂x
)
IL = −πa2
rL
∂V
∂x.
Cm
V = E · d
d E = −∇Φ
ρi Ω
−∆Φ =ρiε0.
ε0
ˆ
∂Ω
−∇Φ · n dν =
ˆ
Ω
ρiε0
dµ ⇔ 2πa∆xE =Q
ε0⇔ E =
Q
2πa∆xε0,
E
V =d
ε0
1
2πa∆x︸ ︷︷ ︸=C−1
m
Q ⇒ Cm =ε0d︸︷︷︸
=:cm
2πa∆x.
cm2πa∆x︸ ︷︷ ︸Cm
∂V
∂t= −πa
2(x)
rL
∂V
∂x(x)
︸ ︷︷ ︸IL(x)
− (−1) · πa2(x+∆x)
rL
∂V
∂x(x+∆x)
︸ ︷︷ ︸IL(x+∆x)
− Im + Ie.
2πa(x)∆xIm Ie im ie
∆x → 0
cm∂V
∂t=
1
2arL
∂
∂x
(a2∂V
∂x
)− im + ie
d ! a
dV
dX= 0.
V = 0.
VL
V = V .
V (·, t0) ≡ V .
x∗1 . . . n x∗
V1(x∗) = V2(x∗) = · · · = Vn(x∗).
n∑
i=1
Ii(x∗) =n∑
i=1
πa2
rL
∂Vi
∂x x∗
= 0.
a x
im
im =V − V
rm.
v := V − V
cm∂v
∂t=
a
2rL
∂2v
∂x2︸ ︷︷ ︸− v
rm+ ie
︸ ︷︷ ︸.
τm := rmcm λ :=√
arm2rL
τm∂v
∂t= λ2
∂2v
∂x2− v + rmie.
∂v∂t = 0
• v → 0 |x| → ∞
• Ie x = 0
2ε |x| < εie = Ie
2πa·2ε ε → 0
λ2d2v
dx2= v − rmie.
ie ≡ 0 x < −ε x > ε
λ2d2v
dx2= v,
v(x) = B1 (−x
λ) +B2 (
x
λ)
x < ε v(x) → 0 (x → −∞)
v(x) → B1
(−x
λ
)= 0 (x → −∞) ⇒ B1 = 0,
x > ε v(x) → 0 (x → ∞)
v(x) → B2
(xλ
)= 0 (x → ∞) ⇒ B2 = 0.
B1 = B2 =: Bx /∈ [−ε, ε]
v(x) = B
(− |x|λ
).
[−ε, ε]
λ2d2v
dx2= v − rmie.
εˆ−ε
λ2d2v
dx2dx =
εˆ−ε
(v − rmie) dx
⇔ λ2(dv
dx(ε)− dv
dx(−ε)
)=
εˆ−ε
v dx− rmie · 2ε =
εˆ−ε
v dx− rmIe2πa
dvdx(−ε)
dvdx(ε)
t↑−ε
dvdx(t) t↓ε
dvdx(t)
dv
dx(t) =
Bλ
(tλ
), t < −ε
−Bλ
(− t
λ
), t > ε
−2λB(− ελ
)= λ2
(−B
λ
(− ελ
)− B
λ
(−ελ
))=
εˆ−ε
v dx− rmIe2πa
.
vε→ 0 v
−2λB · 1 = 0− rmIe2πa
B =rmIe4πaλ
.
x ∈ R Rλ := rm2πaλ
v(x) =RλIe2
(− |x|λ
).
Lλ(
λ :=√
arm2rL
).
2πaL := SD
λ a a = SD2πL
Lλ SD
→
•
•
• V µ
µ
µ
Cm∂Vµ
∂t= IL
(xµ − 1
2Lµ
)− IL
(xµ +
1
2Lµ
)− Im + Ie
IL
Vµ Vµ+1
IL
(xµ − 1
2Lµ
)=
Φµ − Φµ−1
rL12Lµ−1
πa2µ−1+ rL
12Lµ
πa2µ
, IL
(xµ +
1
2Lµ
)=
Φµ+1 − Φµ
rLLµ
2πa2µ+ rL
Lµ+1
2πa2µ+1
.
Φ V
Cm =ε0d︸︷︷︸
=:cm
2πaµLµ
cm∂Vµ
∂t= −iµm + iµe + gµ−1,µ (Vµ − Vµ−1)− gµ,µ+1 (Vµ+1 − Vµ)
gµ−1,µ =
(rL
Lµ−1
2πa2µ−1
+ rLLµ
2πa2µ
)−1
(2πaµLµ)−1 =
aµa2µ−1
rLLµ
(Lµ−1a2µ + Lµa2µ−1
) ,
gµ,µ+1 =
(rL
Lµ
2πa2µ+ rL
Lµ+1
2πa2µ+1
)−1
(2πaµLµ)−1 =
aµa2µ+1
rLLµ
(Lµa2µ+1 + Lµ+1a2µ
)
gµ,µ+1
µ µ+1
•
•
j
FB(Gj) =1
k
k∑
i=1
Gi.
G k
j
FM (Gj) = Gj1 , . . . , Gjk .
j
j
FG(Gj) =k∑
i=1
g(j, i) ·Gi
g(j, i) =
1
(2π)d2 σ
·(−1
2‖i−j‖2
σ2
)
k∑l=1
1
(2π)d2 σ
·(−1
2‖i−l‖2
σ2
) .
d σ‖i−j‖ i j
g(j, i)
u
,j = −D,∇u,
= ,∇ =
∂/∂x∂/∂y∂/∂z
,
D
V uV V
ˆ
V
∂u
∂t(,x) d,x.
V uV
−ˆ
∂V
,j · ,n ds =
ˆ
V
∂u
∂t(,x) d,x.
,j · ,n
V ,Fˆ
V
,F (,x) d,x =
ˆ
∂V
,F (,x) · ,n ds
u = ,∇ · u
ˆ
V
∂u
∂t(,x) d,x = −
ˆ
∂V
,j · ,n ds = −ˆ
V
,j d,x,
V ,j
∂u
∂t=
(D,∇u
).
D
∂u
∂t= D ,∇u = ∆u
∆ := =n∑
i=1
∂2
∂x2i
n
D
D =
1 0 00 1 00 0 1
⇒
D =
5 0 00 1 00 0 1
⇒
D
: M :=∑
i
mi
: R :=1
M
∑mi,ri ,ri : xi
TR =1
2
3∑
l,m=1
Jlm ωl ωm
J :
ω :
Jlm =∑
i
mi(r2i · δlm − ril rim
),
δlm =
1, l = m0,
ril rim l m
i
J
v1, v2, v3 0 < λ1 ≤ λ2 ≤ λ3
J =(v1 v2 v3
)
λ1
λ2λ3
(v1 v2 v3
)T.
λ1 λ3
λ1 λ2,λ3 →λ1,λ2 λ3 →λ1 ≈ λ2 ≈ λ3 →
λ1λ2
2 1,λ2λ3
≈ 1 : D = DL :=(v1 v2 v3
)
1εε
(v1 v2 v3
)T.
λ1λ3
2 1,λ1λ2
≈ 1 : D = DP :=(v1 v2 v3
)
1
1ε
(v1 v2 v3
)T.
•
•
G∗
G∗
G∗ = 90
⇒→
Ω ⊂ Rd
∂u
∂t= D∆u.
dh
h
∂huh∂t
= −D∆huh Ωh.
∂u∂t
u′(t) = f(t, u(t))f
u′(t) ≈ u(t+ ht)− u(t)
ht
uht(t+ ht)− uht(t)
ht= f(t, uht(t))
⇔ uht(t+ ht) = uht + ht · f(t, uht(t))
uht(t+ ht) u(t+ ht)
uht → u ht → 0.
u′(t) = f(t, u(t))
t+ k · ht, (k = 1 . . . n)
uh(t+ h) = uh(t) + h · Φh(t, uh(t), uh(t+ h))
Φ f
Φh → f
h → 0
uh → u
h → 0
u(t+ h) = u(t) + h · u′(t)︸ ︷︷ ︸+h2
2u′′(t)
︸ ︷︷ ︸
+ . . .+hp
p!u(p)(t) +
∆u
∆u = u′′
−∆u = f′
u′(x) =h→0
u(x+ h)− u(x)
h︸ ︷︷ ︸
u′(x) ≈ u(x+ h)− u(x)
h
u′(x) ≈ u(x)− u(x− h)
h
u′(x) ≈ u(x+ h)− u(x− h)
2h
ξ1 ∈ (x − h, x), ξ2 ∈ (x, x + h)
u(x± h) = u(x)± hu′(x) +h2
2u′′(ξ2/1)
⇒ u(x+ h)− u(x)
h= u′(x) +
h
2u′′(ξ2)
u(x)− u(x− h)
h= u′(x)− h
2u′′(ξ1)
u(x± h) = u(x)± hu′(x) +h2
2u′′(x)± h3
6u′′′(ξ2/1)
⇒ u(x+ h)− u(x− h)
2h= u′(x) +
h2
6
(u′′′(ξ1) + u′′′(ξ2)
)
∂+ ∂−
′′( )
(∂−∂+u)(x) :=u(x+h)−u(x)
h − u(x)−u(x−h)h
h
=u(x+ h)− 2u(x) + u(x− h)
h2
u(x± h) = u(x)± hu′(x) +h2
2u′′(x)± h3
6u′′′(x) +
h4
4!u(4)(ξ2/1)
⇒ u(x+ h) + u(x− h) = 2u(x) + h2u′′(x) +h4
4!
(u(4)(ξ1) + u(4)(ξ2)
)
⇒ (∂−∂+u)(x) = u′′(x) +h2
24
(u(4)(ξ1) + u(4)(ξ2)
)
u′′
u ∈ C4(Ω)
−∆u = f ( Ω)
− ∂−∂+uh(x) = f(x) ( Ωh)
O(h2)
Ωh
n+ 1n − 1
Ωh (0, 1) h = n−1
uh =
uh(h)uh(2h)
uh(1− h)
.
Lhuh = qh
Lh =1
h2
2 −1−1 2 −1
−1 2
−1−1 2
qh =
f(h) + h−2ϕ0f(2h)
f(1− h) + h−2ϕ1
,
ϕ0 ϕ1
∂u
∂t︸︷︷︸= D · (∆u)︸ ︷︷ ︸
Lhuh = qh
u′(t) ≈ u(t+ ht)− u(t)
ht
x∂−∂+
uh(t+ ht, x) = uh(t, x) +htD
h2(uh(t+ ht, x− h)− 2uh(t+ ht, x) + uh(t+ ht, x+ h)) ,
Ωh = (0, 1) h = n−1
htD
h2
2 + h2
htD−1
−1 2 + h2
htD
−1
−1 2 + h2
htD
uh(t+ ht, h)uh(t+ ht, 2h)
uh(t+ ht, 1− h)
=
uh(t, h) +htDh2 ϕ0
uh(t, 2h)
uh(t, 1− h) + htDh2 ϕ1
.
Ω = (0, 1)× (0, 1) = (x, y) : 0 < x < 1, 0 < y < 1 .
Ω Ωh (n−1)× (n−1)∂Ω Γh 4n h
Ωh = (x, y) ∈ Ω : x/h, y/h ∈ Z h =1
n,
Γh = (x, y) ∈ ∂Ω : x/h, y/h ∈ Z .
−∆u = −uxx − uyy = f Ω,
u = ϕ Γ = ∂Ω.
(−∆hu)(x, y) :=(−∂−x ∂+x − ∂−y ∂+y
)u(x, y)
= −h−2(u(x− h, y) + u(x+ h, y)
+u(x, y − h) + u(x, y + h)− 4u(x, y))
−∆h = h−2
−1
−1 4 −1−1
.
(h, h), (2h, h), . . . , (1− h, h); (h, 2h), . . . , (1− h, 2h); . . . ; (h, 1− h), . . . , (1− h, 1− h).
Lhuh = qh
Lh = h−2
T −I
−I T
−I−I T
, T =
4 −1
−1 4
−1−1 4
.
T I (n− 1)× (n− 1) I
(n − 1)
•
•u ∈ C4(Ω)
A ∈ Rn×n AT ∈ Rn×n
u, v ∈ Rn
(Au, v) = (u,AT v)
(·, ·) Rn
Rn
u, v(u, v)
(u, v) :=
1ˆ0
u(x)v(x) dx.
A AT AAT A∗
u, v
(Au, v) = (u,A∗v).
A = ddx
u, v [0, 1]
(Au, v) =
(d
dxu, v
)=
1ˆ0
d
dxu(x)v(x) dx =
1ˆ0
u(x) ·(− d
dxv(x)
)dx + u v
∣∣∣1
0︸︷︷︸= 0
⇒(
d
dx
)∗= − d
dx.
L2(Ω) Ω = [0, 1]
u
− d
dx
(D(x)
d
dxu
)= f.
v(x)
(− d
dx
(D(x)
d
dxu
), v
)= (f, v)
⇔1ˆ
0
− d
dx
(D(x)
d
dxu
)v(x) dx =
1ˆ
0
f(x)v(x) dx
D = 1
1ˆ
0
− d
dx
(du
dx(x)
)· v(x) dx =
1ˆ
0
du
dx(x)
dv
dx(x) dx − du
dx(x)v(x)
∣∣∣1
0
u(x) = v(x) = 0 Γ
−dudx(x)v(x)
∣∣∣1
0= 0
v
vU
uU
Φ1(x), . . .Φn(x) u
u(x) ≈ U(x) = U1Φ1(x) + . . .+ UnΦn(x).
U1 . . . Un
V1 . . . Vn U1 . . . Un
Vi
Vi
Vi = Φi i = 1 . . . n
KU = F
K F
i
1ˆ0
dU
dx(x)
dVi
dx(x) dx =
1ˆ0
f(x)Vi(x) dx
⇔1ˆ
0
n∑
j=1
dUj
dx
dΦjdx
(x)
Vi(x) dx =
1ˆ0
f(x)Vi(x) dx.
i K(Ui)i=1...n i F (i, j)
Kij =
1ˆ
0
dΦidx
(x)dVj
dx(x) dx.
K
K = h−1
2 −1
−1 2
−1−1 2
.