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8
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=;>M_=E`
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021C
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8
6 =D)-/.7C
6 =D)-/.7C
9 6 . M!O
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.76 9[MH1-,
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6 = )-<.7C
6 = )-<.7C
-=6 9dOBC
;
U-/.V;
If
!"$#%&
(u + a)t + (c1(x)u − D1ux)x = 0,
at = γ(φ(x)u − a),
R I
= 6
13@4
0 ≤ x ≤ L, 0 ≤ t ≤ T
9 3:9dO+9 1- MK= 354 9dO+9X= 3 ;
u(0, t)
u(L, t) + βux (L, t)
u(x, 0)
a(x, 0)
=
=
=
=
;]N) $.EM!O
Oa=
):= N3@4I1f6 *
µ(t), 0 ≤ t ≤ T
0, 0 ≤ t ≤ T
0, 0 ≤ x ≤ L
0, 0 ≤ x ≤ L
•
1 6 . ; M51-/. 4 _
f
M =D3@MK.:3VO6l1KO+9X=D3 ; =7R
u(x, t), a(x, t)
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; N)E;iOW135M_.^G6 .V]
; U •
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; 9X=D3 MK=Y\
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D
.
MY9X.73KO 1 3@4
1 U7=;!9dO+9 ?@. (NV0 1 M_
= .
M 9X.73KO Y
•
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<9 ; 1 l6 1VOa. Ue1f6l1C . Oa.:6 13@4
γ
?e1Y6g*9 3 5
1 4:;>=6UKO+9X=D3 f
U 6 = 7
U .:6gO*
9 3 C . 4 9 NC 1
φ(x)
=7R(OB5.
;]UY1KO+9 1--j*
N354.76-j*V\
(u + a)t + (c1 (x)u − D1 ux)x
at
u(0, t)
u(L, t) + βux (L, t)
u(x, 0)
a(x, 0)
9 [email protected]
OBY1KO
Uf6 =D)-/.:C
)+*
9;
1
!"$#
;>= -NVO+9[=D3
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0,
γ(φ(x)u − a),
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0, 0 ≤ t ≤ T
0, 0 ≤ x ≤ L
0, 0 ≤ x ≤ L
u(x, t)
9/?5.73
=
=
=
=
=
=
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4 9 6 .EM!O
.0715MO 4I1KOW1^:;]N5M7B
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Oa=
4.Oa.76C
9 3H.
OB5.
MK=.
Mf9X.:3VO
R N3@M!O+9X= 3
φ(x), x ∈
[0, L].
'
.
=DC
("( 9dO
OB5.
If
Oa.:6C
;
c0(x)
13@4
ut
Oa=
= )VOW1:9 3
c1(x)ux + at = D1uxx
at = γ(φ(x)u − a)
R =I6
0 ≤ x ≤ L, 0 ≤ t ≤ T
13@4
;]N) $.EM!O
Oa=
):= N3@4I1f6 *
9 3:9dO+9 1- MK= 354 9dO+9X= 3 ;
B5.
6J. M!O
6 .V;]N-PO+9 3
OBY1KO
9/?5.73
Uf6 = )-<.7C
9 3_?5.:6 ;>.
UY6 =D)-/.7C
u(x, t)
9 [email protected]
)+*
9;
1
4.Oa.:6C
9 [email protected]
93
9 35.
MK=.
9;
;>= -NVO+9[=D3
^ 13@4
Oa=
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4 9j\
Mf9X.:3VO(R$N35MO+9X=D3
41KOW1
g(x) = u(x, T ), 0 ≤ x ≤ L,
Oa=
u(0, t) = µ(t), 0 ≤ t ≤ T
u(L, t) + βux (L, t) = 0, 0 ≤ t ≤ T
a(x, 0) = 0, 0 ≤ x ≤ L.
A
φ(x), x ∈ [0, L],
φ ∈ C[0, L]
13@4
R =I6
φ(x) > 0
x ∈ [0, L].
c1(x) ∈ C[0, L].
A
B5.
M_=D3 ;iOW13KO ;
µ ∈ C[0, L]
13@4
D1 , γ
µ
9<;
1354
β
1f6 .
35=D3]\ 3H. 1KO+9 ?@.
1354
1
µ ∈ C [0, L], µ(0) = 0
9<;
0
µ
U:=;!9dO+9 ?5.
=D3
[0, T ].
U:=; 9 O+9 ?5.
= 3
(0, T ].
R
R =6
µ0(t) − γ 0t e−γ(t−τ )µ0(τ ) dτ ≥ 0,
0≤t≤
T.
"
, ;a;%NC . UY1Y6 1C .%Oa.76GM_=D3@4 9 O+9X=D3 ; \
8 Be1E; 1 N379N5.
A BH.:3 4 9 6 .EM!O Uf6 = )-<.7C
3H= 3%\ 35. I1VO+9/?5. ; = -NKO+9X= 3
` 9dOB
u(x, t), a(x, t)
ux, uxx , at
` B5.76 .
C[QT ]
QT = [0, L] × [0, T ].
∈
3 Oa. D6l1VOa.
K
Oa= = )VW
O 1:9 3
1354 N ; .
Be1KO
O
R =6 1 --
a(x, t) = γφ(x)
RD`
.
; N) ;iO+9dONVOa.
]
. Ne1KO+9X=D3
OBH.
OBH.
(x, t) ∈ QT ,
Z
t
0
9 3KOa=
c1 (x)ux + γφ(x)u − γ φ(x)
Z
t
^@`
.
=D)KOW1:9 3
0
Z
−uxx + c(x)ux + dφ(x)u − γdφ(x)
t
d = γ/D1,
13@4
e−γ(t−τ ) u(x, τ ) dτ = 0.
0
R =D--<=E`
1-/.:3VO
3H.
;
OBe1KO
OB5.
Oa=
OB5.
Uf6 = )-/.:C
MH13
35=E`
.%0 U-/=9dO
4 9 T.76 .73KO+9 1-
.DNe1VO+9X= 3
3H= 3%\ 35. I
1VO+9/?:9dOQ*
OBH.E=6 .7C
13@4
4 9 6 .EMO
^
UY6J= )-/.7C
Uf6 = U7.:6gO+9X.V;
Oa=
e−γ(t−τ ) u(x, τ ) dτ = D1uxx.
k = `
G
-/.O
13@4
1
OB79<; .DNe1VO+c(x)
9X= 3 ):=
. MKc= 1C (x)/D
.V;
O
e−γ(t−τ ) u(x, τ ) dτ.
1354
2
9 3:9dO+9 1-MK= 354 9dO+9X= 3
Uf6 =7?5.
N379 DN5.735.V;a;
^
1354
M7-[179 C
9 3VOa. 6 =Y\
=7R OB79<;
OB5.
. N:9 ?V\
9;
.%09<;iOa.:35M_.^
;
=2R
OB5.
L =6
-[1KOa.76
6J.]R .:6 .73@MK.^ 6J. 1Y6 4 9 3
Z
dφ(x)u−γdφ(x)
t
e
−γ(t−τ )
1
Oa.76C
u(x, τ ) dτ = dφ(x)
0
BY1KOT4=
ST= 354 9dO+9X= 3
1544 9 O+9X=D3e1-KUY1Y6l1C
*:9X.:-<4:;
"
R
1
C
, ;a;%NC
9;
1
.
1V09 C
; = -NKO+9X= 3
Oa=
R =6
Z
t
e−γ(t−τ ) ut(x, τ ) dτ.
0
.Oa.76MK= 354 9dO+9X= 3 ; *:9X.:-<4
NC
UY1Y6 1C
u, a
OBH.:3
1 3@4
ut ∈ C[QT ]
93
UY6 9 35MY9 U-/.
.%Oa.76GM_=D3@4 9 O+9X=D3 ;
OBH.
4 9 6 .EMO
\
UY6 =D)-/.7C
^
(x, t) ∈ QT
µ0(t)k(x) ≤ ut(x, t) ≤ max µ0(t),
`
B5.76 .
0≤t≤T
9<; OBH. ;>=D-NKO+9X= 3 Oa=
k(x)
−v 00 + c(x)v 0 +
; 1VO+9<; R *:9 3 ):=DN354I1f6 * MK= 3@4 9dO+9[=D3 ;
dφ(x)v = 0
1354
1
k(L) + βk0 (L) = 0.
ST= 354 9dO+9X= 3
"
.%O
4 9 6 . M!O
*:9X.:-<4:;
, ;a;%NC
1
.
M_=DC
k(0) =
UY1Y6 9<;>=D3
UY1Y6 1C
u(x, t; φi), a(x, t; φi)
Uf6 = )-<.7C
)7.
M_=66 .V;]U:=D3@4 9 3
R =I6
φ1 (x) ≤ φ2(x)
x ∈ [0, L],
R =I6
ut(x, t; φ2 )
(x, t) ∈ QT
Uf6 9 3@Mf9 U-/.
.%Oa.76GM_=D3@4 9 O+9X=D3 ;
1
;>= -NVO+9[=D3
Oa=
OB5.73
Oa=
\
OB5.
R
φi, i = 1, 2.
ut(x, t; φ1 ) ≥
'
("( 6J. M!O
9/?5.73
OBe1VO
Uf6 = )-<.7C
9<;
u(x, t)
9 [email protected]
1
)+*
;>= -NKO+9X=D3
^ 13@4
Oa=
OB5.
.%021@M!O
41KOW1
g(x) = u(x, T ), 0 ≤ x ≤ L,
Oa=
4.Oa.76C
[0, L],
O+9X=D3 ;
9 3H.
9 [email protected]
OB5.
MK=.
93
Mf9X.:3VO
4 9j\
.
C
R N3@M!O+9X= 3
1m .
OBH.K;>.
φ(x), x ∈
1 ;a;]NC
U%\
1354
2
g(x) ∈ C [0, L]
O+9X= 3 g(0) = µ(T )
1354
g(x)
9<;
1
U:=;!9dO+9 ?5.
R N35M \
g(L) + βg 0(L) = 0.
13@4
00
0
g (x) − c(x)g (x) ∈ C[0, L]
R N3@M!O+9X= 3 = 3
9<;
1
U:=; 9 O+9 ?5.
[0, L].
.
MH13
-/.7C
)+*
; .E.
13
1UUf6 =I1@M2B
;]N)E;iO+9dONKO+9 3
g(x)
OB79<; .DNe1VO+9X= 3 R = 6
13@4
φ(x)
9 UY1KOa.
NE;!9 3
1
OB5.
0@. 4
U:=9 3KO
R =I6C
N-[1
=DU:.76l1KOa=6
ST.:6 OW1:9 3-j*
9.
Aφ = φ,
e 9<; 1 0f.E4
φ
00
0
):=eOB
)7= N354I1f6 *
e
φ(x)
9<;
1
RT
0
9<; 1
φ(x)
^ 1 0@. 4
U:=9 3KO =7R
g (x)−c(x)g (x) = d
F9 3@MK.
d
Z
N ;>.
.73KO
φ(x)
;>=D-j?9 3
9 3KOa=
1f6 D
NC
R =6
φ = Aφ ≡
Oa=
Oa=
^DOBH.:3
OB:9<;
) *
4. (
35.
4. (
3:9 3
1
t (x, τ ; φ) dτ
; =D- V
>
N O+9X= 3
U:= 9 V
3 O =7R
Oa=
A.
;>=D-j?9 3
kh.0_O
g 00(x) − c(x)g 0(x)
e−γ(T −τ ) u
Uf6 = )%\
13VO+9[M \
13@4
3H= 3- 9 35.51f6
,
O BH. .DNY1KO+9X= 3
ST= [email protected];>.7-j*]^ 9 R
A,
T
0
e)−c(x)ux(x, T ; φe).
e−γ(T −τ ) ut(x, τ ; φ) dτ = uxx(x, T ; φ
13@4
; 1KO+9<; R *
OB5.
e
u(x, T ; φ)
M_=D3@4 9dO+9X= 3E;_^
; =
e >
g(x) = u(x, T ; φ)
g(x)
;>= -NVO+9[=D3
Oa=
OB5.
9 3_?5.76c;>.
UY6J= )-/.7C
; 1C
.
OBe1KO
n
O
R = --/=E`
;
R$6 =DC
A
B5._=I6 .7C
OBY1KO
R =6
1
U:=:;!9dO+9/?5.
φ(x),
g 00(x) − c(x)g 0(x)
≥ RT
≡ h(x)
Aφ = R T
−γ(T
−τ
)
−γ(T
−τ
)
ut(x, τ ; φ) dτ
µM dτ
d 0 e
d 0 e
g 00(x) − c(x)g 0(x)
`
B5.76 .
µ0(t).
µM = max0≤t≤T
4. (
3H.
.
E = {φ(x) ∈ C[0, L] | φ(x) ≥ h(x), x ∈ [0, L]}.
A
B5.73
OBH.
;>.O
MK= 3KO+9 3N5= NE;
1354
=7R
U:=:;!9dO+9 ?@.
R N3@M!O+9X= 3E;
93
E ⊂ P,
A B6 = N D
BH= NKO OB5. R =D--<=E` 9 3 ^ ` . N ;>. OBH.
C[0, L].
N3:9 R =6C
3H=I6C ;
93
13@4
A B5.
k·k
C[0, L]
C[QT ].
=DU:.76l1KOa=6
9<; ; 179[4 Oa= )7. 9 Ri^R =6 R$N3@M+\
A
O+9X=D3 ;
1354
9 3 OBH. 4= C 1:9 3 =2R
` 9 OB
φ(x)
Ψ(x)
A
R =6
` . Be1K?@.
φ(x) ≤ Ψ(x)
x ∈ [0, L],
Aφ(x) ≤
R =6
U:.76l1VOa=I6
9<; x ∈ [0, L].
A
AΨ(x)
9 R 9dO
9<;
Uf6 .EM_=DC
; .:6gO ;
MK= C
UY1@M!O
OBY1KO
;>.O ;
9R
OB5.
Ue15MOV^1354
=6 4.:6
,
`
C
1UE;
.7--
mY35=E`
=DU:.76l1KOa=6
=DU:.:6 1KOa=6,
):=DN354.E4
C
A
9<;
;>.O ;
9 3KOa=
OB5._=I6 .7C
1 ;$\
C
=D35=eOa= 35.
13@4
3
1U ; OB5.
35= 3%\$.7C
9 3KOa.76 ?e1-
UVOQ*
9 3KOa=
0
0
[u0 , u ] = {u | u0 ≤ u ≤ u }
9dO ;>.7-jR ^OB5.73 OB5. ;>.DNH.:35M_.V; =2R 9dOa.76l1KOa.V;
=6
{Anu0}
M = [email protected] . Oa= 0@. 4 U:= 9 3VO ; =7R
K
{Anu0}
A.
n n
9dOB
B:9[M2B
; .%O ;
A
9<; 1
C
A
BH.
` .
MH13
= U:.76l1VOa=I6
MK= 3VO+9 3N5=DN ; 13@4
1UE;
):=DN354.E4
B5._=I6J.:C
R =I6 OB5.
9 35= -/= 2
*
Oa.76C
"
)+*
`
OB:9<;
9<;
N ; . 4
.09<;iOa.73@MK.
;>.O ;
Oa=
;]BH= `
C
= 3H=2Oa=D35.
;]BH= `
R
"
OBH.
=7R ;>= -NVO+9[=D3 ;
41KOW1
MK= 354 9dO+9X= 3E;
.V;a; 1f6 *
1354
Uf6 =D)-/.:C
.09 ; O
`
1
B:9[M2B
8
R
OBY1KO
Oa=
\
Be1K?@.
= U7.:6l1VOa=I6
m .%*
Oa=
.Oa.76 M_=D3@4 9dO+9X= 3E;
; 1VO+9<; .E4 ^ OB5.73
1f6 .
Mf9X.:3VO
U:=:;!9dO+9/?5.
MK= 354 9dO+9X= 3
1
R =I6
;>= -NKO+9X=D3
OBH.E=6 .7C
\
13@4
1
35.EM \
OBH.
9<;
9 3_?5.76c;>.
OBe1VO
OB5.76 .
R$N35MO+9X=D3
φ0(x) ∈ C[0, L]
Aφ0(x) ≤ φ0(x), 0 ≤ x ≤ L.
9<; M2N ;a;!9X=D3
;]N
UY1Y6l1C
9 [email protected]
. N:9[MK= 3KO+9 3N5= NE;
9 3KOa=
A : P → E
BY1 ;
=7R
Uf6 ==7R
1
Aφ = φ,
;>= -NKO+9X=D3
;>=
R =I6
OW1m .
^
φ(x)
φ0 = φ.
` .
Be1K?@.
1-6 .51@4e*
n
ST= [email protected];>.7-j*]^ ;%NUU7=;>. OBe1VO U:=:;!9dO+9/?5. M_=D3KO+9 3NH= NE;
R N35MO+9X= 3
; 1KO+9<; .V; OB5. 9 3H. Ne1- d9 OQ*
φ
(x)
* 0 ` . Be1K?@. OBY1KO R =I6 1 3_* Aφ
U:0
=:(x)
;!9dO+9 ?@≤
.
φ0 (x).13@4
S D
= 3 ;>.DNH.:3VO-j*]^
φ(x)
x
∈
[0,
L],
h(x)
≤
Aφ(x).
OBH. C =D35=eOa= 3:9[Mf9dOQ* =7R
9 C U- 9X.K; OBY1KO
C 1U ; OB5.
A
A
=6 4.:6 9 3VOa.:6g?21-
I = {φ(x) ∈ C[0, L] | h(x) ≤ φ(x) ≤ φ0(x), 0 ≤ x ≤ L}
9 3VOa= 9dO ;>.7- R * A BH.E=6 .7C OB5. = U7.:6l1VOa=I6
A:I→
9<; 1 C = 3H=2Oa= 3H. M_=DC UY1@M!O =DU:.76l1KOa=6 O R = --/= ` ;
I
OBY1KO OBH. ; . N5.73@MK. =7R 9dOa.:6l1VOa.K;
φ0, Aφ0 , A2φ0 , . . .
MK= [email protected] .K; Oa= 1 0f.E4 U:=9 3VO
=7R
` B:9[M2B 1E; ` .
φ
A,
BY1_?5. ;>._.:3 9 3 OB5. 9 3KO6J=4DN5MO+9X= 3 =2R OB79 ; ;>.EMO+9X=D3I^
C N ; O ; 1VO+9 ;iR *
u(x, T ; φ) = g(x), 0 ≤ x ≤ L.
,
MK= C
UNKOW1VO+9X= 3e1-- *
N ;>.%R$N-RQ1@M!O
I
R OBH. 9 [email protected];>.
; = -NKO+9X= 3^HOB5.73 OBH. ;>.DNH.:35M_.
M = [email protected] .K; Oa= 1 ;>=D- NVO+9X= 3 =2R
K
;%N@M2B
35.
MH13
U 6 = )-<.7C
f
=7R 9dOa.76l1KOa.V ;
BY1 ;
1
h, Ah, A2h, . . .
R$N3@M!O+9X= 3 ;
φ (x)
MK= 3_?5.76 . n Oa=
un(x, T ) = u(x, T ; φn)
N3:9 R =6C -j* =D3
*].%O
u0(x, t) = u(x, t; φ0 )
[0, L],
Oa.:354:; Oa=
` 9dOB
A BNE; OBH.
kφn − φ0 k = n
∞
n.
9 [email protected];>. Uf6 = )-<.7C
9 ; 35=eO ;iOW1)-/. OBY1KO
MK= 3E;iO6N5MO
n
D" R =6 OBH.
−uxx+p(x)ux+q(x)u−γq(x)
4. 35.
4. 35.E4
; *H; Oa.:C
Z
t
=D3
QT
e−γ(t−τ ) u(x, τ ) dτ = F (x, t)
0
u(0, t) = µ(t), 0 ≤ t ≤ T
u(L, t) + βux(L, t) = 0, 0 ≤ t ≤ T
U (x, t) =
.%O
Z
t
e−γ(t−τ ) u(x, τ ) dτ.
0
1 354
):. ;iOa.7U ;!9K.V; 9 3
13@4 ∆xOa=
= L/M
):. N ;>.E4 9 3 ∆t
OB5=
. 4 T9 ;>/N
M76 .O+9H1KO+9X=D3 =7R OB79 ;
x
Uf6 =D)-/.:C t ^` B5.76 .
13@4
1f6 . U:=:;!9dO+9 ?@. 9 3KOa. .76c; . ?9X.`
4
135M
N1 ; 13 .%?@=D- NVO+9X= 3e1f6 * ;i*H;$\
Oa.7C
9 3 OB5. N3me35=E` 3E ;
1354
A B5. .%?@=D- NVO+9X= 3e1f6 *
u-/.@1f6 .76 U.) * OB5. =D) ;>.76 ?e1KO+9X= 3
1E;]U:.EMO =2R
9<; C 1@4. M2
U
OBY1KO
U (x, t)
=
Z
t−∆t
e
−γ(t−τ )
u(x, τ )dτ +
0
=
e−γ∆tU (x, t − ∆t) +
Z
Z
t
e−γ(t−τ )u(x, τ )dτ
t−∆t
t
e−γ(t−τ )u(x, τ ) dτ.
t−∆t
n
n
F".O
1354
tn = n∆t, 8 n. (=3H. 1, 2, . . . N
xk = k∆x,
pk = p(k∆x), qk = q(k∆x),
k = 1, 2, . . . M.
µn = µ(n∆t), Fk,n = F (k∆x, n∆t), uk,n ≈ u(xk , tn) =
1354
u(k∆x,
n∆t)
U 135≈
U (x , t ) = U (k∆x, n∆t)
b ;>. M_.73KOa.76 .E4 (6 ;iO k,n
4 ;>.EMK=k 3@4 n 4 9
.:6J.:35M_.V; Oa=
4 9<; 7
M 6J.%O+9K.
4 9<; 7
M 6J.%O+9K.
∆x
− 1 + pk
2
1354
1
O 6l1U7. K
= 9[4I1= )VOW1:9 3
.
n
2
n
Fk,n + γe
−γ∆t
. Ne1VO+9[=D3 ;
)7= N354I1f6 *
1VO
MK= 354 9dO+9X= 3 ;
uM +1,n.
;!9 3 D
-/.
U- 9 Mf9dO
93
O+9 C
C
.
\ OB
Oa=
;>= -j?5.
1Y6 M7B79 3
13@4
O+9 C
R =6G?e1-N5.V;
Oa=
9C
.
=7R
∆x
uk,n− 1 − pk
2
o
.7- 9 C
C
U- 9[MY9dO
\
1VO
OB5.
.%0 U- 9[MY9dO-j*
R =6
9 3e1KOa.
9<;
.OB5=4^h`
93
;]UY1@MK.
OB5.
\ OB
n 35=4.V;
bG; .
Oa.76C
;
3H= `
;>= -j?5. 4
B:9[M7B
n .
n = 1, 2, . . . N.
-/.%[email protected] ^I;>=D- ?5.
u
Oa=
∆t
uk,n−1
Uk,n−1 +
2
13@4
6 =D)-/.:C
;iOa.7U
(n − 1)
o
.OB5=4
∆t −γ∆t
e
uk,n−1 + uk,n .
1@M_M_=DN23VOGR =I6 O5
B . 9 3KOa.76 9X=I6
(xk , tn ), k = 1, . . . , M
)+*
qk
γ∆t
1−
2
C
Uk,n = e−γ∆t Uk,n−1 +
BH.K; .
13@4
2
uk−1,n+ 2 + (∆x) qk
(∆x)
A
u0,n
9<;
9 [email protected]
.0K\
U, u
- 9 3H.@1f6 ;i*H;iOa.7C
-/.%[email protected] ^ OBH.:3
n
1VO OBH.
\ OB
U
n
N ;>.
n
-/.]?5.7
uk+1,n =
2/γ
1354
C
93
Oa=
.%[email protected]
9<; ;>.EM_=D3@4
=I6
; OW1):9 - 9dOQ*
∆x < 2/ kpk
MK= [email protected] .K;
6J=I6c;
U:=:;>. OB5.
9 OBC
OBH.
M513
):.
;iOa.:U
M_=DC
; 9 V.K;_^ R =D--<=E` . 4
.0_O6l1U:=D-[1KO+9X= 3
=7R OB5.
)+*
Oa=
; 93 D
-/.
1
*9X.7-<4
1
9dOB
;iOa.7U
A
O6N35M51VO+9[=D3
;iOa.7U
A
OBe1KO
1354
=2R
=I6 4.76
1- \
BY1-jR ; Oa.:U
9[M7BY1Y6 47;>= 3
C
.%OBH=4
T. MH1--OBe1VO
.
9dO
.0715MOh4I1KOW1
Aφ =
L 9 6c;iOV^
RT
d 0 e−γ(T −τ )ut(x,τ ;φ) dτ
4 9<; M76J.%O+9K. OBH. 4I1VOW1 R$N35MO+9X=D3
13@4 1f6 DNC .73KO
g
?9 1
R N35MO+9X= 3
φ
gk = g(k∆x), k = 0, 1, . . . M + 1,
13@4
. M513 4 9/RJ\
φk = φ(k∆x), k = 1, 2, . . . , M.
R .:6J.:3VO+9 1VOa. . Ne1KO+9X=D3
Oa= =D)KOW179 3 OBe1KO
v(x, t) =
1-d;>= ; 1VO+9 ; .K;T;%N@M2B 13 . DNY1KO+9X= 3^ Oa= .OB5.76
ut(x, t)
` 9dOB ):= N3@4I1f6 * MK= 354 9dO+9X= 3 ;
13@4
v(0, t) = µ0(t)
; = OBe1KO OBH. R$N3@M!O+9[=D3
v(L, t) + βvx(L, t) = 0,
R
M513 )7. 1UUf6 =E0K\
V (x, t) =) 0t e−γ(t−τ ) ut(x, τ ) dτ
9 C 1KOa.E4 )+* OB5. = NKOUNKO =7R OB5.
1- =6 9dOBC ^
1UU- 9X.E4 Oa=
6J.K;]N-PO+9 3 9 3 1UUf6 =E09 C 1KOa. 35=4.
v(x, t),
?e1-NH.K;
Vk,N .
n g 00 (x)−c(x)g 0(x)
, ; ;]NC
.:6 \
B5.76 .%R =6 .^eOB5.
1VO ;iOa.7U
∆t <
1- =I6 9 OBC
;!9K.K;K^ ;>=
R =DN6gOB
`
;!9K.V;
UNKOa.E4
OBH.
; = -NKO+9X= 3
U7= ` .:6 ;
=I6 4.:6 9 3
^ 13@4
6 .K; O6 9[MO+9X= 3E;
9 [email protected]
1f6 D
NC
13
M26 .O+9K. 4
B:9[M2B
1
9<;
R =6
;!9 C
6 9 OBC
U-/.
^
?5.76c;>.
R =6
0f.E4
.:35M_. MK= 3@4 9dO+9[=D3 ;
.OB5=4 ^ `
MK= C
Uf6 = )-<.7C
A
UNVO+9 3
=2R
U:=9 3KO
-<.O
b ;>.
S =6 = --[1f6 *
B:9[M2B
` .
OBH.
Oa.:6C
1 ;
9 Oa.76l1VO+9[=D3
; = -NKO+9X= 3
4. (
3H. 4
=2R
Oa=
OB5.
93
Φ0
) *
4.]?:9<;>.
Oa=
9 379dO+9 1- N5.V;a;
OBH.
h(x)
6 .DN79 6 .K;
B:9<;
k = 1, 2, . . . M.
` B:[9 M7B 1f6 . 4.%Oa.76C 9 3H. 4
gM +1, g0,
4 9<; M76J.%O+9H1VO+9[=D3
NE;>.
)+*
R =6
C
9 [email protected]
4 9; \
dVk,N (∆x)2
):= N3@4I1f6 *
1
OBH.
∆x )g
)g
−
2g
+
(1
−
c
(1 + ck ∆x
k−1
k
k
k+1
2
2
?e1- 9[4
?e1-N5.
OBH.
Φ = (φ1 , . . . , φM ),
=DU:.:6 1KOa=6 9<;
(A M Φ ) k =
`
.73KO
1-=Y\
OBH.
9 3]\
)7.
OB5.
A
BH.:3
N3VO+9 - M_=D3_?5.:6 \
n
# .O
L = 2, T = 2, γ = β = 1, c(x) =
1 + cos(3x)/3, φ(x) = 2 − x/2 + x2 sin(3x)/3
1 3@4
` B5.76 .
1f6l1C \
µ(t) = (1 − e−αt )/α,
α > 0.
.Oa.76
MK= 354 9dO+9X= 3
α = 0.2
O+9X=D3
Oa=
4.:C
`
9dOB
.:6 .
Oa=
OB5.
- 9 35.51f6 6 1KOa.
9dOa.76l1KO+9X=D3I^`
0f.E4
U7=9 3KO
1Y6 1C
9R
.%Oa.76
.
Oa=
=6 4.:6
1@M_M2N6l15M%*
A
BH.:3
=D)KOW179 3
=2R
N ; .
OB79<;
9 3_?5.76c;>.
9 3:9dO+9 1- D
NH.K; ;
; .%O
=7R
.
M_=D3_?5.:6 .73@MK.
=2R
BH.:6J.
OB5.
9dOa.76l1KO+9X=D3
4I1VOW1
R$N3@M+\
15MEM2N6l15M%*
OBY13
; = -NKO+9X= 3
φ(x)
;>._.
13
OB5.
9 3]\
93
OW13]\
OB5.
Oa=
M_=DC
UNKOa.
9;
OBH.
R N35MO+9X= 3
1UUY6 =E09 C
1KOa.7-j*
R
N3VO+9 -1):=DNKO OB5.
4 9<; M76J.%O+9H1VO+9[=D3
Oa=
Oa=
1- =I6 9 OBC
UY6J= )-/.7C
R =I6
OBH.
=2R
N3H.%0_O6l1U:= -[1KOa.E4
φ0 (x) = h(x)
B:9 DBH.:6
1
Uf6 = )-<.7C
; = -NKO+9X= 3
1-=6 9dOBC
.%0 U:.EMOa.E4
?5.76c;>.
α0 < 0.2031.
bG;>. OB5. .0EO6l1U]\
∆x = ∆t = 0.01.
1354
g(x)
; 1KO+9<; .E4
1Y6J.
; 1KO+9<; .E4
9<;
=D-[1KOa.E4
OBH.
\
MK= 354 9dO+9X= 3 ;
;iOW1--
OB
.:66 =I6hMH1NE;>.K;
n Example 5.1 iterates with α=0.2.
2
Iterate 1
Iterate 2
Iterate 3
Iterate 4
Exact Solution
1.8
1.6
1.4
φ(x)
1.2
1
0.8
0.6
0.4
0.2
0
,GO
0
0.2
0.4
OB5.
R = N6gOB
0.6
0.8
1
x
9dOa.76l1KO+9X= 3^OB5.
1.2
35=6C
1.4
=2R
1.6
OB5.
1.8
.:66 =I6
9<;
A B5. C = 3H=2Oa= 3H. ):.7Be1K?9X=6
.
kg − φ4 k = 0.0021.
=2R OBH. =DU:.:6 1KOa=6
9<; .%?9[4.73KO B5.73 ` . Be1-j?5. 4
A
)7=2OB
13@4
OBH. (3Y1-e.:66J=I6 ` 1 ; 4.EM26 .@1E;>.E4
∆x
∆t,
n
2
)+*
1
R1@M!Oa=I6 =7R
1-=I6 9dOBC
1):=DNKO
9;
R = N6 ^ MK= 3 (
6C
;>.EM_=D3@4
93
OBY1KO
=I6 4.:6 1@M_M7N6l1KOa.
OB5.
Fixed point iterates for Example 5.1 with α=3.0.
2.5
Iterate 1
Iterate 2
Iterate 3
Iterate 4
Exact Solution
2
φ(x)
1.5
1
0.5
0
0
0.2
1Y6l1C
0,
OBH.
R =I6
0.4
0.8
.Oa.76MK= 354 9dO+9X= 3
0≤t≤T
U79 M!ON6J.
R =I6 OB5.
0.6
C
^ RQ179 -d;
OBe1VO
= 35=eOa= 3H.
OB:9<;
^
1
x
1.2
1.4
1.6
1.8
2
R t −γ(t−τ ) 0
0
µ (t)−γ 0 e
µ (τ ) dτ ≥
R =6
. ;>._. R$6 =DC
α=3
M_=D3@4 9 O+9X=D3
Uf6 =DU:.:6 OQ*
=7R
9;
A
6J.@1- -j*
35._. 4. 4
. MO =2R 3H=9<;>. A BH.K; . .
E
.EMO ; 1Y6 . ;iO6 = 3 F9 35M_. =DU:.76l1KOa=6
9<; M_=DC UY1@M!OV^ = 3H. 35._. 47; 1 6J. @\
A
N-[1Y6 9H1VO+9X= 3 ; O6l1KOa. 2* 8 9<; M26 .%O+9H1VO+9X= 3 =7R OBH. 4.%R \
9 379 O+9X=D3
)+* ` 1+* =2R
9<; 9dO ;>.:-jR 1 6J. @\
=7R
N-[1Y6 9 H1VO+ 9X= 3 ;iO6l1KA
Oa. 2* ` 9dOB 6J. N-[1Y6 9 1KO+9X= 3 Ue1f6l1C \
.Oa.76
F9 3@MK. OB5. =DU:.:6 1KOa=6
9<; 35=D3- 9 35.51Y6 ^ 13
∆x.
A
.021@M!O 13Y1-j*H; 9 ; =7RhOBH. .766 =6 9<; 4 9
M2N-PO B5=E` .%[email protected] ^
=D35. .%0 U:.EMO ; OBe1VO OB5. Uf6 9 3@Mf9 Ue1- ;>=DN6 MK. =7R 4 9 \
M2N-POQ* 9<; 9 3 OB5. 4 9<; M26 .%O+9 H1VO+9X= 3 =2R OB5. 3NC .76l1VOa=I6
93
, ;!9 C U-/. A 1+* -/=6 ;>.:6 9[.V; 13e1-j*H;!9<; ;]BH= ` ;
OBY1K O 9 R(OBH. 35= 9<;>. -/.]?5.:-79<; ):=DN354.E4 )+*
OB5.73 OB5.
δ,
4 9<; M76J.%O+9 H1VO+9[=D3 .766 =6 9<; ):=DN3@4. 4 )+* 1 R$N3@M!O+9X= 3 =7R
OBH. R =6C
δ
c1 δ
2
+
c
(∆x)
+
2
(∆x)2
∆x
R =I6 ;]N:9dOW1)-/. MK= 3 ; OW13KO ;
4.:U:.73@4 9 3 = 3
c1 , c 2
1 3@4 OBH. ;>.EM_=D3@4 1354 OB:9 6 4 4.:6 9 ?e1KO+9 ?5.K; 2
= R
c(x).
A
B:9<;
=6 4.:6 9 3
.%0 Uf6 .K; ;!9X= 3
∆x
1KO
6 . N-[1f6 9H1KO+9X= 3
1
NE;>.
A
9 meB5=D35=2?
.DNY1KO+9X= 3
=2R
41KOW1
;>=DC
93
;>=6gO
OBH.
9 3:9 C
9K. 4
ut
-/.@154 9 3
Oa=
6 . D
N- 1f6 9H1VO+9[=D3
ϕ = Aφ.
.
C
13@4
` B:9[M2B f
U 6 =7?:9[4.V;
1/4
∆x = (c2 δ)
,
;iO6l1VOa. e* , -POa.:63e1KOa.7-j*]^5= 5
3 . M_=DN-<4
= N6 MH1-<M7N-[1KO+9X= 3E;
NE;>.
9<;
c(x)
=7R ;]C
A
B:9<;
=D3
` = N-<4
; 9
3:9 MH13VO- *
==eOB:9 3
4 9<; M26 .%O+9H1VO+9X= 3
`
OB5.
1-jOa.76
= U7.:6l1VOa=I6
OBH.
6 =D35.
R =6C
.
=6 (
-PO6l1VO+9[=D3
MK= N- 4
.7C
U-/=H*
93
=7RGOB5.
n
a
,
;!9 C
=2R
U-/.
OBH.
4. 6 ._.
b ;>.
4I1VOW1
OBH.
9<;
; 1C
.
MH1-<M7N-[1KO+9X= 3^
3
?:9X.%`
U-/.^
= 3H.
R =I6
1-/= 3
;%C
`
==eOB5.E4
=2R(1
Oa.:3VOB
M7-/.51Y6- *
9<;
1
N ; 93
OB5.73
13
= UVO+9 C
13@4
F9 3@MK.
OB5.
4I1VOW1
`
9dOB
OBe1KO
A
1f6 .
OB79<;
.%021C
1Y6J.
-[1@M2m@9 3
.:66 =I6
UNKOW1VO+9[=D3 ;
M_=DC
UY1Y6 .^e` .
; Ne1f6 .K;
9dOV^fOBH.:3
A
\
9 -- NE;iO6l1VOa. 4^
-/.51 ;iO
9 1-WOa=
= 3
δ = 0.0001.
S =DC
1
BH.
9 1-;%C
R =6
O
1UU- 9X. 4
U-/=2O$Oa.E4
U:= -j*35=DC
6J. N-[1Y6 9 1KO+9X= 3
B5.
6J. N-[1Y6 9 1KO+9X= 3
A =
∆x = 0.01.
9 3@4 9[M51VOa.K;
;]NU:.:6 9[=6
4= 9 3
15M \
35=9<;>.
` .
∆x = 0.1
U:= -j*35=DC
B:9 DBH.:6
^ MH1- \
1KO+9X= 3^ O6 9 1-G13@4
.%021@M!O ;>=D-NKO+9X= 3
4. D
6J.E.
.
4 9 ; O6 9 )NKOa.E4
NC
O
;>= C
U-<.
1
-j*
1- ?21- NH.
)+*
Oa==Dm
= UVO+9 C
9 3_R =6C
071C
UY6J. MK. 4 9 3
∆x ≈ 0.1.
9 1- =7R
1@44
.
OBH.
1-=6 9dOBC
;%N-PO
):.
93
; D
NY1Y6J.K;
-<.51 ; O
N379/R =I6C
OB5.
4.76 9 ?21VO+9 ?@.
9dOB
1 ;
1354
4 9<; M2N ;a;!9X=D3
∆x = 0.05
1
U:= -j*35= C
.Oa.:6 ;
1354
.0U:.EM!O ;
.K;iO
9 - -(;]N
1
[−δ/2, δ/2].
.%Oa.76Oa=
93
6l1354=DC
=7R OB5.
UY1Y6 1C
`
4I1KOW1
9 3KOa.76 ?21-
4=
.0715MO-j*
M2N6 1@M*
OBH.
Oa=
Ue1f6l1C
OBH.
9<;
Oa=
U:=9 3KO ;
M2N-[1VOa.
3H=9<;>.
1UUf6 = 15M7B
6 . \
==eOB]\
Fifth iterates
Fifth iterate, polynomial smoothed data
2
2.2
Dx = 0.01
Solution
1.8
2
1.6
1.8
Dx = 0.05
Dx = 0.1
Solution
1.4
1.6
1.4
φ(x)
φ(x)
1.2
1
1.2
0.8
1
0.6
0.8
0.4
0.6
0.2
0.4
0
0
0.5
∆x = 0.01
6l1UB 1
x
`
1.5
1 ;
Oa==
0.2
2
)e154
Oa=
0
):.
0.5
UNKO
1
x
93
1.5
6c;iO
OB5.
2
$
.
1f6 .
6J. M!O
8
OBH.
A =
13@4
.735.76l1 -"Uf6
;>.E.
`
OB5.
4.:6 9/?e1KO+9 ?5.
A
B5.
-/.7C
9 R1
R =6
OBY1KO
C
;>=DN6 MK.
;!9[4.
93
u
;
8
= 3
9\
13e1- * ;!9<;
=7R
M_=D335.EM!O+9[=D3 ;
Oa=
N@M2B
=7R = N6G13Y1-j*H; 9 ;
Oa.:6C
=2R
.735.76l1-8
OB5.
N6
U:=I6 OW13KO
;
35=eOa.
BY13@4
Oa=
9<;
t)
6 =4N@MK.
3VfO(x,
U-[1@MK. 4
1
O+9 C
M_=66 .V;]U:=D3@4 9 3
=DU:.76l1KOa=6R =6OB5.
9 3_?5.:6c; .
Uf6 = )%\
9<;
B5.V;>.
R =I6C
9dOa.76l1KO+9X=D3
4 9 6 . M!O
9 [email protected];>.
;
g 00(x) − c(x)g 0(x) − ut(x, T ; φ)
RT
d 0 e−γ(T −τ ) ut(x, τ ; φ) dτ
3Y1KON6 1--j*
; M7BH.:C
UY6 =D)-/.7C
Uf6 = )-<.7C
.V;
1354
R =I6
.K;iO 15M35=2? 4.76 9 ?9 3 ;>=D-NKO+9X= 3 Oa=
;]N
0f.E4
.
= )VOW1:9 3
−D1uxx + c1 (x)ux + at = −ut
at = γ(φ(x)u − a)
φ = Aφ ≡
A
9C
=D)-/.:C
B_*]^` .
6 9 DBKO
6 = )-/.:C
BY1_?5.
MH1f66 9X.K; OB6J= N B
=D3
6 .K; .@1f6 M2B
3_?5.:6c; .
1354
MK= 3VO+9 3N:9 3
I
U:=9 3KO
9dOa.:6l1VO+9X= 3
OQ*U:.
OB5.
R =6"OB5.