Báo cáo nghiên cứu khoa học: "Sự hội tụ của nghiệm xấp xỉ của bài toán xác định nguồn." pptx






u
t
− Lu = f(x, t); (x, t) ∈ Ω × (0, T ),
u|
∂Ω×(0,T ]
= 0,
u|
t=0
= ϕ.
1.





u
t
− Lu = f(x, t); (x, t) ∈ Ω × (0, T ),
u|
∂Ω×(0,T ]
= 0,
u|
t=0
= ϕ.
(1.1)
u(x, T ; f, ϕ) = ψ

T
(x), x ∈ Ω (1.2)
u(x, T ; f, ϕ) f ϕ
n

i,j=1
(a
ij
(x, t)u
x
j
+ a
i
(x, t)u)
x
i
+
n

i=1
b
i
(x, t)u
x
i
+ a(x, t)u.
f u u f
u
ψ
T

(x)
f ∈ L
2
(Q
T
)
J(f) =
1
2
u(., T ; f, ϕ) − ψ
T
(x)
2
L
2
(Ω)
(1.3)
1
J(f) + αf
2
L
2
(Q
T
)
(1.4)
α > 0 ψ
T
f(x, t) = f(x) ∈ L
2

(Ω)
2.
Ω R
n
, n ≥ 2; T > 0
Q
T
= Ω × (0, T )
L
2
(Ω) Ω
(u, v)
L
2
(Ω)
=

Ω
u(x)v(x)dx u
L
2
(Ω)
= (u, u)
L
2
(Ω)
; ∀u, v ∈ L
2
(Ω) H
1,0

(Q
T
) =
L
2
((0, T ); H
1
(Ω)) u(x, t) L
2
(Q
T
)
∂u/∂x
i
, i = 1, , n Q
T
(u, v)
H
1,0
(Q
T
)
=

Q
T
(uv + u
x
v
x

)dxdt; u
H
1,0
(Q
T
)
= (u, u)
H
1,0
(Q
T
)
; ∀u, v ∈ H
1,0
(Q
T
).
H
1,0
0
(Q
T
) = L
2
((0, T ); H
1
0
(Q
T
)) H

1,0
(Q
T
)
∂Q
T
Q
T
L
∞
(Q
T
)
Q
T
a
ij
(x, t), i, j = 1, 2, n; a
i
(x, t), b
i
(x, t), i = 1, 2, , n; a(x, t)
a
ij
, a
i
, b
i
∈ L
∞

(Q
T
); a
ij
= a
ji
, i, j = 1, 2, , n.
ν, µ
νξ
2
≤
n

i,j=1
(a
ij
(x, t)ξ
i
ξ
j
≤ µξ
2
.




n

i=1

a
2
i
,




n

i=1
b
2
i
, |a| ≤ µ.
u(x, t) H
1
,
0
0
(Q
T
)
u ∈ H
1,0
0
(Q
T
)


Q
T
(−uη
t
+
n

i,j=1
a
ij
u
x
j
η
x
i
+
n

i=1
a
i
uη
x
i
+
n

i=1
b

i
u
x
i
η + auη)dxdt =

Ω
ϕη(x, 0)dx +

Q
T
fηdxdt,
∀η ∈ H
1,0
(Q
T
).
ϕ ∈ L
2
(Ω), f ∈ L
2
(Q
T
)
u(x, t) C([0, T ]; L
2
(Ω)) ∩ H
1,0
(Q
T

).
u
H
1,0
(Q
T
)
≤ C(ϕ
L
2
(Ω)
+ f
L
2
(Q
T
)
),
C Ω
ψ
k
(
x
) H
1
0
(Ω) L
2
(Ω)
u

N
(x, t) =
N

k=1
u
N
k
(t)ψ
k
(x), (2.1)
(u
N
t
, ψ
l
) +
n

i=1
(
n

j=1
a
ij
u
N
x
j

+ a
i
u
N
, ψ
lx
i
) +
n

i=1
(b
i
u
N
x
i
+ au
N
, ψ
l
) = (f
N
, ψ
l
); l = 1, 2, , N. (2.2)
u
N
l
(0) = (ϕ, ψ

l
); l = 1, 2, , N, (2.3)
L
2
(Ω) f
f
N
=
N

k=1
(f, ψ
k
)ψ
k
, (2.4)
ψ
k
(f
N
, ψ
l
) = (f, ψ
l
) (2.5)
u
N
u
N


L
2
(Q
T
)
≤ C,
C N {u
N
k
} {u
N
} u
N
k
L
2
(Q
T
) u
N
k
x
, u ∈ H
1,0
0
(Q
T
), k → ∞.
u ∈ H
1,0

0
(Q
T
)
a
i
= b
i
= 0; i =
1, 2, n,
J(f
N
) =
1
2

N

k=1
u
N
k
(T )ψ
k
− ψ
δ
T

2
, (2.6)

ψ
δ
T
ψ
T
f
N
∗
J
∗
N
≤ J(f
N
∗
) ≤ J
∗
N
+ ε
N
(2.7)
ε
N
> 0 ε
N
→ 0 N → ∞
J
∗
N
= inf
f

N
∈H
N
J(f
N
)
H
N
(ψ
1
, ψ
2
, , ψ
N
).
3.
f ∈ L
2
(Ω) f
N
=
N

k=1
f
k
ψ
k
→ f L
2

(Ω) N → ∞.
|J(f) − J(f
N
)| → 0, N → ∞.
J(f) − J(f
N
) = u(., T ; f, ϕ) − ψ
δ
T

2
L
2
(Ω)
− u(., T ; f
N
, ϕ) − ψ
δ
T

2
L
2
(Ω)
= u(., T ; f, ϕ) − u(., T; f
N
, ϕ) + u(., T ; f
N
, ϕ) − ψ
δ

T

2
L
2
(Ω)
− u(., T ; f
N
, ϕ) − ψ
δ
T

2
L
2
(Ω)
= u(., T ; f, ϕ) − u(., T; f
N
, ϕ)
2
L
2
(Ω)
+ 2(u(., T ; f, ϕ) − u(., T ; f
N
, ϕ), u(., T ; f
N
, ϕ) − ψ
δ
T

)
= u(., T ; f − f
N
, 0)
2
L
2
(Ω)
+ 2(u(., T ; f − f
N
, 0), u(., T ; f
N
, ϕ) − ψ
δ
T
)
f
N
→ f L
2
(Ω), u(., T ; f − f
N
, 0)
2
L
2
(Ω)
→ 0,
u(., T ; f
N

, ϕ) −ψ
δ
T
 ≤ u(., T ; f
N
, ϕ)  +  ψ
δ
T

|J(f) − J(f
N
)| → 0 N → ∞.

f
N
∗
(u
∗
, f
∗
)
f
N
∗
f
∗
L
2
(Ω) u(., t; f
N

∗
, ϕ) u
∗
L
2
(Q
T
).
ε > 0 f
ε
∈ H
1
(Ω)
J
∗
≤ J(f
ε
) ≤ J
∗
+
ε
2
.
J
∗
= inf
f∈L
2
(Ω)
J(f) N

∗
N ≥ N
∗
|J(f
ε
) − J(f
εN
)| <
ε
2
.
−
ε
2
< J(f
ε
) − J(f
εN
) <
ε
2
J(f
εN
) < J(f
ε
) +
ε
2
.
J

∗
N
≤ J(f
εN
) < J(f
ε
) +
ε
2
≤ J
∗
+ ε.
lim sup
N→∞
J
∗
N
≤ J
∗
.
J
∗
≤ J
∗
N
,
lim inf
N→∞
J
∗

N
≥ J
∗
.
lim
N→∞
J
∗
N
lim
N→∞
J
∗
N
= J
∗
. (3.1)
0 ≤ J(f
N
∗
) − J
∗
≤ |J(f
N
∗
) − J
∗
N
| + |J
∗

N
− J
∗
|.
|J
∗
N
− J
∗
| |J(f
N
∗
) − J
∗
N
|
J(f
N
∗
) → J
∗
N → ∞,
f
N
∗
∈ H
N
J(f). J(f)
f
N

∗
f
∗
, H
1
0
(Ω) L
2
(Ω), f
N
∗
f
∗
L
2
(Ω). u(x, t; f
N
∗
, ϕ) u(x, t; f
N
∗
, ϕ)
L
2
(Q
T
).







u
t
− Lu = f(x, t); (x, t) ∈ Ω × (0, T ),
u|
∂Ω×(0,T ]
= 0,
u|
t=0
= ϕ.