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Приближенное решение обратной граничной задачи для уравнения теплопроводности нелинейным методом проекционной регуляризации.

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1
. .
,
-
.
.
: , , , , .
[1]
C [ 0, ∞ ) ,
h (t )
,
,
2
[2]
.
.
h (t ) ∈ ,
ε >0
3
(1−ε )
W22
( −∞, ∞ )
.
,
[2],
.
∂u ( x, t )
∂t
h (t ) –
∂ 2 u ( x, t )
; 0 < x < 1, t > 0 ,
∂x 2
u ( x, 0) = 0; 0 ≤ x ≤ 1,
u (0, t ) = h(t ); t ≥ 0,
-
h′ ( t ) ,
u ( x, t ) ∈ C ([0,1] × [ 0, ∞ ) ) C
,
2,1
(2)
(3)
h(0) = 0
t ≥ t0
h(t ) = 0;
u (1, t ) = 0; t ≥ 0.
u ( x, t )
(4)
(5)
(6)
(1)–(3),
(6),
. .
,
( ( 0,1) × ( 0, ∞ ) ).
,
u ( x, t ) .
(1)
,
t0 > 0
[3, . 424]
-
1.
:
1
=
t.
Φ ( t ) ∈ C [ 0, ∞ )
.
-
% !" $#& '()*((" – +),- .)(.!"(#, /0() "12+(# !- (/, 34 !-5)#+/- *!+6)+"(
1- 6 "()+(.
-
E-mail: KamaltdinovaTS@mail.ru
26
.
«
.
.
»
..
∞
∞
º
∂ ª
′
u
x
,
t
t
dt
u ( x, t ) Φ ( t ) dt »
Φ
=
(
)
(
)
«
³ x
³
∂x «¬ 0
»¼
0
∞
³ u′′xx ( x, t ) Φ ( t ) dt =
0
u ( x, t ) –
2.
∞
º
∂2 ª
u x, t ) Φ ( t ) dt » .
«
2 ³ (
∂x «¬ 0
»¼
(1)–(3), (6).
∞
∞
lim ³ u ( x, t ) − h ( t ) dt = lim ³ u ( x, t ) dt = 0 .
x →0
x →1
0
0
,
(3),
(1)–(6)
,
h(t ) ,
x1 ∈ ( 0,1) ,
f (t )
,
-
u ( x1 , t ) = f ( t ) ; t ≥ 0 .
(7)
(1)–(2), (6)–(7)
Z ∈ L2 [ 0, ∞ ) .
h(t ) ∈ Z
,
n
h ( t ) = ¦ ϕ j ( t ) + ψ ( t ),
j =1
­aj ª
t − cj
°
ϕ j ( t ) = ® 2 ¬«
° 0;
¯
(
a j , b j , c j > 0, c j ≥ b j
)
2
− b2 j º ; c j − b j ≤ t ≤ c j + b j
,
¼»
t < c j − bj , t > c j + bj
j ≠ k , ψ ( t ) ∈W23 2 [ 0, ∞ ) .
c j ≠ ck
f (t ) = f 0 ( t ) ,
,
h0 (t ) ,
(7),
f0 ( t )
Z,
fδ ( t ) ∈ L2 [ 0, ∞ ) L1 [ 0, ∞ )
,
δ >0
,
fδ − f 0 ≤ δ .
fδ , δ ,
,
Z,
hδ − h0
(6)–(8)
H = L2 [ 0, ∞ ) + iL2 [ 0, ∞ ) –
L2 [ 0, ∞ ) L1 [ 0, ∞ )
hδ (t )
hδ (t )
L2
1
π
∞
³ h (t ) e
− iτ t
dt; τ ≥ 0 .
(9),
(9)
.
-
:
∂ uˆ ( x,τ )
∂x 2
1
,
0
F,
. [1].
2
5,
F–
H
1 2
(1)–(2), (6)–(8).
,
2013,
(1)–(2),
h0 (t ) .
,
F ª¬ h ( t ) º¼ =
1.
(8)
= iτ uˆ ( x,τ ) ; x ∈ ( 0,1) , τ ≥ 0,
(10)
27
uˆ ( x,τ ) = F ª¬u ( x, t ) º¼ .
(6)–(7)
,
fˆ (τ ) = F ª¬ f ( t ) º¼ .
2
uˆ (1,τ ) = 0; τ ≥ 0
(11)
uˆ ( x1 ,τ ) = fˆ (τ ) ; τ ≥ 0,
(12)
uˆ ( x,τ )
,
[0,1] × [0, ∞ ) .
(10)–(12)
(10)
uˆ ( x, t ) = A (τ ) e μ0 x
1
(1 + i ) ,
2
(11)–(13)
A (τ )
μ0 =
+ B (τ ) e − μ0 x
τ
,
(13)
fˆ (τ ) , τ ≥ 0.
(14)
B (τ ) –
.
,
hˆ (τ ) =
x1 ≤
,
sh μ0 τ
sh μ0 (1 − x1 ) τ
1
, τ ∈ [ 0, 2] .
2
2
sh μ0 τ
sh μ0 (1 − x1 ) τ
(15)
τ
=
ch 2τ − cos 2τ
,
ch (1 − x1 ) 2τ − cos (1 − x1 ) 2τ
(15)
,
2
sh μ0 τ
=
sh μ0 (1 − x1 ) τ
ch λ − cos λ
,
ch (1 − x1 ) λ − cos (1 − x1 ) λ
(16)
λ = 2τ .
,
(16)
,
sh μ0 τ
(
≤
sh μ0 (1 − x1 ) τ
e−2
sh μ0 τ
sh μ0 (1 − x1 ) τ
,
,
=
ch
ch 2τ − cos 2τ
ch (1 − x1 ) 2τ − cos (1 − x1 ) 2τ
1
x1 ≤ , τ ≥ 2 ,
2
2≤e≤e
0 ≤τ ≤ 2 .
(17)
τ ≥ 2.
sh μ0 (1 − x1 ) τ
sh μ0 τ
)
2 e2 + 1
2(1− x1 )
τ
2
sh (1 − x1 )
τ
2e
≤
e(
τ
1− x1 ) 2
2
2
− 1− x
− e ( 1)
τ
,
2
τ
2
,
sh μ0 τ
sh μ0 (1 − x1 ) τ
(
≤
τ
≤ 4e
x1
τ
2
τ ≥2.
(18)
) >4,
2 e2 + 1
(17)
28
(18)
e−2
.
«
.
.
»
..
sh μ0 τ
≤ ae
sh μ0 (1 − x1 ) τ
a=
(
x1
τ
2
, τ ≥0,
(19)
).
2 e2 + 1
e−2
(20)
(14)
,
L2 [ 0, ∞ )
(1)–(2), (6)–(7)
L2 [ 0, ∞ )
Tfˆ (τ ) =
μ0 =
sh μ0 τ
sh(1 − x1 ) μ0 τ
fˆ (τ ) , τ ≥ 0,
1
(1 + i ) , fˆ , Tfˆ ∈ H .
2
(14)
F
:
(22)
Ẑ = F [ Z ] ,
(23)
-
L2 [ 0, ∞ ) ,
(9),
T
(22)
.
fˆ (τ ) = fˆ0 (τ )
,
fˆ0 (τ )
(21)
hˆ (τ ) = Tfˆ (τ ) ; τ ≥ 0 .
Ẑ
hˆ0 (τ ) ∈ Zˆ ,
fˆδ (τ ) ∈ H
,
δ >0
,
fˆδ (τ ) − fˆ0 (τ ) ≤ δ .
fˆδ (τ )
,
ĥδ (τ )
-
T,
(24)
δ,
ĥ0 (τ )
,
Ẑ ,
hˆδ (τ ) − hˆ0 (τ ) .
{Tα : α ≥ 0} ,
:
°­Tfˆ (τ ) ; τ ≤ α
Tα fˆ (τ ) = ®
, α > 0,
°̄ 0; τ > α
(21).
ĥδα (τ )
T
,
hˆδα = Tα fˆδ (τ ) ,
Tα
T
(26)
(
α = α fˆδ , δ
(25).
(25)
)
(26)
2
T −1hˆδα (τ ) − fˆδ (τ ) = 9δ 2 .
,
1
2
T hδ (τ ) − fˆδ (τ ) ∈ C [ 0, ∞ )
−1 ˆα
2013,
5,
1
α,
(27)
[4],
,
fˆδ
2
α →∞
29
α →0.
fˆδ
,
2
> 9δ 2
(27).
,
α ∈ [α1 , α 2 ]
hˆδα
=
[α1,α 2 ]
(27)
hˆδα1
,
.
.
(27).
T
,
ĥδ (τ )
(22), (24)
ª α ( fˆδ ,δ )
º
hˆδ (τ ) = pr « hˆδ
(τ ) ; H 0 » ,
¬
¼
(28)
H 0 = F ª¬ L2 [ 0, ∞ ) º¼ .
hˆδ (τ ) − hˆ0 (τ )
(22), (24)
ĥδ
hˆδ (τ ) − hˆ0 (τ )
hˆ0 (τ ) ∈ Zˆ ,
ĥ0 (τ ) .
τ ≥0
ĥδ (τ )
c>0
(23)
c
hˆ0 (τ ) ≤
∞
³ ª¬1 + τ
(29)
§
¼
1º
ε ∈ ¨ 0, »
© 2¼
,
3(1−ε ) º
0
.
1+τ 4
d >0
(29)
,
2
d
hˆ0 (τ ) dτ ≤ .
(30)
ε
hˆ0 (τ ) ∈ H 0 ,
,
(31)
[2],
(28), (31)
,
d
hˆδ (τ ) − hˆ0 (τ ) ≤ 6 Gε ª¬α ( δ , ε ) º¼ ,
ε
Gε (σ ) ,
(14), (19), (20)
σ = ae
α (δ , ε )
x1
τ
2
(32)
(30)
:
3 1−ε
, Gε (τ ) = ª1 + τ ( ) º
¬
¼
d Gε (α )
ε
α
−
1
2
; τ ≥ 0,
(33)
= δ.
§
(34)
1º
ε (δ ) ,
ε ∈ ¨ 0, » ,
© 2¼
(32)
-
,
6
(32)–(35)
d
d
Gε (δ ) ¬ªα (δ , ε ( δ ) ) ¼º = min 6 Gε ª¬α ( δ , ε ) º¼ .
ε (δ )
ε
ε ∈( 0, 1 2 º¼
ĥδ (τ )
hˆδ (τ ) − hˆ0 (τ ) ≤ 6
30
(35)
d
ε (δ )
Gε (δ ) ª¬α (δ , ε (δ ) ) º¼ .
.
«
(36)
.
.
»
..
§
(36),
.
,
1º
ε ∈ ¨ 0, »
© 2¼
ª § ·3(1−ε )
º
2
6 1−ε § σ ·
Gε (σ ) = «1 + ¨ 2 ¸
ln ( ) ¨ ¸ »
« ¨x ¸
© a ¹»
¬ © 1 ¹
¼
(37)
σ >a
,
§
(37)
(38)
,
(1−ε )
σ −1
ª x1 º
« »
¬2¼
3(1−ε )
3(1−ε )
(38)
(39)
σ ≥ σ1
,
ªx º
Gε (σ ) ≥ « 1 »
¬« 2 »¼
σ →∞
(37)
1º
σ1 > a
lim
.
ª x2 º 2
−3 1−ε § σ ·
ln ( ) ¨ ¸ .
Gε (σ ) ≤ « 1 »
©a¹
«¬ 2 »¼
§ 1º
σ > a ε ∈ ¨ 0, »
© 2¼
Gε (σ )
lim
=1,
3 (1−ε )
σ →∞
2
2
ª x1 º
−3 1−ε § σ ·
ln ( ) ¨ ¸
« »
©a¹
¬2¼
(39)
,
1
2
ε ∈ ¨ 0, »
© 2¼
3
(37)
−
1º
−3 1−ε § σ ·
ln ( ) ¨ ¸ .
©a¹
(40)
σ 2 > σ1
=0
−3 1−ε § σ ·
ln ( ) ¨ ¸
©a¹
§
ε ∈ ¨ 0, »
© 2¼
,
σ >σ2
ªx
σ <« 1
«¬ 2
(40),
º
»
»¼
3(1−ε )
σ >σ2
1
σ
α (δ , ε )
2
<
1
σ
(34),
d
(42)
ε
,
−3 1−ε § σ ·
ln ( ) ¨ ¸ ,
©a¹
Gε (σ ) .
αˆ (δ , ε )
α −2 = δ .
αˆ (δ , ε ) = 4
(34), (41)
δ ∈ ( 0, δ 0 )
2013,
5,
(42)
,
,
(32), (38)
§
1º
(44)
(41)
d
(42)
.
(43)
αˆ (δ , ε ) ≤ α (δ , ε ) .
(44)
εδ 2
δ0 > 0
,
-
ε ∈ ¨ 0, »
© 2¼
1
31
hˆδ (τ ) − hˆ0 (τ )
(44)
(45)
2
d ªx
≤ 36 « 1
ε «¬ 2
3(1−ε )
§
δ ∈ ( 0, δ 0 )
,
º
»
»¼
−3 1−ε § αˆ (δ , ε ) ·
ln ( ) ¨
¸.
© a ¹
(45)
1º
ε ∈ ¨ 0, »
© 2¼
2
d 3 1−ε −3 1−ε § 1 d ·
hˆδ (τ ) − hˆ0 (τ ) ≤ 36 x1 ( ) ln ( ) ¨¨ 4 2 ¸¸ .
ε
© a εδ ¹
ε (δ ) =
(46)
1
hˆδ (τ ) − hˆ0 (τ )
,
δ
§
d
≤ 36 d ln ln ln ¨ 4 4
¨ a ε (δ ) δ 2
δ
©
δ1 > δ 0
1
−3
δ12 d
a
δ ∈ ( 0, δ1 )
,
ln ln 1
,
2
4
ln ln
1
δ1
· ª ε (δ ) §
dδ 2
¨4
¸ « ln
¸
¨ a 4ε ( δ ) δ 4
¹ «¬
©
3
·º
¸» .
¸»
¹¼
(47)
<1.
(47)
§ d ln ln 1
2
1
δ
hˆδ (τ ) − hˆ0 (τ ) ≤ 36d ln ln ln −3 ¨ 4
4 2
¨¨
δ
aδ
©
δ ∈ ( 0, δ1 )
ln ln
(49)
(46)
ε (δ )
·
3
¸ ªln ε (δ ) 1 º .
¸¸ «¬
δ »¼
¹
(48)
1
1
§1·
ln ln = 1 ,
¨ ¸=
1
δ
© δ ¹ ln ln
(49)
δ
,
3
ª ε (δ ) § 1 · º
3
¨ ¸» = e ,
«ln
δ
©
¹
¬
¼
(48)
(50),
§ d ln ln 1
2
1
δ
hˆδ (τ ) − hˆ0 (τ ) ≤ 36 d e3 ln ln ln −3 ¨ 4
4 2
¨¨
δ
a δ
©
1 a4
,
δ 2 < δ1 ln ln >
δ ∈ ( 0, δ 2 )
δ2 d
,
ln ln
d
a
,
(53)
,
(51)
(52)
,
4
·
¸.
¸¸
¹
(51)
1
δ > 1 .
δ
δ2
δ ∈ ( 0, δ 2 )
(52)
2
2
1
§ 1 ·
hˆδ (τ ) − hˆ0 (τ ) ≤ 36 d e3 ln ln ln −3 ¨
¸.
δ
© δ ¹
δ ∈ ( 0, δ 2 )
1
hˆδ (τ ) − hˆ0 (τ ) ≤ 17 d e3 ln ln ln
δ
32
(50)
.
−
3
2
«
(53)
§1·
¨ ¸.
©δ ¹
.
.
»
..
1.
, . .
/ . .
. – 2010. – . 16, 2. – . 238–252.
2.
, . .
/ . .
, . .
. – 2012. – . 18, 1. – . 281–288.
3.
, . .
/ . .
, . .
4.
. .
, . .
//
– . 353–368.
,
. .
//
,
.–
. .
.:
//
, 1968. – 576 .
. – 2006. – . 9,
/
4.
APPROXIMATE SOLUTION OF INVERSE BOUNDARY PROBLEM
FOR THE HEAT CONDUCTIVITY EQUATION BY NONLINEAR METHOD
OF PROJECTION REGULARITY
T.S. Kamaltdinova
1
Inverse boundary problem is solved in the hypothesis that the required solution is a piecewise
smooth function, estimates of above approximate solution are given. The estimates are considerably superior to the known estimates by the accuracy.
Keywords: operator equation, regularity, optimal method, error estimation, ill-posed problem.
References
1. Tanana V.P., Sidikova A.I. O garantirovannoj ocenke tochnosti priblizhennogo resheniya odnoj
obratnoj granichnoj zadachi teplovoj diagnostiki [On the guaranteed accuracy estimate of an approximate solution of one inverse problem of thermal diagnostics]. Trudy Instituta Matematiki I Mekhaniki
URO RAN. 2010. Vol. 16, no. 2. pp. 238–252. (in Russ.).
2. Tanana V.P., Bredikhina A.B., Kamaltdinova T.S. Trudy Instituta Matematiki I Mekhaniki URO
RAN. 2012. Vol. 18, no. 1. pp. 281–288. (in Russ.).
3. Mikhlin S.G. Kurs matematicheskoj fiziki [Course of mathematical physics]. Moscow: Nauka,
1968. 576 p. (in Russ.).
4. Tanana V.P., Yaparova N.M. Ob optimal'nom po poryadku metode resheniya uslovnokorrektnykh zadach [The optimum in order method of solving conditionally-correct problems]. Sib. Zh.
Vychisl. Mat. 2006. Vol. 9, no. 4. pp.353–368. (in Russ.).
!"#$% & '()%*+#, 29 -./'. 2012 0.
1
Kamaltdinova Tatyana Sergeevna is a Senior Lecturer, Numerical Mathematics Department, South Ural State University.
E-mail: KamaltdinovaTS@mail.ru
2013,
5,
1
33
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