Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 260941, 13 pages
doi:10.1155/2009/260941
Research Article
Solving the Axisymmetric Inverse Heat Conduction
Problem by a Wavelet Dual Least Squares Method
Wei Cheng1 and Chu-Li Fu2
1
2
College of Science, Henan University of Technology, Zhengzhou 450001, China
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Correspondence should be addressed to Chu-Li Fu, fuchuli@lzu.edu.cn
Received 17 August 2008; Revised 23 January 2009; Accepted 10 March 2009
Recommended by Ugur Abdulla
We consider an axisymmetric inverse heat conduction problem of determining the surface
temperature from a fixed location inside a cylinder. This problem is ill-posed; the solution if it
exists does not depend continuously on the data. A special project method—dual least squares
method generated by the family of Shannon wavelet is applied to formulate regularized solution.
Meanwhile, an order optimal error estimate between the approximate solution and exact solution
is proved.
Copyright q 2009 W. Cheng and C.-L. Fu. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Inverse heat conduction problems IHCP have become an interesting subject recently, and
many regularization methods have been developed for the analysis of IHCP 1–13. These
methods include Tikhonov method 1, 2, mollification method 3, 4, optimal filtering
method 5, lines method 6, wavelet and wavelet-Galerkin method 7–11, modified
Tikhonov method 12 and “optimal approximations” 13, and so forth. However, most
analytical and numerical methods were only used to dealing with IHCP in semiunbounded
region. Some works of numerical methods were presented for IHCP in bounded domain 14–
19.
Chen et al. 14 applied the hybrid numerical algorithm of Laplace transform
technique to the IHCP in a rectangular plate. Busby and Trujillo 15 used the dynamic
programming method to investigate the IHCP in a slab. Alifanov and Kerov 16 and
Louahlia-Gualous et al. 17 researched IHCP in a cylinder. However to the authors’
knowledge, most of them did not give any stability theory and convergence proofs.
In this paper, we will treat with a special IHCP whose physical model consists of
an infinitely long cylinder of radius R. It is considered axisymmetric and a thermocouple
2
Boundary Value Problems
measurement equipment of temperature is installed inside the cylinder at the radius r1 ,
0 < r1 < R. The correspondingly mathematical model of our problem can be described by
the following axisymmetric heat conduction problem:
∂2 u 1 ∂u
∂u
Δu 2 , 0 < r ≤ R, t > 0,
∂t
r ∂r
∂r
ur, 0 0, 0 ≤ r ≤ R,
ur1 , t gt,
ur, t
1.1
t ≥ 0,
bounded in r 0, t > 0,
where the functions ur, · and g· belong to L2 0, ∞ for every fixed r ∈ 0, R, r is the
radial coordinate, gt denotes the temperature history at one fixed radius r1 0 < r1 < R
of cylinder. We want to recover ur, · for r1 < r ≤ R. This problem is ill-posed problem; a
small perturbation in the data may cause dramatically large errors in the solution ur, · The
details can be seen in Section 2.
To the authors’ knowledge, up to now, there is no regularization theory with error
estimate for problem 1.1 in the interval r1 < r ≤ R. The major objective of this paper is to do
the theoretic stability and convergence estimates for problem 1.1.
Xiong and Fu 11 and Regińska 20 solved the sideways heat equation in semiunbounded region by applying the wavelet dual least squares method, which is based
on the family of Meyer wavelet. In this paper, we will apply a wavelet dual least
squares method generated by the family of Shannon wavelet to problem 1.1 in bounded
domain for determining surface temperature. According to the optimality results of general
regularization theory, we conclude that our error estimate on surface temperature is order
optimal.
2. Formulation of Solution of Problem 1.1
As we consider problem 1.1 in L2 R with respect to variable t, we extend ur , ·, g· :
ur1 , ·, f· : uR , ·, and other functions of variable t appearing in the paper to be zero for
t < 0. Throughout the paper, we assume that for the exact g the solution u exists and satisfies
an apriori bound
f· : uR, · ≤ E,
p
p
p ≥ 0,
2.1
where f·p is defined by
f· :
p
∞
−∞
p 2 dξ
1 ξ2 fξ
1/2
.
2.2
Since g is measured by the thermocouple, there will be measurement errors, and we would
actually have as data some function gδ ∈ L2 R, for which
gδ · − g· ≤ δ,
2.3
Boundary Value Problems
3
where the constant δ > 0 represents a bound on the measurement error, and · denotes the
L2 R norm and
1
hξ
√
2π
∞
−∞
e−iξt htdt
2.4
is the Fourier transform of function ht. The problem 1.1 can be formulated, in frequency
space, as follows:
iξu
r, ξ ur, ξ
∂2 u
r, ξ 1 ∂
, r ∈ 0, R, ξ ∈ R,
r ∂r
∂r 2
u
r1 , ξ g ξ, ξ ∈ R,
u
0, ξ < ∞, ξ ∈ R.
2.5
2.6
2.7
Then we have the following lemma.
Lemma 2.1. Problem 2.5–2.7 has the solution given by
I0 iξr
u
r, ξ g ξ,
I0 iξr1
r ∈ 0, R, ξ ∈ R,
2.8
where I0 z denotes modified spherical Bessel function which given by [21]
2k
∞
1
z
I0 z .
2
2
k0 k!
2.9
Proof. Due to 21, we can solve 2.5, in the frequency domain, to obtain
u
r, ξ AξI0
iξr BξK0
iξr
ξ ∈ R,
2.10
where K0 z denotes also modified spherical Bessel function which is given by
2k
∞
1
1
z
1
z
·
·
·
.
1
K0 z −I0 z ln C 2
2
2
k
2
k1 k!
2.11
Combining limz → 0 K0 z ∞ with condition2.7, we obtain Bξ 0, that is,
u
r, ξ AξI0
iξr ,
r ∈ 0, R, ξ ∈ R.
2.12
According to 21, there holds
4k 1/2
∞
|ξ|r/2
iξr ber
|ξ|r iσ bei
,
|ξ|r I0
2
k0 k! 2k!
2.13
4
Boundary Value Problems
where σ sgnξ, both berx and beix denote the Kelvin functions. Since 0 ≤ r ≤ R, |ξ| ≥ 0,
we have
∞
4k
|ξ|r/2
k0
k!2 2k!
1
8
|ξ|r/2
2!2 4!
12
|ξ|r/2
3!2 6!
· · · ≥ 1.
2.14
Therefore, for 0 ≤ r ≤ R, ξ ∈ R,
4k 1/2
∞
|ξ|r/2
iξr ≥ 1.
I0
2
k0 k! 2k!
2.15
Solving the systems 2.6 and 2.12 using 2.15 we get
Aξ I0−1
iξr1 g ξ.
2.16
Substitution of Aξ in 2.16 into 2.12, we obtain 2.8.
Applying an inverse Fourier transform to 2.8, problem 1.1 has the solution
1
ur, t √
2π
I0 iξr
e
g ξdξ,
I0 iξr1
−∞
∞
iξt
r, t ∈ 0, R × R.
2.17
In order to obtain ill-posedness of problem 1.1 for r, t ∈ r1 , R × R, we need the
following lemma.
Lemma 2.2. If function |I0 iξr| satisfies 2.15, then there exist positive constants ck , k 1, 2, 3, 4, such that, for r ∈ r1 , R
I0 iξr c1 exp
|ξ|/2 r − r1 ≤ |ξ|/2r − r1 ,
≤ c2 exp
I0 iξr1 I0 iξr |ξ|/2r − R ≤ ≤ c4 exp
|ξ|/2r − R ,
c3 exp
I0 iξR ξ ∈ R,
2.18
ξ ∈ R.
2.19
Proof. First, due to 21 and 2.15, we have, for r ∈ r1 , R and |ξ| → ∞,
1/2 exp |ξ|/2r 1
2
2
1O
iξr ber
|ξ|r bei
|ξ|r
,
I0
|ξ|
2πr |ξ|
2.20
Boundary Value Problems
5
then there exist positive constants ck , k 1, 2, 3, 4, such that, for |ξ| large enough, say |ξ| ≥ ξ0
exp
|ξ|/2r
exp
|ξ|/2r
c1 iξr ≤ c2 ≤ I0
,
r ∈ r1 , R,
2πr |ξ|
2πr |ξ|
exp
|ξ|/2r1
exp
|ξ|/2r1
c3 ≤ I0
.
iξr1 ≤ c4 2πr1 |ξ|
2πr1 |ξ|
2.21
From these we know that there exist positive constants c5 and c6 such that, for r ∈ r1 , R and
|ξ| ≥ ξ0 ,
c5 exp
I0 iξr |ξ|/2r − r1 ≤ |ξ|/2r − r1 .
≤ c6 exp
I0 iξr1 2.22
Then, since function |I0 iξr/I0 iξr1 | is continuous in the closed region r1 , R × −ξ0 , ξ0 .
Threrfore, there exist constants c7 and c8 such that, for r ∈ r1 , R and |ξ| ≤ ξ0 ,
c7 exp
I0 iξr |ξ|/2 r − r1 ≤ |ξ|/2 r − r1 .
≤ c8 exp
I0 iξr1 2.23
Finally, combining inequalities 2.22 with 2.23, we can see that there exist others constants
c1 and c2 such that, for r ∈ r1 , R, inequalities 2.18 are valid. Similarly, we obtain
inequalities 2.19.
In order to formulate problem 1.1 for r1 < r ≤ R in terms of an operator equation in
the space X L2 R, we define an operator Kr : ur, · → g·, that is,
∀ur, · ∈ X,
Kr ur, t gt,
r1 < r ≤ R.
2.24
From 2.8, we have
I0 iξr1
r, ξ g ξ.
Kr ur, ξ u
I0 iξr
2.25
r, ξ, and we can see that K
r : L2 R → L2 R is a multiplication
Denote K
r ur, ξ : Kr u
operator:
r u
r, ξ K
iξr1
r, ξ.
u
I0 iξr
I0
From 2.26, we can prove the following lemma.
2.26
6
Boundary Value Problems
Lemma 2.3. Let Kr∗ be the adjoint to Kr , then Kr∗ corresponds to the following problem where the
left-hand side ∂u/∂t of problem 1.1 is replaced by −∂U/∂t, says
−
∂2 U 1 ∂U
∂U
ΔU , 0 < r ≤ R, t ≥ 0,
∂t
r ∂r
∂r 2
Ur, 0 0, 0 ≤ r ≤ R,
Ur1 , t gt,
Ur, t
t ≥ 0,
bounded in r 0, t > 0,
I0 iξr1
∗
r K
.
I0 iξr
2.27
2.28
Proof. Via the the following relations, combining with 2.26,
∗ ∗ r u
, υ u
, K r υ u, K ∗r υ u
, K r υ ,
Kr u, υ K
2.29
we can get the adjoint operator Kr∗ of Kr in frequency domain
r∗ K
r ∗ K
iξr1
.
I0 iξr
I0
2.30
On the other hand, the problem 2.27 can be formulated, in frequency space, as
follows:
ξ −iξUr,
ξ
ξ 1 ∂Ur,
∂2 Ur,
,
r
∂r
∂r 2
r ∈ 0, R, ξ ∈ R,
1 , ξ g ξ, ξ ∈ R,
Ur
U0,
ξ < ∞, ξ ∈ R.
2.31
ξ u
Taking the conjugate operator for problem 2.5–2.7, we realize that Ur,
r, ξ.
Therefore, by Lemma 2.1, we conclude that
I0 iξr
ξ u
r, ξ Ur,
g ξ,
I0 iξr1
2.32
Boundary Value Problems
7
that is,
iξr1
∗
r∗ Ur,
ξ K
ξ : K
g ξ Ur,
r U.
I0 iξr
I0
2.33
Hence the conclusion of Lemma 2.3 is proved.
The Parseval formula for the Fourier transform together with inequality 2.18, there
holds
2
ur, ·2 u
r, ·
∞
2
u
r, · dξ
−∞
∞ √ 2 I iξr 2
r−r1 |ξ|/2 0
g ξe
dξ
I
iξr
−∞
0
1
∞ √ 2
≥ c12
g ξer−r1 |ξ|/2 dξ.
2.34
−∞
This implies that g ξ, which is Fourier transform of exact data gt, must decay rapidly at
high frequencies since r1 < r. But such a decay is not likely to occur in the Fourier transform
of the measured noisy data gδ t at r r1 . So, small perturbation of gt in high frequency
components can blow up and completely destroy the solution ur, t given by 2.17 for r ∈
r1 , R.
3. Wavelet Dual Least Squares Method
3.1. Dual Least Squares Method
A general projection method for the operator equation Ku g, K : X L2 R → X L2 R is
generated by two subspace families {Vj } and {Yj } of X and the approximate solution uj ∈ Vj
is defined to be the solution of the following problem:
Kuj , y g, y,
∀y ∈ Yj ,
3.1
where ·, · denotes the inner product in X. If Vj ⊂ RK ∗ and subspaces Yj are chosen in such
a way that
K ∗ Yj Vj .
3.2
Then we have a special case of projection method known as the dual least squares method. If
{ψλ }λ∈Ij is an orthogonal basis of Vj and yλ is the solution of the equation
K ∗ yλ kλ ψλ ,
yλ 1,
3.3
8
Boundary Value Problems
then the approximate solution is explicitly given by the expression
uj λ∈Ij
g, y λ
1
ψλ .
kλ
3.4
3.2. Shannon Wavelets
In 22, the Shannon scaling function is φ sin πt/πt and its Fourier transform is
1,
φξ
0,
|ξ| ≤ π,
otherwise.
3.5
The corresponding wavelet function ψ is given by its Fourier transform
ψξ
e−iξ/2 ,
π ≤ |ξ| ≤ 2π,
0,
otherwise.
3.6
Let us list some notation: φj,k t : 2j/2 φ2j t − k, ψj,k t : 2j/2 ψ2j t − k, j, k ∈ Z, Ψ−1,k : φ0,k
and Ψl,k : ψl,k for l ≥ 0, the index set
I J {j, k} : j −1, 0, . . . , J − 1; k ∈ Z ⊂ Z2 .
I {j, k} : j, k ∈ Z ⊂ Z2 ,
3.7
Because VJ VJ−1 ⊕ WJ−1 VJ−2 ⊕ WJ−2 ⊕ WJ−1 · · · V0 ⊕ W1 ⊕ · · · ⊕ WJ−1 , hence we can
define the subspaces VJ
VJ span{Ψλ }λ∈IJ .
3.8
Define an orthogonal projection PJ : L2 R → VJ :
PJ ϕ ϕ, Ψλ Ψλ ,
∀ϕ ∈ L2 R,
3.9
λ∈IJ
then from 3.4 we easily conclude uJ PJ u. From the point of view of an application to
the problem 1.1, the important property of Shannon wavelets is the compactness of their
support in the frequency space. Indeed, since
−j
ψj,k ξ 2−j/2 e−i2
kξ
−j ψ 2 ξ ,
−j −j
φj,k ξ 2−j/2 e−i2 kξ φ 2 ξ ,
3.10
it follows that for any k ∈ Z
supp ψj,k ξ : π2j ≤ |ξ| ≤ π2j1 ,
supp φj,k ξ : |ξ| ≤ π2j .
3.11
Boundary Value Problems
9
From 3.9, PJ can be seen as a low-pass filter. The frequencies with greater than π2J1 are
filtered away.
Theorem 3.1. If ur, t is the solution of problem 1.1 satisfying the condition uR, ·p ≤ E, then
for any fixed r ∈ r1 , R
√
−p
ur, · − PJ ur, · ≤ c−1 2J1 er−R 1/2π2J E.
3
3.12
Proof. From 3.9, we have
ur, · ur, ·, Ψλ Ψλ ,
λ
PJ ur, · ur, ·, Ψλ Ψλ .
3.13
λ∈IJ
Due to Parseval relation and 2.8, 2.19, and 2.1, there holds
ur, · − PJ ur, · u
r, · − P
J ur, ·
λ −
λ
u
, Ψλ Ψ
u
, Ψλ Ψ
λ∈I
λ∈IJ
λ
u
, Ψλ Ψ
λ∈Ij≥J1
" !
I0 i·r
λ
3.14
g ·, Ψλ Ψ
I0 i·r1
λ∈Ij≥J1
" !
I0 i·r
λ
f·, Ψλ Ψ
I0 i·R
λ∈Ij≥J1
I
#
$ iξr 2 p/2 −p 0
≤
sup
|ξ| f·, Ψλ Ψλ 1 ·
J
J1
I
iξR
π2 ≤|ξ|≤π2
0
λ∈Ij≥J1
√
√
−p
J1 −p r−R 1/2π2J
≤
sup c4 ξ er−R |ξ|/2 E ≤ c4 2
e
E.
π2J ≤|ξ|≤π2J1
Hence the conclusion of Theorem3.1 is proved.
4. Error Estimates via Dual Least Squares Method Approximation
Before giving error estimates, we present firstly subspaces Yj . According to K ∗ Yj Vj , the
subspaces Yj are spanned by ρλ , λ ∈ IJ , where
K ∗ ρλ Ψ λ ,
kλ ρλ −1 ,
yλ ρλ
kλ ρ λ .
ρλ 4.1
10
Boundary Value Problems
ρλ can be determined by solving the following parabolic equation see Lemma 2.3:
−
∂2 U 1 ∂U
∂U
ΔU , 0 < r ≤ R, t ≥ 0,
∂t
r ∂r
∂r 2
Ur, 0 0, 0 ≤ r ≤ R,
Ur1 , t Ψj,k t,
Ur, t
4.2
t ≥ 0,
bounded in r 0, t > 0.
Since supp ψj,k is compact, the solution exists for any t ∈ 0, ∞. Similarly the solution of the
adjoint equation is unique. Therefore for a given Ψλ , ρλ can be uniquely determined according
to 4.2, furthermore
I0 iξr
I0 iξr
ρλ Ψλ ξ ⇐⇒ yλ kλ Ψλ ξ,
I0 iξr1
I0 iξr1
λ {j, k}.
4.3
The approximate solution for noisy data gδ is explicitly given by
PJ uδ r, t uδJ 1
Ψλ .
uδ , Ψλ Ψλ gδ , y λ
kλ
λ∈IJ
4.4
λ∈IJ
Now we will devote to estimating the error PJ uδ − PJ u.
Theorem 4.1. If gδ is noisy data satisfying the condition g· − gδ · ≤ δ, then for any fixed
r ∈ r1 , R
PJ uδ − PJ u ≤ c4 er−r1 √
1/2π2J
δ.
4.5
λ . Note that PJ uδ given by 4.4, PJ u
Proof. From 4.3, we have yλ I0 iξr/I0 iξr1 kλ Ψ
given by 3.4 and 2.18, for r1 < r ≤ R, there holds
1
δ
PJ u r, · − PJ ur, · Ψλ gδ − g, yλ
kλ λ∈IJ
1
λ
Ψ
gδ − g , yλ
kλ λ∈IJ
"
I0 i·r
1
gδ − g , Ψ
Ψ
k
λ λ kλ λ I0 i·r1
λ∈IJ
I0 iξr ≤
sup gδ − g , Ψλ Ψλ · J−1
J
π2 ≤|ξ|≤π2 I0
iξr1
λ∈IJ
Boundary Value Problems
11
I0 iξr ≤
sup gδ − g · PJ iξr1
π2J−1 ≤|ξ|≤π2J I0
I0 iξr ≤
sup · δ
iξr1 π2J−1 ≤|ξ|≤π2J I0
√
≤ c2 sup er−r1 |ξ|/2 δ
π2J−1 ≤|ξ|≤π2J
≤ c2 er−r1 √
1/2π2J
δ.
4.6
Hence the conclusion of Theorem 4.1 is proved.
The following is the main result of this paper.
Theorem 4.2. Let u be the exact solution of 1.1 and let PJ uδ be given by 4.4. If g − gδ ≤ δ and
J Jδ is such that
2 2
1
E −2p
E
,
ln
ln
J log2
π R − r1
δ
δ
4.7
then for any fixed r ∈ r1 , R
ur, · − PJ uδ r, ·
≤E
1−R−r/R−r1 R−r/R−r1 δ
E
ln
δ
−2p1−R−r/R−r1 C o1
4.8
for δ −→ 0,
where C c4 R − r1 2p c2 .
Proof. Combining Theorem 4.1 with Theorem 3.1, and noting the choice rule 4.7 of J, we can
obtain
ur, · − PJ uδ r, ·
√
J1 −p r−R√1/2π2J
J
≤ c4 2
e
E c2 er−r1 1/2π2 δ
!−2p
!r−R/R−r1 2p
E
E
E −2p
E −2p
ln
ln
ln
≤ c4 E R − r1
δ
δ
δ
δ
−2p !r−r1 /R−r1 E
E
ln
c2 δ
δ
δ
2p
%
&
c4 R − r1 lnE/δ
E −2p1−R−r/R−r1 1−R−r/R−r1 R−r/R−r1 ≤E
δ
c
ln
.
2
2p
δ
ln E/δlnE/δ−2p
4.9
12
Boundary Value Problems
Note that
lnE/δ
lnE/δ
−→ 1 for δ −→ 0,
−2p lnE/δ
−
2p ln lnE/δ
ln E/δ lnE/δ
4.10
thus, there holds, for δ → 0
ur, · − PJ uδ r, ·
≤ E1−R−r/R−r1 δR−r/R−r1 E
ln
δ
−2p1−R−r/R−r1 2p
c2 o1 .
c4 R − r1
4.11
Hence the conclusion of Theorem 4.2 is proved.
Remark 4.3. i When p 0 and r1 < r < R, estimate 4.8 is a Hölder stability estimate given
by
ur, · − PJ uδ r, · ≤ c4 c2 E1−R−r/R−r1 δR−r/R−r1 .
4.12
ii When p > 0, r1 < r < R, estimate 4.8 is a logarithmical Hölder stability estimate.
iii When p > 0, r R, estimate 4.3 becomes
−2p
uR, · − uδ R, · ≤ E ln E
c4 R − r1 2p c2 o1 −→ 0 for δ −→ 0.
δ
4.13
This is a logarithmical stability estimate.
Remark 4.4. In general, the a-priori bound E is unknown in practice, in this case, with
2 2
1
1 −2p
1
J log2
,
ln
ln
π R − r1
δ
δ
4.14
then
−2p1−R−r/R−r1 ur, · − PJ uδ r, · ≤ δR−r/R−r1 ln 1
o1 for δ −→ 0,
C
δ
4.15
c4 R − r1 2p E c2 .
where C
Acknowledgments
The work is supported by the National Natural Science Foundation of China No. 10671085,
the Hight-level Personnel fund of Henan University of Technology 2007BS028, and the
Boundary Value Problems
13
Fundamental Research Fund for Natural Science of Education Department of Henan Province
of China No. 2009B110007.
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