A modified regularization method for an inverse heat conduction problem with only boundary value
Cheng and Ma Boundary Value Problems (2016) 2016:100
DOI 10.1186/s13661-016-0606-7
RESEARCH
Open Access
A modified regularization method for an
inverse heat conduction problem with only
boundary value
Wei Cheng1* and Yun-Jie Ma2
*
Correspondence:
1
College of Science, Henan
University of Technology,
Zhengzhou, 450001, P.R. China
Full list of author information is
available at the end of the article
Abstract
This paper aims to solve an inverse heat conduction problem with only boundary
value in a bounded domain, where the boundary data is given for x = 0. The solution
is sought in the interval 0 < x ≤ 1. The problem is seriously ill posed in the Hadamard
sense. Using the Hölder inequality and some inequalities, a conditional stability is
proved for this problem. A modified Tikhonov regularization method is proposed to
recover the stability of the solution. An order optimal error estimate between the
approximate solution and the exact solution is obtained with a suitable choice of
regularization parameter. Numerical results are presented to illustrate the accuracy
and efficiency of the proposed method.
MSC: 65M30; 35R25; 35R30
Keywords: ill-posed problem; inverse heat conduction problem; regularization; error
estimate
1 Introduction
In this paper we consider the following inverse heat conduction problem with only boundary value:
ut = uxx ,
< x < , < t < π,
u(, t) = f (t),
≤ t ≤ π,
ux (, t) = g(t),
≤ t ≤ π,
(.)
where f and g are given. This problem is ill posed []. We want to recover the temperature
distribution u(x, ·) for < x ≤ from the boundary data f and g.
The inverse heat conduction problem (IHCP) arises from many physical and engineering disciplines. It is well known that the problem is severely ill posed in the Hadamard
sense that the solution (if it exists) does not depend continuously on the given data, i.e.,
a small measurement error in the given data can cause an enormous error in the solution [–]. To overcome such difficulties, some regularization techniques are required [].
The IHCP has been considered by many authors using different methods. These methods
include the wavelet and wavelet-Galerkin method [–], the Tikhonov method [], the
© 2016 Cheng and Ma. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Cheng and Ma Boundary Value Problems (2016) 2016:100
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mollification method [–], the fundamental solution method [], the Fourier method
[], and so on.
To the best of the knowledge of the authors, the results available in the literature are
mainly devoted to the IHCP with known initial-boundary value. However, in practical
real-life problems we cannot know the initial condition because the heat process has already started before we estimate the problem. A few works are developed for the IHCP
without initial value [, ]. Ginsberg [] used a cutoff method for an IHCP with only
boundary value and gave a Hölder type error estimate. Recently, Liu and Wei [] used a
quasi-reversibility regularization method for solving an IHCP without initial data. Yang
and Fu [] applied a simplified Tikhonov regularization method for determining the heat
source. In this paper, we will use a modified Tikhonov regularization method to deal with
the IHCP without initial value (.) and obtain an order optimal error estimate between
the approximate solution and the exact solution.
The paper is organized as follows. In Section , we give the formulation of the solution for problem (.) and present some preliminary results. In Section , we prove the
conditional stability for the IHCP (.) by using the Hölder inequality. Section proposes
a modified Tikhonov regularization method. An order optimal error estimate for the approximate solution is obtained with a suitable choice of regularization parameter. To verify
the efficiency and accuracy of the proposed method for problem (.), we give two numerical examples in Section . A brief conclusion is given in Section .
2 Mathematical formulation and preliminaries
Throughout this paper, we use the following formulation and lemmas. For the IHCP (.),
we want to determine the temperature distribution u(x, ·) for < x ≤ from the Cauchy
data f and g. Since the Cauchy data f and g are measured, there will be measurement
errors, and we would actually have measured Cauchy data f δ , g δ ∈ L [, π], for which
f – f δ ≤ δ,
g – g δ ≤ δ,
(.)
where the constant δ > represents a bound on the measurement error, · and (·, ·)
denote the norm and inner product on L [, π], respectively.
In the following, we split the IHCP (.) into two independent IHCPs:
vt = vxx ,
< x < , < t < π,
v(, t) = f (t),
≤ t ≤ π,
vx (, t) = ,
≤ t ≤ π,
(.)
and
wt = wxx ,
w(, t) = ,
< x < , < t < π,
≤ t ≤ π,
wx (, t) = g(t),
(.)
≤ t ≤ π.
Let v(x, t) and w(x, t) be the solution of problems (.) and (.), respectively. Then u =
v + w is the solution of problem (.). Therefore, we only need solve problems (.) and
(.), respectively.
Cheng and Ma Boundary Value Problems (2016) 2016:100
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By the method of separation of variables, the exact solutions of problems (.) and (.)
are given by
+∞
v(x, t) =
√
f (t), eint eint cosh( inx)
(.)
n=–∞
and
w(x, t) =
+∞
√
√ g(t), eint eint sinh( inx).
in
n=–∞
(.)
Then the exact solution of problem (.) is given by
u(x, t) =
+∞
√
√
(g(t), eint ) int
f (t), eint eint cosh( inx) + √
e sinh( inx) .
in
n=–∞
(.)
We assume also that there exists an a priori condition for problem (.):
max v(, ·)p , w(, ·)p ≤ E,
p ≥ ,
(.)
p/
in(·) in(·)
where v(, ·)p = +∞
)e .
n=–∞ ( + n ) (v(, ·), e
In order to give an error estimate for the regularized solution, we need the following
lemma whose proof is similar to that of Lemma . in [].
Lemma . Let < x ≤ , < α < /e
√
. We have the following inequalities:
exs
≤ α –x ,
s
s≥ + α e
(.)
sup
p
– p
e(+x)s ( + s )–
≤ α –(+x) – ln(α) p+ .
s
+α e
s≥
(.)
sup
We need also the following results.
Lemma . Let < x ≤ , then there holds []:
√
sinh( inx)
= x,
√
n→o
in
√
|n|
cosh( inx) ≤ e x ,
√
√
|n|
sinh( inx)
≤ xe x , n ∈ Z,
√
in
√
|n|
sinh( inx) ≤ e x , n ∈ Z,
√
cosh( in) ≥ ce
√
sinh( in) ≥ ce
lim
|n|
,
|n|
,
|n| ∈ N+ ,
(.)
(.)
(.)
√
where c = ( – e– )/.
3 Conditional stability
In this section, we will provide the conditional stabilities for problems (.), (.), and (.),
respectively.
Cheng and Ma Boundary Value Problems (2016) 2016:100
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Theorem (...truncated)
