Recovery the interior temperature of a nonhomogeneous elliptic equation from boundary data
Nguyen and Tran Journal of Inequalities and Applications 2014, 2014:19
http://www.journalofinequalitiesandapplications.com/content/2014/1/19
RESEARCH
Open Access
Recovery the interior temperature of a
nonhomogeneous elliptic equation from
boundary data
Tuan H Nguyen1* and Binh Thanh Tran2
*
Correspondence:
1
Faculty of Mathematics and
Statistics, Ton Duc Thang University,
No. 19 Nguyen Huu Tho Street, Tan
Phong Ward, District 7, Ho Chi Minh,
Vietnam
Full list of author information is
available at the end of the article
Abstract
We consider the problem of finding a function u from the boundary data u(x, 1) and
uy (x, 1), satisfying a nonhomogeneous elliptic equation
u = f (x, y),
x ∈ R, 0 < y < 1.
The problem is shown to be ill-posed. In this paper, we apply the Fourier transform to
get an integral equation and give a regularized solution by directly perturbing this
equation in combination with truncating high frequencies. The error estimate
between the regularization solution and the exact solution is established. Finally, we
present a numerical result which shows the effectiveness of the proposed method.
MSC: 31A25; 34K29; 35J05; 35J25; 35J99; 42A38; 44A35
Keywords: Fourier transform; ill-posed problem; quasi-boundary value methods;
truncation method
1 Introduction
In this paper, we consider a problem of recovering the interior temperature from surface
data (or boundary data). In fact, the interior temperature of a body (e.g., the skin of a
missile) cannot be determined in several engineering contexts (see, e.g., [–]) and many
industrial applications. Hence, in order to get the distribution of interior temperature, we
have to use the measured temperature outside the surface. In optoelectronics, the determination of a radiation field surrounding a source of radiation (e.g., a light emitting diode)
is a frequently occurring problem. As a rule, experimental determination of the whole radiation field is not possible. Practically, we are able to measure the electromagnetic field
only on some subset of physical space (e.g., on some surfaces). So, the problem arises how
to reconstruct the radiation field from such experimental data (see, for instance, []). In
the paper of Reginska [], the authors considered a physical problem which is connected
with the notion of light beams. Some applications of this model can be established in more
detail in [].
Precisely, we consider a two-dimensional body represented by the domain R × (, ). Let
u(x, y) be the temperature of the body at (x, y) ∈ R × (, ), and let f ≡ f (x, y) be a given
source, we have the following nonhomogeneous equation:
u = f (x, y),
x ∈ R, < y < ,
()
©2014 Nguyen and Tran; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Nguyen and Tran Journal of Inequalities and Applications 2014, 2014:19
http://www.journalofinequalitiesandapplications.com/content/2014/1/19
where =
∂
∂x
+
∂
.
∂y
Page 2 of 13
We assume that the temperature on the line y = is known, i.e.,
u(x, ) = ϕ(x),
()
and that
∂u
(x, ) = ψ(x),
∂y
()
where ϕ(x), ψ(x) are given functions in L (R). The problem can be referred to as a sideways
elliptic problem and the interior measurement ϕ(x) is also called (in geology) the borehole
measurement.
The latter problem is a Cauchy elliptic problem in an infinite strip and is well known as
an ill-posed problem, i.e., solutions of the problem do not always exist and, whenever they
do exist, there is no continuous dependence on the given data. This makes the numerical
computations become difficult. So, ill-posed problems need to be regularized.
The homogeneous problem (f ≡ ) was studied with various methods in many papers.
Using the boundary element method, the homogeneous problems were considered in [,
, ] etc. Similarly, many methods have been investigated to solve the Cauchy problem
for a linear homogeneous elliptic equation such as the method of successive iterations
[, ], the optimization method [, ], the quasi-reversibility method [–], fourthorder modified method [, ], Fourier truncation regularized (or spectral regularized
method) [–], etc. The number of papers devoted to the Cauchy problem for linear
homogeneous elliptic equation are very rich, for example, [, –] and the references
therein.
Although there are many papers on homogeneous cases, we only find a few papers on
nonhomogeneous sideways problems (for both parabolic and elliptic equations). The main
aim of this paper is to present a simple and effective regularization method, and investigate
the error estimate between the regularization solution and the exact solution. In a sense,
this paper is an extension of recent results in [, –, , ].
The paper is organized as follows. In Section , we present the formulation of the Cauchy
problem for the elliptic equation and propose a modified regularization method. The error
estimate is given based on two different a priori assumptions for the exact solution. Finally,
in Section , we give a numerical example to demonstrate the effectiveness of our proposed
method.
2 Regularization and error estimate
+∞
Let
f (ξ ) = √π –∞ f (x)e–iξ x dx be the Fourier transform of function f ∈ L (R). By taking
Fourier transformation with respect to variable x ∈ R, we transform problem ()-() to the
following form:
ϕ (ξ ) e(–y)|ξ | + e(y–)|ξ |
u(ξ , y) =
(ξ ) e(y–)|ξ | – e(–y)|ξ |
ψ
+
|ξ |
(η–y)|ξ | –|y–η||ξ |
f (ξ , η) dη
+
e
–e
y |ξ |
Nguyen and Tran Journal of Inequalities and Applications 2014, 2014:19
http://www.journalofinequalitiesandapplications.com/content/2014/1/19
Page 3 of 13
– e(–y)|ξ | (–y)|ξ |
(ξ )
ψ
ϕ (ξ ) e(–y)|ξ | + e(y–)|ξ | +
=
e
|ξ |e(–y)|ξ |
(η–y)|ξ |
e
– (η–y)|ξ |
f (ξ , η) dη.
e
+
|ξ |e(η–y)|ξ |
y
()
In the present paper, by approximating (), we have a regularized solution u (x, y), the
Fourier transform of which is
u (ξ , y) =
ϕ (ξ )
+ e(y–)|ξ |
α( ) + e(y–)|ξ |
(ξ )
– e(–y)|ξ |
ψ
(–y)|ξ
|
|ξ |e
α( ) + e(y–)|ξ |
+
+
y
e(η–y)|ξ | –
f (ξ , η)
dη χ[–β(
|ξ |e(η–y)|ξ | α( ) + e(y–η)|ξ |
),β( )] (ξ )
()
iξ x
dξ .
),β( )] (ξ )e
()
or
u (x, y) = √
π
+
+∞
–∞
ϕ (ξ )
+ e(y–)|ξ |
α( ) + e(y–)|ξ |
(ξ )
ψ
– e(–y)|ξ |
|ξ |e(–y)|ξ | α( ) + e(y–)|ξ |
e(η–y)|ξ | –
f (ξ , η)
dη
χ[–β(
|ξ |e(η–y)|ξ | α( ) + e(y–η)|ξ |
+
y
Here, α( ) and β( ) are positive numbers (called regularization parameters) which depend
on . They will be chosen later such that α( ) ∈ (, ) and β( ) → +∞ when
→ . For
convenience, from now on, we denote α( ) by α, and β( ) by β.
In practice, the exact data (ϕex , ψex ) ∈ L (R) × L (R) is given only by mea (...truncated)
