Recovery the interior temperature of a nonhomogeneous elliptic equation from boundary data
© Nguyen and Tran; licensee Springer. 2014
Received: 16 August 2013
Accepted: 10 December 2013
Published: 14 January 2014
We consider the problem of finding a function u from the boundary data and , satisfying a nonhomogeneous elliptic equation
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.
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., [1–5]) 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 .
where , are given functions in . The problem can be referred to as a sideways elliptic problem and the interior measurement 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 () was studied with various methods in many papers. Using the boundary element method, the homogeneous problems were considered in [2, 7, 8]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 [8, 9], the optimization method [10, 11], the quasi-reversibility method [12–14], fourth-order modified method [15, 16], Fourier truncation regularized (or spectral regularized method) [17–19], etc. The number of papers devoted to the Cauchy problem for linear homogeneous elliptic equation are very rich, for example, [7, 20–23] 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 [4, 9–11, 22, 24].
The paper is organized as follows. In Section 2, 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 3, we give a numerical example to demonstrate the effectiveness of our proposed method.
2 Regularization and error estimate
Here, and are positive numbers (called regularization parameters) which depend on ϵ. They will be chosen later such that and when . For convenience, from now on, we denote by α, and by β.
We first have the following lemma.
Lemma 1 (The stability of a solution of problem (5))
for all .
This completes the proof of Lemma 1. □
This completes the proof of Theorem 1. □
Remark Theorem 1 gives a good approximation not only in the case but also in the case .
Let and , we have Theorem 1.
3 Numerical experiment
Let , be the disturbed measure data such that , .
In the numerical experiment, we always fix the interval .
Here, regularized solutions are calculated by the first regularization solution () with , and the second regularization () with , .
Error estimations for the first regularization solution ( ) and the second regularization solution ( )
ϵ = 10−1
ϵ = 10−3
ϵ = 10−5
ϵ = 10−7
Relative error estimations for the first method ( ) and second regularization solution ( )
ϵ = 10−1
ϵ = 10−3
ϵ = 10−5
ϵ = 10−7
This project was supported by Ton Duc Thang University (FOSTECH). The authors would like to thank the anonymous referees for their valuable suggestions and comments leading to the improvement of our manuscript. We would like to thank Truong Trong Nghia in School of Computing, the University of Utah, USA for his most helpful comments on numerical results.
- Beskos DE: Boundary element method in dynamic analysis: Part II (1986-1996). Appl. Mech. Rev. 1997, 50: 149-197. 10.1115/1.3101695View ArticleGoogle Scholar
- Chen JT, Wong FC: Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J. Sound Vib. 1998., 217: Article ID 7595Google Scholar
- Harari I, Barbone PE, Slavutin M, Shalom R: Boundary infinite elements for the Helmholtz equation in exterior domains. Int. J. Numer. Methods Eng. 1998, 41: 1105-1131. 10.1002/(SICI)1097-0207(19980330)41:6<1105::AID-NME327>3.0.CO;2-0View ArticleGoogle Scholar
- Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X: Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations. Comput. Mech. 2003,31(3-4):367-377.MathSciNetGoogle Scholar
- Marin L, Lesnic D: The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput. Struct. 2007,83(4-5):267-278.MathSciNetView ArticleGoogle Scholar
- Reginska T, Tautenhahn U: Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation. Numer. Funct. Anal. Optim. 2009, 30: 1065-1097. 10.1080/01630560903393170MathSciNetView ArticleGoogle Scholar
- Cheng J, Hon YC, Wei T, Yamamoto M: Numerical computation of a Cauchy problem for Laplace equation. Z. Angew. Math. Mech. 2001, 81: 665-674.MathSciNetView ArticleGoogle Scholar
- Denisov AM, Zakharov EV, Kalinin AV, Kalinin VV: Numerical methods for some inverse problems of heart electrophysiology. Differ. Equ. 2009,45(7):1034-1043. 10.1134/S0012266109070106MathSciNetView ArticleGoogle Scholar
- Kozlov VA, Mazya VG, Fomin AV: An iterative method for solving the Cauchy problem for elliptic equations. Zh. Vychisl. Mat. Mat. Fiz. 1991,31(1):64-74.MathSciNetGoogle Scholar
- Hao DN, Lesnic D: The Cauchy problem for Laplaces equation via the conjugate gradient method. IMA J. Appl. Math. 2000, 65: 199-217. 10.1093/imamat/65.2.199MathSciNetView ArticleGoogle Scholar
- Kabanikhin SI, Karchevsky AL: Optimizational method for solving the Cauchy problem for an elliptic equation. J. Inverse Ill-Posed Probl. 1995,3(1):21-46.MathSciNetGoogle Scholar
- Qian Z, Fu CL, Li ZP: Two regularization methods for a Cauchy problem for the Laplace equation. J. Math. Anal. Appl. 2008,338(1):479-489. 10.1016/j.jmaa.2007.05.040MathSciNetView ArticleGoogle Scholar
- Qian A, Xiong X-T, Wu YJ: On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation. J. Comput. Appl. Math. 2010,233(8):1969-1979. 10.1016/j.cam.2009.09.031MathSciNetView ArticleGoogle Scholar
- Tuan NH, Trong DD, Quan PH: A new regularization method for a class of ill-posed Cauchy problems. Sarajevo J. Math. 2010,6(19)(2):189-201.Google Scholar
- Qian Z, Fu CL, Xiong XT: Fourth-order modified method for the Cauchy problem for the Laplace equation. J. Comput. Appl. Math. 2006,192(2):205-218. 10.1016/j.cam.2005.04.031MathSciNetView ArticleGoogle Scholar
- Shi R, Wei T, Qin HH: A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation. Numer. Math., Theory Methods Appl. 2009, 2: 326-340.MathSciNetGoogle Scholar
- Fu CL, Feng XL, Qian Z: The Fourier regularization for solving the Cauchy problem for the Helmholtz equation. Appl. Numer. Math. 2009,59(10):2625-2640. 10.1016/j.apnum.2009.05.014MathSciNetView ArticleGoogle Scholar
- Qian A, Mao J, Liu L: A spectral regularization method for a Cauchy problem of the modified Helmholtz equation. Bound. Value Probl. 2010., 2010: Article ID 212056Google Scholar
- Tuan NH, Trong DD, Quan PH: A note on a Cauchy problem for the Laplace equation: regularization and error estimates. Appl. Math. Comput. 2010,217(7):2913-2922. 10.1016/j.amc.2010.09.019MathSciNetView ArticleGoogle Scholar
- Berntsson F, Eldren L: Numerical solution of a Cauchy problem for the Laplace equation. Inverse Probl. 2001, 17: 839-853. 10.1088/0266-5611/17/4/316View ArticleGoogle Scholar
- Chakib A, Nachaoui A: Convergence analysis for finite-element approximation to an inverse Cauchy problem. Inverse Probl. 2006., 22: Article ID 1191206Google Scholar
- Klibanov MV, Santosa F: A computational quasi-reversibility method for Cauchy problems for Laplaces equation. SIAM J. Appl. Math. 1991, 51: 1653-1675. 10.1137/0151085MathSciNetView ArticleGoogle Scholar
- Takeuchi T, Imai H: Direct numerical simulations of Cauchy problems for the Laplace operator. Adv. Math. Sci. Appl. 2003, 13: 587-609.MathSciNetGoogle Scholar
- Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X: BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method. Eng. Anal. Bound. Elem. 2004,28(9):1025-1034. 10.1016/j.enganabound.2004.03.001View ArticleGoogle Scholar
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