- Open Access
Identification of the pollution source of a parabolic equation with the time-dependent heat conduction
© Tuan et al.; licensee Springer. 2014
Received: 30 January 2014
Accepted: 25 March 2014
Published: 6 May 2014
We consider the problem of identifying the pollution source of a 1D parabolic equation from the initial and the final data. The problem is ill posed and regularization is in order. Using the quasi-boundary method and the truncation Fourier method, we present two regularization methods. Error estimates are given and the methods are illustrated by numerical experiments.
and the boundary condition . To consider a more general case, we will replace D in (2) by a given function which is defined later.
In the simplest case, one reduces this approximation to its first term , where the function φ is given. Source terms of this form frequently appear, for example, as a control term for the parabolic equation.
In another context, this problem is called the identification of heat source; it has received considerable attention from many researchers in a variety of fields using different methods since 1970. If the pollute source has the form of , the inverse source problem was studied in . In , the authors considered the heat source as a function of both space and time variables, in the additive or separable forms. Many researchers viewed the source as a function of space or time only. In [4, 5], the authors determined the heat source dependent on one variable in a bounded domain by the boundary-element method and the iterative algorithm. In , the authors investigated the heat source which is time-dependent only by the method of a fundamental solution.
and u satisfies the condition (3). This kind of equation (5) has many applications in groundwater pollution. It is a simple form of advection-convection, which appears in groundwater pollution source identification problems (see ). Such a model is related to the detection of the pollution source causing water contamination in some region.
The remainder of the paper is divided into three sections. In Section 2, we apply the quasi-boundary value method and truncation method to solve the problem (2)-(3). Then we also estimate the error between an exact solution and the regularization solution with the logarithmic order and Hölder order. Finally, some numerical experiments will be given in Section 3.
2 Identification and regularization for inhomogeneous source depending on time variable
and , , , where the constant ϵ represents a noise level and .
We continue to estimate the term .
2.1 Regularization by a quasi-boundary value method
Now we will give an error estimate between the regularization solution and the exact solution by the following theorem.
We divide the proof into three steps.
Remark 1 If we choose , , then (21) holds.
Remark 2 In this theorem, with the assumption , we have an error of logarithmic order. In the next section, we introduce a truncation method which improves the order of the error. We present the error of Hölder estimates (the order is , ) with a weaker assumption of f, i.e., .
2.2 Regularization by a truncation method
where . □
3 Numerical results
with , a discretization of function f, .
where , with , works as the amplitude of noise.
and we have convergence to zero.
The error estimation between exact solution and regularized solution by quasi-reversibility method
3.33236 × 10−1
4.82402 × 10−2
9.20728 × 10−4
1.11864 × 10−5
The error estimation between exact solution and regularized solution by truncation method
1.74326 × 10−2
This research is funded by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST HCMC) under the project name ‘Inverse parabolic equation and application to groundwater pollution source’.
- Atmadja J, Bagtzoglou AC: Marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: application to contaminant plume spatial distribution recovery. Water Resour. Res. 2003, 39: 1038-1047.Google Scholar
- Cannon JR, Duchateau P: Structural identification of an unknown source term in a heat equation. Inverse Probl. 1998, 14: 535-551. 10.1088/0266-5611/14/3/010MathSciNetView ArticleMATHGoogle Scholar
- Savateev EG: On problems of determining the source function in a parabolic equation. J. Inverse Ill-Posed Probl. 1995, 3: 83-102.MathSciNetView ArticleMATHGoogle Scholar
- Farcas A, Lesnic D: The boundary-element method for the determination of a heat source dependent on one variable. J. Eng. Math. 2006, 54: 375-388. 10.1007/s10665-005-9023-0MathSciNetView ArticleMATHGoogle Scholar
- Johansson T, Lesnic D: Determination of a spacewise dependent heat source. J. Comput. Appl. Math. 2007, 209: 66-80. 10.1016/j.cam.2006.10.026MathSciNetView ArticleMATHGoogle Scholar
- Yan L, Fu C-L, Yang F-L: The method of fundamental solutions for the inverse heat source problem. Eng. Anal. Bound. Elem. 2008, 32: 216-222. 10.1016/j.enganabound.2007.08.002View ArticleMATHGoogle Scholar
- Yang F, Fu C-L: Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation. J. Inverse Ill-Posed Probl. 2009,17(8):815-830.MathSciNetView ArticleMATHGoogle Scholar
- Cheng W, Fu C-L: Identifying an unknown source term in a spherically symmetric parabolic equation. Appl. Math. Lett. 2013, 26: 387-391. 10.1016/j.aml.2012.10.009MathSciNetView ArticleMATHGoogle Scholar
- Yang F, Fu C-L: A simplified Tikhonov regularization method for determining the heat source. Appl. Math. Model. 2010, 34: 3286-3299. 10.1016/j.apm.2010.02.020MathSciNetView ArticleMATHGoogle Scholar
- Yang F, Fu C-L: A mollification regularization method for the inverse spatial-dependent heat source problem. J. Comput. Appl. Math. 2014, 255: 555-567.MathSciNetView ArticleMATHGoogle Scholar
- Trong DD, Tuan NH: A nonhomogeneous backward heat problem: regularization and error estimates. Electron. J. Differ. Equ. 2008., 2008: Article ID 33Google Scholar
- Trong DD, Quan PH, Alain PND: Determination of a two dimensional heat source: uniqueness, regularization and error estimate. J. Comput. Appl. Math. 2006, 191: 50-67. 10.1016/j.cam.2005.04.022MathSciNetView ArticleMATHGoogle Scholar
- Qian A, Li Y: Optimal error bound and generalized Tikhonov regularization for identifying an unknown source in the heat equation. J. Math. Chem. 2011,49(3):765-775. 10.1007/s10910-010-9774-3MathSciNetView ArticleMATHGoogle Scholar
- Hasanov A: Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time. Int. J. Heat Mass Transf. 2012, 55: 2069-2080. 10.1016/j.ijheatmasstransfer.2011.12.009View ArticleGoogle Scholar
- Denche M, Bessila K: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 2005, 301: 419-426. 10.1016/j.jmaa.2004.08.001MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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 credited.