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’.
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