Research Article | Open Access
Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements
We consider an inverse problem for simultaneously determining the space-dependent source and the initial distribution in heat conduction equation. First, we study the ill-posedness of the inverse problem. Then, we construct a regularization problem to approximate the originally inverse problem and obtain the regularization solutions with their stability and convergence results. Furthermore, convergence rates of the regularized solutions are presented under a prior and a posteriori strategies for selecting regularization parameters. Results of numerical examples show that the proposed regularization method is stable and effective for the considered inverse problem.
In the past decades, various classes of inverse heat conduction equation problems have been studied by many scholars including recovery of the initial temperature [1–5], reconstruction of the heat source [6–12], and identification of thermal diffusion coefficients [13, 14]. The inverse problems of heat equations such as the backward problems and the source reconstruction problems arise from various scientific and engineering fields, including heat conduction, hydrology, environmental controlling. It is worth noting that most of the existing literature considers recovery of only one unknown term or parameter. However, in many applications we hope to simultaneously reconstruct more than one unknown term from some overspecified conditions, which makes inverse problems very complicated. To the authors’ knowledge, papers devoted to the simultaneous recovery problems are very limited. In , a numerical algorithm based on the fundamental solutions method is proposed to reconstruct the space-dependent heat source and the initial value simultaneously in an inverse heat conduction problem, which is transformed into a homogeneous backward problem and a Dirichlet boundary value problem for Poisson’s equation. In , an iterative algorithm is proposed for reconstructing both the space-dependent source term and the initial value based on solving a sequence of well-posed direct problems for the heat equation. In [17, 18], the unknown initial temperature and heat source are reconstructed simultaneously from the temperature data at the final time and at a fixed internal location over the time interval. In , the identification of the space-dependent heat source and the heat flux at the left endpoint is studied by the Tikhonov regularization method with generalized cross validation criterion for one-dimensional inverse heat conduction problem. In , the authors studied the inverse problem of reconstructing the time- and space-dependent heat source and the Robin boundary condition from the measured final data. In , the authors considered an inverse problem to simultaneously reconstruct the time- and space-dependent heat source and the initial temperature distribution and established the conditional stability and uniqueness of the inverse problem, which is solved by the variational regularization method. In , the inverse problem of simultaneous determination of the time-dependent source term and the time-dependent coefficients in the heat equation is studied by the overspecified conditions of integral type. Recently, there has been a growing interest in inverse problems with fractional derivatives. In , the authors studied the inverse problem of the time-fractional diffusion equation in one-dimensional spatial space for determining the initial value and the heat flux on the boundary simultaneously and proved the uniqueness of the inverse problem by using the Laplace transform and the unique extension technique.
Motivated by the idea of [3, 5] for solving the backward problem of heat equation, we consider the inverse problem of heat equation to simultaneously determine the space-dependent source and the initial distribution from two final temperature measurements at two terminal times. This paper is organized as follows. The inverse problem is formulated in Section 2 with its ill-posedness. A regularization approximation problem is constructed to approximate the inverse problem, and regularized solutions are obtained with their stability and convergence rates in Section 3. Numerical examples are given in Section 4 to show the feasibility and validity of the proposed method. Finally, some conclusions are drawn in Section 5.
2. Problem Formulation and Its Ill-Posedness
2.1. Problem Formulation
Consider the initial-boundary problem of a nonhomogeneous heat conduction equation:where the initial distribution satisfies the compatibility condition: . As we all know, the direct problem is solving problem (1) for yielding when the source and the initial distribution are known. However, the inverse problem considered in this paper is to reconstruct and from problem (1) and two additional final measurementsandwhere .
By the method of separation of variables, we obtain the solution to problem (1):where and are Fourier coefficients of and , respectively. And they are defined by From the Fourier expansions of and with respect to the eigenfunctions , it follows thatwhere By simple calculations, from system (6) we obtain that Thus, the solution to the inverse problem is obtained in the following form:Hence, the inverse problem has a unique solution in if and only if the Fourier coefficients of and satisfy thatandIn other words, and must decay faster than as for guaranteeing the existence of the inverse solution.
Unfortunately, the two additional measurements are always contaminated by noise in practical applications. In the context, we denote the contaminated measurements of and by and , which satisfy thatHere, represents the -norm on the interval , and is the noise level. Although and belong to the space , the Fourier coefficients and no longer satisfy inequalities (11) and (12) since the random noise does not decay and will be dramatically amplified by and . Therefore, the inverse problem is ill-posed for determining the heat source and the initial distribution simultaneously in .
3. Regularization Method
First, we construct a regularization approximation problem for simultaneously determining the heat source term and the initial distribution:and satisfies thatwhere is a regularization parameter. The superscript is used to denote the dependency of the approximate solution on the regularization parameter. By using the method of separation of variables, we can easily derive the solution to the approximate problem (14):where
We firstly prove the posedness of problem (14)–(16). Since we have obtained the solution to problem (14)–(16), the existence and uniqueness of the solution are obvious. Therefore, we only need to prove the stability of the solution.
Before we prove the main theoretical results, we introduce a lemma.
Lemma 1. For , , and , the following inequalities are valid: (a).(b).(c)
Proof. (a) We define the function by the formula By simple computations, the function gets the maximum at the point ; that is(b)(c) Set . After a simple calculation we can say , so the conclusion is true.
Theorem 2. Let and satisfy (13), and thenandi.e., regularized solutions and are consecutively dependent on the final value data and , where and are regularized solutions corresponding to and , respectively.
Theorem 3. Let , and . and are the true solutions of the inverse problem (1). If there exists a constant such that then
Theorem 3 illustrates that and are really regularized solutions, since and approximate the true solutions and as , respectively.
Theorem 4 gives a priori strategy for selecting regularization parameter. However, it is often undesirable to have the value of constant in practical applications. It is well known that regularization parameter plays a very important role in solving ill-posed problems, and the effectiveness of a regularization method depends strongly on the choice of the regularization parameter. Here, we adopt the Morozov discrepancy principle [24–26], a posteriori strategy, to select the regularization parameter; that is, selecting the regularization parameter makeswhere is a given constant. For regularization solutions and , the deviation equation (46) means that
Theorem 5. Let and be the true solutions of (1)–(3) with their Fourier coefficients and , respectively. If there exist constants and such that and , the a posteriori strategy (47) for selecting the regularization parameter leads to the following estimates:where and are regularization solutions with respect to a regularization parameter selected by the a posteriori strategy (47).
Proof. (1) By careful computations and the Hölder inequality, we can get From (13), we can get and In summary, (2) Based on the Hölder inequality, we can similarly obtain that From inequalities (13), we get