# Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

Authors: Shufang Qiu; Wen Zhang; Jianmei Peng
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

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 [15], 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 [16], 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 [19], 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 [20], 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 [21], 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 [22], 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 [23], 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:(1)ut-uxx=fx,x,t∈0,π×0,T,u0,t=uπ,t=0,t∈0,T,ux,0=φx,x∈0,π,where the initial distribution φ(x) satisfies the compatibility condition: φ(0)=φ(π)=0. As we all know, the direct problem is solving problem (1) for yielding u(x,t) when the source f(x) and the initial distribution φ(x) are known. However, the inverse problem considered in this paper is to reconstruct f(x) and φ(x) from problem (1) and two additional final measurements(2)ux,T1=gx,x∈0,π,and(3)ux,T2=hx,x∈0,π,where T1<T2.


### 2.2. Ill-Posedness

By the method of separation of variables, we obtain the solution to problem (1):(4)ux,t=∑k=1∞e-k2tφk+1-e-k2tk2fksin⁡kx,where φk and fk are Fourier coefficients of φ(x) and f(x), respectively. And they are defined by (5)φk=2π∫0πφxsin⁡kxdx,fk=2π∫0πfxsin⁡kxdx. From the Fourier expansions of g(x) and h(x) with respect to the eigenfunctions sin⁡(kx), it follows that(6)e-k2T1φk+1-e-k2T1k2fk=gk,e-k2T2φk+1-e-k2T2k2fk=hk,where (7)gk=2π∫0πgxsin⁡kxdx,hk=2π∫0πhxsin⁡kxdx. By simple calculations, from system (6) we obtain that (8)fk=k2hk-gke-k2T2-T11-e-k2T2-T1,φk=ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-1. Thus, the solution to the inverse problem is obtained in the following form:(9)fx=∑k=1∞k2hk-gke-k2T2-T11-e-k2T2-T1sin⁡kx,(10)φx=∑k=1∞ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-1sin⁡kx.Hence, the inverse problem has a unique solution (f(x),φ(x)) in L2(0,π) if and only if the Fourier coefficients of g(x) and h(x) satisfy that(11)∑k=1∞k2hk-gke-k2T2-T11-e-k2T2-T12<∞and(12)∑k=1∞ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-12<∞.In other words, gk and hk must decay faster than e-k2T1 as k→+∞ 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 ofg(x) and h(x) by gδ(x) and hδ(x), which satisfy that(13)gδx-gx≤δ,hδx-hx≤δ.Here, ∥·∥ represents the L2-norm on the interval (0,π), and δ is the noise level. Although gδ(x) and hδ(x) belong to the space L2(0,π), the Fourier coefficients {gkδ} and {hkδ} no longer satisfy inequalities (11) and (12) since the random noise does not decay and will be dramatically amplified by k2 and ek2T1. Therefore, the inverse problem is ill-posed for determining the heat source and the initial distribution simultaneously in L2(0,π).


## 2.1. Problem Formulation

Consider the initial-boundary problem of a nonhomogeneous heat conduction equation:(1)ut-uxx=fx,x,t∈0,π×0,T,u0,t=uπ,t=0,t∈0,T,ux,0=φx,x∈0,π,where the initial distribution φ(x) satisfies the compatibility condition: φ(0)=φ(π)=0. As we all know, the direct problem is solving problem (1) for yielding u(x,t) when the source f(x) and the initial distribution φ(x) are known. However, the inverse problem considered in this paper is to reconstruct f(x) and φ(x) from problem (1) and two additional final measurements(2)ux,T1=gx,x∈0,π,and(3)ux,T2=hx,x∈0,π,where T1<T2.


## 2.2. Ill-Posedness

By the method of separation of variables, we obtain the solution to problem (1):(4)ux,t=∑k=1∞e-k2tφk+1-e-k2tk2fksin⁡kx,where φk and fk are Fourier coefficients of φ(x) and f(x), respectively. And they are defined by (5)φk=2π∫0πφxsin⁡kxdx,fk=2π∫0πfxsin⁡kxdx. From the Fourier expansions of g(x) and h(x) with respect to the eigenfunctions sin⁡(kx), it follows that(6)e-k2T1φk+1-e-k2T1k2fk=gk,e-k2T2φk+1-e-k2T2k2fk=hk,where (7)gk=2π∫0πgxsin⁡kxdx,hk=2π∫0πhxsin⁡kxdx. By simple calculations, from system (6) we obtain that (8)fk=k2hk-gke-k2T2-T11-e-k2T2-T1,φk=ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-1. Thus, the solution to the inverse problem is obtained in the following form:(9)fx=∑k=1∞k2hk-gke-k2T2-T11-e-k2T2-T1sin⁡kx,(10)φx=∑k=1∞ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-1sin⁡kx.Hence, the inverse problem has a unique solution (f(x),φ(x)) in L2(0,π) if and only if the Fourier coefficients of g(x) and h(x) satisfy that(11)∑k=1∞k2hk-gke-k2T2-T11-e-k2T2-T12<∞and(12)∑k=1∞ek2T1hk1-e-k2T1-gk1-e-k2T2e-k2T2-T1-12<∞.In other words, gk and hk must decay faster than e-k2T1 as k→+∞ 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 ofg(x) and h(x) by gδ(x) and hδ(x), which satisfy that(13)gδx-gx≤δ,hδx-hx≤δ.Here, ∥·∥ represents the L2-norm on the interval (0,π), and δ is the noise level. Although gδ(x) and hδ(x) belong to the space L2(0,π), the Fourier coefficients {gkδ} and {hkδ} no longer satisfy inequalities (11) and (12) since the random noise does not decay and will be dramatically amplified by k2 and ek2T1. Therefore, the inverse problem is ill-posed for determining the heat source and the initial distribution simultaneously in L2(0,π).


## 3. Regularization Method

First, we construct a regularization approximation problem for simultaneously determining the heat source term and the initial distribution:(14)utα-uxxα=fαx,x,t∈0,π×0,T,uα0,t=uαπ,t=0,t∈0,T,uαx,0=φαx,x∈0,π,and uα(x,t) satisfies that(15)uαx,T1=∑k=1∞e-k2T1αek2+e-k2T1gksin⁡kx,x∈0,π,(16)uαx,T2=∑k=1∞e-k2T2αek2+e-k2T2hksin⁡kx,x∈0,π,where α∈(0,+∞) is a regularization parameter. The superscript α 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):(17)uαx,t=∑k=1∞1-e-k2tk2fkα+φkαe-k2tsin⁡kx,where (18)fkα=2π∫0πfαxsin⁡kxdx,φkα=2π∫0πφαxsin⁡kxdx.From (15) and (16), we have(19)1-e-k2T1k2fkα+φkαe-k2T1=e-k2T1αek2+e-k2T1gk,1-e-k2T2k2fkα+φkαe-k2T2=e-k2T2αek2+e-k2T2hk.Thus, the approximate solutions of the inverse problem are(20)fαx=∑k=1∞k2e-k2T2/αek2+e-k2T2hk-e-k2T1/αek2+e-k2T1gke-k2T2-T11-e-k2T2-T1sin⁡kx,(21)φαx=∑k=1∞ek2T1e-k2T2/αek2+e-k2T2hk1-e-k2T1-e-k2T1/αek2+e-k2T1gk1-e-k2T2e-k2T2-T1-1sin⁡kx.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.
ForT>0,  0<T1<T2, and  α∈(0,+∞), the following inequalities are valid: (a)
1/αeλ+e-Tλ≤T/α1+T(T/α)1/1+T,λ∈(-∞,+∞).(b)
λ/eλT2-eλT1≤1/T2-T1,λ≥1.(c)
eλT2/eλT2-eλT1≤eT2/eT2-eT1,λ≥1.Proof.
(a) We define the functionh(λ) by the formula (22)hλ=1αeλ+e-Tλ,λ>0. By simple computations, the function h(λ) gets the maximum h(ln⁡(T/α)/1+T) at the point λ=ln⁡(T/α)/1+T; that is(23)hλ≤hln⁡T/α1+T≤Tα1+TT/α1/1+T.
(b)(24)λeλT2-eλT1=λeλT2-T1-1e-λT1≤λλT2-T1+1-1≤1T2-T1.
(c) SetF(λ)=eλT2/eλT2-eλT1. After a simple calculation we can say F′(λ)<0, so the conclusion is true.Theorem 2.
Letg(x),h(x),gδ(x),hδ(x)∈L2(0,π) and satisfy (13), and then(25)fαx-fα,δxL20,π≤2πT2-T1T2α1+T2T2/α1/1+T2δ,and(26)φαx-φα,δxL20,π≤2πeT2eT2-eT1T2α1+T2T2/α1/1+T2δ;i.e., regularized solutions fα(x) and φα(x) are consecutively dependent on the final value data g(x) and h(x), where fα,δ(x) and φα,δ(x) are regularized solutions corresponding to gδ(x) and hδ(x), respectively.Proof.
From the perturbation datagδ(x) and hδ(x), by simple calculations we obtain(27)fα,δx=∑k=1∞k2e-k2T2/αek2+e-k2T2hkδ-e-k2T1/αek2+e-k2T1gkδe-k2T2-T11-e-k2T2-T1sin⁡kx,(28)φα,δx=∑k=1∞ek2T1e-k2T2/αek2+e-k2T2hkδ1-e-k2T1-e-k2T1/αek2+e-k2T1gkδ1-e-k2T2e-k2T2-T1-1sin⁡kx,where (29)gkδ=2π∫0πgδxsin⁡kxdx,hkδ=2π∫0πhδxsin⁡kxdx. From formulas (20) and (27) and Lemma 1, we can get(30)fαx-fα,δxL20,π2=∑k=1∞k21-e-k2T2-T1e-k2T2αek2+e-k2T2hk-hkδ-e-k2T1αek2+e-k2T1e-k2T2-T1gk-gkδsin⁡kx2=π2∑k=1∞k2ek2T2-ek2T11αek2+e-k2T2hk-hkδ-1αek2+e-k2T1gk-gkδ2≤π∑k=1∞k2ek2T2-ek2T11αek2+e-k2T22hk-hkδ2+gk-gkδ2≤π1T2-T1T2α1+T2T2/α1/1+T22h-hδL20,π2+g-gδL20,π2; i.e., (31)fαx-fα,δxL20,π≤2πT2-T1T2α1+T2T2/α1/1+T2δ. From formulas (21) and (28) and Lemma 1, we obtain (32)φαx-φα,δxL20,π2=∑k=1∞ek2T1e-k2T2-T1-1e-k2T21-e-k2T1αek2+e-k2T2hk-hkδ-e-k2T11-e-k2T2αek2+e-k2T1gk-gkδsin⁡kx2=π2∑k=1∞1ek2T2-ek2T1ek2T1-1αek2+e-k2T2hk-hkδ-ek2T2-1αek2+e-k2T1gk-gkδ2≤π∑k=1∞ek2T2ek2T2-ek2T11αek2+e-k2T22hk-hkδ2+gk-gkδ2≤πeT2eT2-eT1T2α1+T2T2/α1/1+T22h-hδL20,π2+g-gδL20,π2; i.e., (33)φαx-φα,δxL20,π≤2πeT2eT2-eT1T2α1+T2T2/α1/1+T2δ.Theorem 3.
Letg(x),h(x)∈L2(0,π), and  α∈(0,T). f(x) and φ(x) are the true solutions of the inverse problem (1). If there exists a constant A such that (34)∑k=1∞ek21+T2hk2+ek21+T1gk2≤A2, then(35)fx-fαxL20,π≤πT2-T1T2A1+T2T2/α1/1+T2.(36)φx-φαxL20,π≤πeT2eT2-eT1T2A1+T2T2/α1/1+T2.Proof.
From formulas (9) and (20) and Lemma 1, we derive that (37)fx-fαxL20,π2=∑k=1∞k21-e-k2T2-T11-e-k2T2αek2+e-k2T2hk-1-e-k2T1αek2+e-k2T1e-k2T2-T1gksin⁡kx2=π2∑k=1∞k2ek2T2-ek2T1ααek2+e-k2T2ek21+T2hk-ααek2+e-k2T1ek21+T1gk2≤π2∑k=1∞k2ek2T2-ek2T1ααek2+e-k2T2ek21+T2hk-ek21+T1gk2≤π2∑k=1∞k2ek2T2-ek2T1ααek2+e-k2T2ek21+T2hk-ek21+T1gk2≤π1T2-T1T21+T2T2/α1/1+T22A2.i.e.,(38)fx-fαxL20,π≤πT2-T1T2A1+T2T2/α1/1+T2. From formulas (10) and (21) and Lemma 1, we can obtain (39)φx-φαxL20,π2=∑k=1∞ek2T1e-k2T2-T1-1αek2e-k2T1-1αek2+e-k2T2hk+αek21-e-k2T2αek2+e-k2T1gksin⁡kx2=π2∑k=1∞ek2T1+T2ek2T2-ek2T1αek2e-k2T1-1αek2+e-k2T2hk+αek21-e-k2T2αek2+e-k2T1gk2≤π∑k=1∞ek2T2ek2T2-ek2T1ααek2+e-k2T22ek2T1+1hk2+ek2T1+1gk2≤πeT2eT2-eT1T21+T2T2/α1/1+T22A2;i.e., (40)φx-φαxL20,π≤πeT2eT2-eT1T2A1+T2T2/α1/1+T2.Theorem3 illustrates that fα(x) and φα(x) are really regularized solutions, since fα(x) and φα(x) approximate the true solutions f(x) and φ(x) as α→0, respectively.Theorem 4.
Under the conditions of Theorems2 and 3, the choice α=2δ/A leads to asymptotically optimal estimates of the regularization solutions:(41)fx-fα,δxL20,π=Oδ1/1+T2and(42)φx-φα,δxL20,π=Oδ1/1+T2.Proof.
From conclusions of Theorems2 and 3, we have (43)fx-fα,δxL20,π≤fx-fαxL20,π+fαx-fα,δxL20,π≤πT2-T1T2A1+T2T2/α1/1+T2+2πT2-T1δT2α1+T2T2/α1/1+T2and(44)φx-φα,δxL20,π≤φx-φαxL20,π+φαx-φα,δxL20,π≤πeT2eT2-eT1T2A1+T2T2/α1/1+T2+2πeT2eT2-eT1δT2α1+T2T2/α1/1+T2. From the above inequalities, the choice α=2δ/A leads to the asymptotically optimal estimates: (45)fx-fα,δxL20,π=Oδ1/1+T,φx-φα,δxL20,π=Oδ1/1+T2.Theorem4 gives a priori strategy for selecting regularization parameter. However, it is often undesirable to have the value of constant A 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 α makes(46)max⁡uα,δx,T1-gδx2,uα,δx,T2-hδx2=Cδ2,where C is a given constant. For regularization solutions fα,δ(x) and φα,δ(x), the deviation equation (46) means that(47)max⁡∑k=1∞e-k2T1αek2+e-k2T1gkδ-gkδ2,∑k=1∞e-k2T2αek2+e-k2T2hkδ-hkδ2=Cδ2.Theorem 5.
Letf(x) and φ(x) be the true solutions of (1)–(3) with their Fourier coefficients fk and gk, respectively. If there exist constants P>0 and E such that (48)∑k=1∞ek2T1Pfk2≤E,∑k=1∞ek2T1Pφk2≤E, and PT1<2, the a posteriori strategy (47) for selecting the regularization parameter leads to the following estimates:(49)fx-fα,δxL20,π2=Oδmin⁡2P/P+2,2,(50)φx-φα,δxL20,π2=Oδmin⁡2P/P+2,2,where fα,δ(x) and φα,δ(x) 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(51)fx-fα,δxL20,π2=∑k=1∞k21-e-k2T2-T1hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ2=∑k=1∞k21-e-k2T2-T12hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ4/P+2hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ2-4/P+2≤∑k=1∞k21-e-k2T2-T12hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ4/P+2P+2/22/P+2∑k=1∞hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ2-4/P+2P+2/PP/P+2. From (13), we can get (52)∑k=1∞k21-e-k2T2-T12hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ4/P+2P+2/22/P+2=∑k=1∞k21-e-k2T2-T1P+2hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ22/P+2=∑k=1∞k21-e-k2T2-T1P+21-e-k2T2αek2+e-k2T2hk+e-k2T2αek2+e-k2T2hk-hkδ-e-k2T2-T1gk1-e-k2T1αek2+e-k2T1-e-k2T2-T1e-k2T1αek2+e-k2T1gk-gkδ22/P+2≤2∑k=1∞k21-e-k2T2-T1P+2hk-e-k2T2-T1gk2+e-k2T2αek2+e-k2T2hk-hkδ+e-k2T2αek2+e-k2T1gk-gkδ22/P+2≤2∑k=1∞k21-e-k2T2-T1Pfk2+4∑k=1∞k2P+2e-2k2T2δ22/P+2≤2P+1E+4C1δ22/P+2and(53)∑k=1∞hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ2-4/P+2≤∑k=1∞hk-e-k2T2αek2+e-k2T2hkδ-e-k2T2-T1gk-e-k2T1αek2+e-k2T1gkδ2-4/P+2P+2/PP/P+2≤2∑k=1∞hk-hkδ-e-k2T2αek2+e-k2T2hkδ-hkδ2+2∑k=1∞e-k2T2-T1gk-gkδ-e-k2T1αek2+e-k2T1gkδ-gkδ2P/P+2≤8Cδ2+8δ2P/P+2=C2δ2P/P+2. In summary, (54)fx-fα,δxL20,π2≤2P+1E+4C1δ22/P+2C2δ2P/P+2≤C3Eδ2P/P+2+C4δ2=Oδmin⁡2P/P+2,2.
(2) Based on the Hölder inequality, we can similarly obtain that(55)φx-φα,δxL20,π2=∑k=1∞ek2T11-e-k2T2-T11-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk2=∑k=1∞ek2T11-e-k2T2-T121-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk4/P+21-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk2-4/P+2≤∑k=1∞ek2T11-e-k2T2-T121-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk4/P+2P+2/22/P+2∑k=1∞1-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk2-4/P+22/P+2P/P+2 From inequalities (13), we get (56)∑k=1∞ek2T11-e-k2T2-T121-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk4/P+2P+2/22/P+2≤2∑k=1∞ek2T11-e-k2T2-T1P+2e-k2T2αek2+e-k2T2hkδ-hk2+e-k2T1αek2+e-k2T1gkδ-gk22/P+2≤2∑k=1∞ek2T11-e-k2T2-T1P+2e-k2T1αek2+e-k2T1gkδ-gk-1-e-k2T1αek2+e-k2T1gk2+e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2αek2+e-k2T2hk22/P+2≤4∑k=1∞ek2T2ek2T2-ek2T1P+2ek2PT1-2α2δ2+ek2T1P+2gk2+ek2PT1-2α2δ2+ek2T1P+2hk22/P+2≤4∑k=1∞ek2T2ek2T2-ek2T1P+22ek2PT1-2α2δ2+ek2T1P+2gk2+ek2T1P+2hk22/P+2≤4∑k=1∞ek2T2ek2T2-ek2T1P+22C1δ2+ek2T1P+2e-k2T1φk+1-e-k2T1k2fk2+ek2T1P+2e-k2T2φk+1-e-k2T2k2fk22/P+2≤4∑k=1∞ek2T2ek2T2-ek2T1P+22C1δ2+2ek2T1P+22e-2k2T1φk2+2e2k2T2-T1k4fk22/P+2≤4∑k=1∞ek2T2ek2T2-ek2T1P+22C1δ2+4ek2T1Pφk2+4ek2T1P+2T2k4fk22/P+2≤4∑k=1∞eT2eT2-eT1P+22C1δ2+4E+4C2E2/P+2≤4eT2eT2-eT122C1δ2+C3E2/P+2,and(57)∑k=1∞1-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk2-4/P+2≤∑k=1∞1-e-k2T1e-k2T2αek2+e-k2T2hkδ-hk-1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gk2-4/P+22/P+2P/P+2≤2∑k=1∞1-e-k2T1e-k2T2αek2+e-k2T2hkδ-hkδ-hk-hkδ2+2∑k=1∞1-e-k2T2e-k2T1αek2+e-k2T1gkδ-gkδ-gk-gkδ2P/P+2≤8Cδ2+8δ2P/P+2=C4δ2P/P+2.
To sum up, we have(58)φx-φα,δxL20,π2≤4eT2eT2-eT122C1δ2+C3E2/P+2C4δ2P/P+2≤C5Eδ2P/P+2+C6δ2=Oδmin⁡2P/P+2,2.


## 4. Numerical Examples

In this section, we give two numerical examples to show effectiveness and stability of the proposed method: the first one has an analytical expression of the temperature concentration for the direct problem of heat conduction equation; the second one does not have an explicit solution of the direct problem, which must be solved by numerical methods such as the finite element method and the finite difference method.In addition, the infinite summation in (27) and (28) must be truncated in practical computations. In the absence of confusion, we still denote the truncated solutions of regularization solutions by fα,δ(x) and φα,δ(x); that is,(59)fα,δx=∑k=1mk2e-k2T2/αek2+e-k2T2hkδ-e-k2T1/αek2+e-k2T1gkδe-k2T2-T11-e-k2T2-T1sin⁡kx,(60)φα,δx=∑k=1mek2T1e-k2T2/αek2+e-k2T2hkδ1-e-k2T1-e-k2T1/αek2+e-k2T1gkδ1-e-k2T2e-k2T2-T1-1sin⁡kx.Here, the Fourier coefficients gkδ and hkδ are computed by the trapezoid rule of numerical integral as follows: (61)gkδ=2π∫0πgδxsin⁡kxdx≈2hπ∑i=1n-1gδxisin⁡kxi,and(62)hkδ=2π∫0πhδxsin⁡kxdx≈2hπ∑i=1n-1hδxisin⁡kxi, where gδ(xi)=g(xi)+ρ(2rand(xi)-1)g(xi),  hδ(xi)=g(xi)+ρ(2rand(xi)-1)h(xi), ρ is a relative error level, and rand is a random function in MATLAB that generates a uniformly distributed number over (0,1). In all numerical experiments, we execute the Morozov discrepancy principle (46) for selecting regularization parameters as follows.Letr∈(0,1) be a fixed constant and α0>0 be an initial guess of the regularized parameter. Here, we consider regularization parameters as a geometric sequence form: (63)αk=α0rk,k=0,1,2,⋯.The a posteriori strategy for selecting regularization parameters is to select the regularization parameter αk∗-1 so that in the first step k∗ has(64)uαk∗,δx,T1-gδxL20,π2≤Cδ2,uαk∗,δx,T2-hδxL20,π2≤Cδ2where C≥1 is a constant and uαk,δ(x,T1) and uαk,δ(x,T2) are also the truncated solutions as the following forms: (65)uαk,δx,T1=∑k=1me-k2T1αkek2+e-k2T1gkδsin⁡kx,uαk,δx,T2=∑k=1me-k2T2αkek2+e-k2T2hkδsin⁡kx.We always taken=100 and m=20 and use two strategies for selecting regularization parameters in all numerical examples: (a) the a priori selection strategy given in Theorem 4 and (b) the a posteriori selection strategy 3.17 in which α0=1, r=1/2, and C=1.0. In order to compare the truncated regularization solutions with the true solutions, we introduce E1 and E2 representing the relative errors of regularization solutions and defined by (66)E1=∑i=0nfα,δxi-fxi21/2∑i=0nfxi21/2,E2=∑i=0nφα,δxi-φxi21/2∑i=0nφxi21/2.Example 1.
Consider simultaneous determination of the source function and the initial distribution in the problem of heat conduction equation:(67)ut-uxx=fx,x,t∈0,π×0,1,u0,t=uπ,t=0,x,t∈0,π×0,1,ux,0=φx,x∈0,π,ux,T1=gx,x∈0,π,ux,T2=hx,x∈0,π, where T1=0.5,  T2=1,  g(x)=2-e-0.5sin⁡x, and  h(x)=2-e-1sin⁡x. In this case, f(x)=2sin⁡x and φ(x)=sin⁡x. Numerical results of this example are shown in Table 1. Behaviors of the true solutions and the regularization solutions are shown in Figures 1 and 2.Table 1
Numerical results of Example1.
δ α E 1 E 2 a priori selection strategy 0.05 4.4678e-05 2.9108e-03 1.0786e-02 0.10 8.9357e-05 5.1605e-03 1.1633e-02  a posteriori selection strategy 0.05 ( 1 / 2 ) 12 2.3366e-03 4.5157e-03 0.10 ( 1 / 2 ) 11 3.8406e-03 1.0433e-02Figure 1
Behaviors of the true and the regularization solution whenρ=0.05.
(a)
Results with the a priori strategy for selecting regularization parameters (b)
Results with the a posteriori strategy for selecting regularization parametersFigure 2
Behaviors of the true and the regularization solution whenρ=0.10.
(a)
Results with the a priori strategy for selecting regularization parameters (b)
Results with the a posteriori strategy for selecting regularization parametersExample 2.
Consider simultaneous determination of the source function and the initial distribution in the problem of heat conduction equation:(68)ut-uxx=fx,x,t∈0,π×0,1,u0,t=uπ,t=0,x,t∈0,π×0,1,ux,0=φx,x∈0,π, where we take f(x)=x(π-x) and φx=xsin⁡(x). Obviously, the direct problem does not have an analytical solution. So the temperature distribution functions u(x,T1) and u(x,T2) are obtained by using the inner function “pdepe” in MATLAB to solve the direct problem. Numerical results of this example are shown in Table 2. Behaviors of the true solutions and the regularization solutions are shown in Figures 3 and 4.Table 2
Numerical results of Example2.
δ α E 1 E 2 a priori selection 0.05 5.9942e-05 3.8019e-02 2.8939e-02 0.10 1.1988e-04 3.8028e-02 2.8573e-02  a posteriori selection 0.05 ( 1 / 2 ) 12 3.8293e-02 3.1005e-02 0.10 ( 1 / 2 ) 10 3.9655e-02 7.2344e-02Figure 3
Behaviors of the true and the regularization solution whenρ=0.05.
(a)
Results with the a priori strategy for selecting regularization parameters (b)
Results with the a posteriori strategy for selecting regularization parametersFigure 4
Behaviors of the true and the regularization solution whenρ=0.10.
(a)
Results with the a priori strategy for selecting regularization parameters (b)
Results with the a posteriori strategy for selecting regularization parametersFrom Tables1 and 2 and Figures 1–4, we can see that the proposed method is effective and stable for the noise data especially with large noise level. And the relative errors generated by using the a posteriori strategy for selecting regularization parameters are almost equivalent to the ones generated by using the a priori strategy for selecting regularization parameters, which also indicates that regularization parameters selected by the a posteriori strategy are reasonable for obtaining the meaningful inversion solutions.Remark 6.
It is well known that the backward heat conduction problem is a severely ill-posed problem. To show the influence of the final time valuesT1 and T2 on the numerical inversion results, we solve the inverse problem in Examples 1 and 2 by our proposed method with different large final time values and fixed values n=200, m=20, and δ=0.10. Since the numerical results for Example 2 have the same behavior as Example 1, we only give the numerical results of Example 1 in Table 3, which show that the inversion results of the source term are better than that of the initial value with the increase of final time values T1 and T2. That is to say, the heat source can be reconstructed very well from the data measured at large final times, but the initial value is not available. It shows that the ill-posedness of the backward heat conduction problem is stronger than that of the inverse source problem.Table 3
Numerical results of Example1 with different final time values.
T 1 T 2 α E 1 E 2 a priori selection strategy 1.0 1.5 7.3143e-05 1.7461e-03 2.3633e-03 2.0 2.5 7.8949e-05 4.4939e-03 3.1173e-02 3.0 3.5 8.1085e-05 1.1924e-02 2.6646e-01 0.5 5.0 8.2050e-05 3.2691e-02 4.0627e-02 1.0 5.0 8.2050e-05 3.2922e-02 1.0862e-01 3.5 5.0 8.2050e-05 3.9430e-02 2.0275e+00 5.0 6.0 8.2226e-05 1.1243e-01 2.3531e+01  a posteriori selection strategy 1.0 1.5 ( 1 / 2 ) 14 5262e-03 2.2503e-03 2.0 2.5 ( 1 / 2 ) 16 1.2060e-03 3.4426e-02 3.0 3.5 ( 1 / 2 ) 17 1.5073e-03 1.0573e-01 0.5 5.0 ( 1 / 2 ) 19 1.1903e-03 2.4252e-02 1.0 5.0 ( 1 / 2 ) 19 1.1958e-03 4.8508e-02 3.5 5.0 ( 1 / 2 ) 19 1.3532e-03 4.2210e-01 5.0 6.0 ( 1 / 2 ) 21 1.1297e-03 1.7172e+00


## 5. Conclusions and Discussion

In this paper, we study an inverse problem for simultaneously determining the space-dependent source and the initial distribution in heat conduction equation. For obtaining stable solutions of the inverse problem, we construct a regularization problem to approximate the originally inverse problem. Stability and convergence of the regularization solutions are proved, and convergence rates of the regularized solutions are also obtained under the a priori and a posteriori strategies for selecting regularization parameters, respectively. Numerical results show that the proposed regularization method is stable and effective for simultaneously determining the space-dependent source and the initial distribution from the measurements at suitable final time values.



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