Abstract
In this paper, we consider the reconstruction of heat field in one-dimensional quasiperiodic media with an unknown source from the interior measurement. The innovation of this paper is solving the inverse problem by means of two different homotopy iteration processes. The first kind of homotopy iteration process is not convergent. For the second kind of homotopy iteration process, a convergent result is proved. Based on the uniqueness of this inverse problem and convergence results of the second kind of homotopy iteration process with exact data, the results of two numerical examples show that the proposed method is efficient, and the error of the inversion solution is given.
1. Introduction
The energy demand is the hotspot in theory research, and many studies have been conducted on energy loss reduction and heat transfer enhancement [1–4]. Conduction is heat transfer by means of molecular agitation within a material without any motion of the material as a whole. Convection is heat transfer by mass motion of a fluid such as air or water when the heated fluid is caused to move away from the source of heat, carrying energy with it. The heat equation is a parabolic partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. In order to achieve an optimal thermal control system, the accurate prediction of the thermal conductivity is particularly necessary. In the process of transportation, diffusion, and conduction of natural materials, the following heat equation is a suitable approximation:where represents the state variable, is a bounded domain in , and the right-hand-side denotes physical laws in our case source terms. However, there are many inverse problems for heat equations, and it is well known that they are generally ill posed; i.e., the existence, uniqueness, and stability of their solutions are not always guaranteed. So, the regularization numerical methods have been investigated for the inverse problems.
2. Problem Description
Consider the heat conduction modelfor temperature field with the unknown describing the strength of heat source, where and are known and is a known constant representing the heat flux exchanges at the boundary . The boundary conditions in (2) arise quite often; for an example, see its nature in mathematical biology in [5]. Our aim is to determine the heat field in from the given dataat some interior point . In [6], the uniqueness of reconstruction in for suitable and is given by introducing the system
Lemma 1. assumed that the known function pair meets(i), both and meet the boundary conditions in (4)(ii), for any fixed and , both and meet the boundary conditions in (4)If the observation point is chosen such that , then there exists a unique solution for inverse problems (2) and (3).
Also, in [7], the inverse problem has been solved by a regularization scheme with an explicit selection strategy for the regularization parameter. Based on the uniqueness of the inverse problem, we mainly discuss the method based on series solution in this paper. To overcome these numerical difficulties, rather than considering (2) and (3) as an inverse problem for , here we propose to firstly solve in from a nonstandard initial boundary value problem by deleting in (2) using the extra data , and then, the unknown source can be obtained from the function if required. Although the uniqueness of this nonstandard problem is still established by the uniqueness of for the inverse problem, its solution will be given explicitly by the homotopy analysis method (HAM) for suitably constructed homotopy transform starting from a quite general initial guess.
The homotopy analysis method is developed by Liao in 1992 [8, 9] and has been successfully applied to solve many types of nonlinear problems in science and engineering by many authors [10–12]. Additional, it has been found that the homotopy schemes can also be applied to solve some inverse problems [7, 13, 14]. Liu et al. [15] propose a homotopy-based iterative regularizing scheme to solve a backward heat conduction problem, and the convergence results of the homotopy sequence are given for noisy input data. Wang et al. [16] proposed a homotopy-based iterative regularizing scheme to solve a source distribution in heat conduction.
3. Homotopy Scheme and Iteration Process
The uniqueness of the inverse problems which recovers from input data (3) ensures that we can solve in in terms of (2) and (3), noticing for known , (2) is a well-posed problem. Here, instead of identifying , we firstly solve the field directly in terms of the extra data . Taking in the equation and using the fact in , we have
Also, inserting (5) into (2) again, a following nonstandard initial boundary value problem for function is obtained as follows:
We introduce the function spaceand the homotopy transform by the functions such thatwith some known initial guess for the unknown .
For the function , we define the linear operator and operator in two forms, respectively. The first form is
This kind of definition is frequently used to solve some partial differential equations [17–19]. We construct the zero-order deformation equationwith some nonzero constant as the convergence controlling parameter for . The homotype scheme aims to generate the zero point at from the known zero point at from (10). To compute for from the known initial guess , we expand the function in terms of as a power seriesand insert it into (10). By comparing the coefficients of for , we obtain the mth-order deformation equationswith the constantswhere
Now, the solution of the mth-order deformation equation (12) for becomes
However, in Section 3, we will illustrate the divergence of in Example 1 by means of the form in (15). So, we discuss the second formand consider the zeroth-order decomposition equationwith some nonzero constant as the convergence controlling parameter for . By the same method, we obtainfor by noticing the linearity of operators and .
Here, we first prove the following result.
Theorem 1. If the seriesconverges, where is governed by (18), it must be an exact solution of (6) with initial condition .
Proof. If the series is convergent, we can writeand it holdsand then, using (16) and (18), we havewhich gives, since ,Thus, according to the definition of , we haveFrom the initial and [2], it holds . So, satisfies the equation in (6), and therefore, it is an exact solution of problem (6). This ends the proof.
We defineThe convergence of the iteration process is stated in [6].
Lemma 2. Assume either or , where and
Then, for any initial guess and , the iteration process will give the exact solution as
Moreover, has the representationwhich means we can compute in terms of , noticing . Also, at iteration step, the error can be obtained as follows:
Remark 1. The means of symbols , , , , , and are the same as those in [6].
4. Test Examples
In this section, we present two examples to show the efficiency of the present method described in the previous part. The inverse solution is shown in the figures. We define the numerical relative error in the form
All the computations were performed using MATLAB 2016a on a personal computer with Intel Core i5 and 8.00 GB memory.
Example 1. The inverse problem is considered in the form [20]with the exact solutionWe set , , then , and the coefficient .
According to (15), when selecting the initial guess , some terms of series solution are as follows:So,when set , , , are all the exact solutions of problem (31). Thus, the limit of the sum (34) does not exist. Even if takes other values, it is impossible to get convergence results. So, the form in (15) is unsuitable for calculating , .
According to the convergence results in Lemma 2, the form in (18) can be used to calculate , . Similarly, we select the initial guess . By (18), we can obtainIt should be noted that we truncate to construct in (25) and in (26):We set , in (36), and in (37) and substitute , , and (36) into (28). For the exact input of , we can approximate using our iterative process by directly. Our results of in using exact input data with are shown in Figure 1. Therefore, according to (5), we can obtain the approximate solution of the inverse problem by means of the approximate solution when , see Figure 2. By (30), the error of inverse solution is shown in Table 1. Moreover, the running time of the scheme based on the homotopy method in this paper is 1m30s, while the algorithm in [20] takes 7m10s.
(a)
(b)
Example 2. We consider the inverse problem in the form in [20], which is the same as with Example 1 with the exact solutionWe set , , then , and the coefficient .
Same as Example 1, when selecting the initial guess , we can obtainFor the exact input of , we can approximate using our iterative process by directly. Our results of in using exact input data with are shown in Figure 3. Similarly, according to (5), the approximate solution of the inverse problem can be obtained by means of the approximate solution when , see Figure 4. By (30), the error of inverse solution is shown in Table 2. Moreover, the running time of the scheme in this paper is 1m48s, while the algorithm in [20] takes 7m25s.
(a)
(b)
5. Discussion and Conclusions
In fact, the inverse problem is frequently proposed by using the noisy data satisfying . For noisy inversion input data, the number of iterations in the homotopy scheme is taken as the regularizing parameter, for which Wang and Liu [6] proposed a choice strategy in terms of the noisy level, and then, the convergence rate for the regularizing solution is given. The choice of the convergence parameter can also be determined by means of the results of Lemma 2. In [8], Liao showed that whatever a solution series converges, it will be one of the solutions of the considered problem. To investigate the influence of convergence parameter on the convergence of the solution series given by the HAM, according to the -curves, it is easy to find the valid region of , which corresponds to the line segment nearly parallel to the horizontal axis [8]. Additionally, some modified homotopy analysis methods are for dealing with different types of problems with strong nonlinearity [21–24].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of China (grant no. 11 501 236), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 18KJD110002), and the Scientific Research Foundation of Huaiyin Normal University (no. 31WBX00).