# The Method Based on Series Solution for Identifying an Unknown Source Coefficient on the Temperature Field in the Quasiperiodic Media

Authors: Bingxian Wang; Chuanzhi Bai; M. Xu; L. P. Zhang
Journal: International Journal of Differential Equations
 (2021)

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## 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 solutionrt is given.


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## Body

## 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:(1)ut−Δu=fx,t;u,x,t∈Ω×0,tmax,where u represents the state variable, Ω is a bounded domain in Rd, and the right-hand-side f 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 model(2)ut=uxx+rtfx,t,x,t∈Q:=0,1×0,Tu0,t=u1,t,−ux0,t+ηu0,t=0,t∈0,Tux,0=φx,x∈0,1.for temperature field ux,t with the unknown rt describing the strength of heat source, where fx,t and φ are known and η>0 is a known constant representing the heat flux exchanges at the boundary x=0. 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 rt in QT from the given data(3)ux0,t=ht,t∈0,T,at some interior point x0∈0,1. In [6], the uniqueness of reconstruction rt in C0,1 for suitable fx,t and φx is given by introducing the system(4)X″x+λXx=0,0≤x≤1,X0=X1,−X′0+ηX0=0.Lemma 1.
assumed that the known function pairfx,t,φx meets(i)
A1ϕx∈C40,1, both φx and φ″x meet the boundary conditions in (4)(ii)
A2f⋅,t∈C40,t, fx,⋅∈C0,T for any fixed t∈0,T and x∈0,1, both f⋅,t and ∂xx2f⋅,t meet the boundary conditions in (4)
If the observation pointx0∈0,1 is chosen such that fx0,t|t∈0,T≠0, then there exists a unique solution rt∈C0,T 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 rt, here we propose to firstly solve ux,t in DT from a nonstandard initial boundary value problem by deleting rt in (2) using the extra data ht, and then, the unknown source rt can be obtained from the function ux,t if required. Although the uniqueness of this nonstandard problem is still established by the uniqueness of rt 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 recoversrt from input data (3) ensures that we can solve ux,t in Q in terms of (2) and (3), noticing for known rt, (2) is a well-posed problem. Here, instead of identifying rt, we firstly solve the field ux,t directly in terms of the extra data ht. Taking x=x0∈0,1 in the equation and using the fact fx0,t≠0 in 0,T, we have(5)rt=1fx0,th′t−uxxx0,t.Also, inserting (5) into (2) again, a following nonstandard initial boundary value problem for function ux,t is obtained as follows:(6)ut−uxx−fx,tfx0,th′t−uxxx0,t=0,x,t∈Q,u0,t=u1,t,−ux0,t+ηu0,t=0,0≤t≤T,ux,0=φx,0≤x≤1.We introduce the function space(7)Dφ=ωx,t∈C2,1Q:ω0,t=ω1,t,−ωx0,t+ηω0,t=0,ωx,0=φxand the homotopy transform by the functions ϕx,t;p:p∈0,1 such that(8)ϕx,t;0=u0x,t,ϕx,t;1=ux,t,with some known initial guess u0x,t∈Dφ for the unknown ux,t∈Dφ.For the functionωx,t, we define the linear operator L and operator ℕ in two forms, respectively. The first form is(9)L1ωx,t≔ωtx,t,ℕ1ωx,t≔ωtx,t−ωxxx,t−fx,tfx0,th′t−ωxxx0,t.This kind of definition is frequently used to solve some partial differential equations [17–19]. We construct the zero-order deformation equation(10)1−pL1ϕx,t,p−u0x,t=ℏℕ1ϕx,t,p,with some nonzero constant ℏ as the convergence controlling parameter for ϕx,t;p:p∈0,1⊂Dφ. The homotype scheme aims to generate the zero point ux,t at p=1 from the known zero point u0x,t at p=0 from (10). To compute umx,t for m=1,2,…, from the known initial guess u0x,t, we expand the function ϕx,t;p in terms of p as a power series(11)ϕx,t;p=ϕx,t;0+∑m=1∞1m!∂ϕ∂pm|p=0pm=u0x,t+∑m=1∞umx,tpm,and insert it into (10). By comparing the coefficients of pm for m=1,2,…,, we obtain the mth-order deformation equations(12)L1umx,t−χmum−1x,t=ℏℝu⟶m−1with the constants(13)χm=0,m=0,1,m=1,2,…,where(14)ℝu⟶m−1=∂um−1∂t−∂2um−1∂x2−fx,tfx0,th′t−∂2um−1∂x2x0,t.Now, the solution of themth-order deformation equation (12) for m≥1 becomes(15)umx,t=χm−1um−1x,t+ℏL1−1ℝu⟶m−1.However, in Section3, we will illustrate the divergence of ∑n=0∞unx,t in Example 1 by means of the form in (15). So, we discuss the second form(16)L2ωx,t≔ωtx,t−ωxxx,t,ℕ2ωx,t≔ωtx,t−ωxxx,t+fx,tfx0,tωxxx0,tand consider the zeroth-order decomposition equation(17)1−pL2ϕx,t;p−u0x,t=ℏpℕ2ϕx,t;p−fx,tfx0,th′t,with some nonzero constant ℏ as the convergence controlling parameter for ϕx,t;p:p∈0,1⊂Dφ. By the same method, we obtain(18)L2um−χm−1um−1x,t=ℏℕ2um−1x,t−ℏ1−χm−1fx,tfx0,th′t,m=1,2,…,for um∈D0 by noticing the linearity of operators L2 and ℕ2.Here, we first prove the following result.Theorem 1.
If the series(19)u0x,t+∑m=1∞umx,t,converges, where umx,tm=1,2,…, is governed by (18), it must be an exact solution of (6) with initial condition ux,0=φx.Proof.
If the series is convergent, we can write(20)s=∑m=0∞umx,t,and it holds(21)limn⟶∞unx,t=0,and then, using (16) and (18), we have(22)ℏ∑m=1∞ℕ2um−1x,t−1−χm−1fx,tfx0,th′t=limn⟶∞∑m=1nL2umx,t−χm−1um−1x,t,=L2limn⟶∞∑m=1numx,t−χm−1um−1x,t,=L2limn⟶∞unx,t,=0,which gives, since ℏ≠0,(23)∑m=1∞ℕ2um−1x,t−1−χm−1fx,tfx0,th′t=0.
Thus, according to the definition ofℕ2, we have(24)∑m=1∞ℕ2um−1x,t−1−χm−1fx,tfx0,th′t,=∑m=1∞∂um−1∂t−∂2um−1∂x2+fx,tfx0,t∂2um−1x0,t∂x2−1−χm−1fx,tfx0,th′t,=∂s∂t−∂2s∂x2−fx,tfx0,t∂2sx0,t∂x2−h′t,=0.
From the initialumx,0=0 and u0x,t=φx [2], it holds sx,0=φx. So, sx,t satisfies the equation in (6), and therefore, it is an exact solution of problem (6). This ends the proof.
We define(25)K3f=∑n=0∞e−λnt−τf⋅,τ,Ynfx0,τXn.
The convergence of the iteration process is stated in [6].Lemma 2.
Assume either0<T<2 or A<1, where Agt=∫0tFt−τ,τ,x0gτdτ and(26)Ft−τ,τ,x0=∑n=0∞λnXnx0e−λnt−τf⋅,τ,Ynfx0,τ.Then, for any initial guessu0x,t∈Dφ and h∈−1,0, the iteration process umx,t:m=0,1,2,3,…, will give the exact solution as(27)ux,t=∑m=0∞umx,t.Moreover,ux,t has the representation(28)ux,t=u0x,t−1ℏu1x,t+1ℏK3ft−τ,τ,xℐ+A−1∂xx2u1x0,⋅τdτ,which means we can compute ux,t in terms of u0, noticing u1=ℏℳu0−Q. Also, at N iteration step, the error can be obtained as follows:(29)ux,t−∑m=0Numx,t,=1+ℏNℳu0x,t−Qf,hx,t+1+ℏN∫0tK3ft−τ,τ,xA−1W1τ−∑k=0NCNkℏ1+ℏkAkℛτdτ.Remark 1.
The means of symbolsA, ℐ, K, ℳ, W1, and ℛ 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 solutionrt is shown in the figures. We define the numerical relative error in the form(30)Error=rt−rnumtL20,1rtL20,1.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](31)ut=uxx+rtsin2πx+π4,u0,t=u1,t,−ux0,t+ηu0,t=0,φx=0,with the exact solution(32)ux,t=sin2πx+π4sin8t.
We setx0=1/4, η=2π, then ht=u1/4,t=2/2sin8t, and the coefficient rt=8cos8t+4π2sin8t.
According to (15), when selecting the initial guess u0x,t=φx=0, some terms of series solution ux,t are as follows:(33)u0x,t=0,u1x,t=−ℏsin2πx+π4sin8t,u2x,t=−ℏ2+2ℏsin2πx+π4sin8t,u3x,t=−ℏ3+3ℏ2+3ℏsin2πx+π4sin8t,u4x,t=−ℏ4+4ℏ3+6ℏ2+4ℏsin2πx+π4sin8t.⋮
So,(34)∑k=0nukx,t=1+ℏ1−1+ℏn+nℏℏsin2πx+π4sin8t,when set ℏ=−1, u1x,t, u2x,t,…, unx,t are all the exact solutions of problem (31). Thus, the limit of the sum (34) does not exist. Even if ℏ takes other values, it is impossible to get convergence results. So, the form in (15) is unsuitable for calculating unx,t, n=1,2,…,.
According to the convergence results in Lemma2, the form in (18) can be used to calculate unx,t, n=1,2,…,. Similarly, we select the initial guess u0x,t=φx=0. By (18), we can obtain(35)u1x,t=−ℏsin2πx+π4sin8t−24π2cos8t,∂xx2u1x0,t=ℏ22π2sin8t−42cos8t.
It should be noted that we truncate to constructK3f in (25) and Ft−τ,τ,x0 in (26):(36)K3f=∑n=0M1e−λnt−τf,˙τ,Ynfx0,τXn,(37)Ft−τ,τ,x0=∑n=0M2λnXnx0e−λnt−τf⋅,τ,Ynfx0,τ.
We setℏ=−0.5, M1=10 in (36), and M2=10 in (37) and substitute u0x,t, u1x,t,∂xx2u1x0,t, and (36) into (28). For the exact input of ux0,t=2/2sin8t, we can approximate ux,t using our iterative process by ∑mNumx,t directly. Our results of ux,t in 0,1×0,1 using exact input data with N=5,15,20 are shown in Figure 1. Therefore, according to (5), we can obtain the approximate solution rt of the inverse problem by means of the approximate solution ux,t when N=20, see Figure 2. By (30), the error of inverse solution rtt∈0,1 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.Figure 1
Reconstructions of Example1. (a) First line: exact solution of ux,t (left); approximate solution of ux,t when N=5 (right); (b) second line: approximate solutions of ux,t when N=15 (left) and N=20 (right), respectively.
(a)(b)Figure 2
Reconstructions ofrt of Example 1.Table 1
The error of solutionrtt∈0,1 of Example 1.
x0.10.20.30.40.5Error0.032 0330.021 2540.063 2160.011 2080.037 257x0.60.70.80.91Error0.020 0580.030 6740.004 5830.034 3960.046 099Example 2.
We consider the inverse problem in the form in [20], which is the same as with Example 1 with the exact solution(38)ux,t=sin2πx+π4e−π2−3/4t.
We setx0=1/4, η=2π, then ht=u1/4,t=2/2e−π2−3/4t, and the coefficient rt=e−π2t.
Same as Example1, when selecting the initial guess u0x,t=φx=sin2πx+π/4, we can obtain(39)u1x,t=ℏ1−14π2π2−34sin2πx+π4e−π2−3/4t,∂xx2u1x0,t=22ℏ3π2+34e−π2−3/4t.
For the exact input ofux0,t=2/2sin8t, we can approximate ux,t using our iterative process by ∑mNumx,t directly. Our results of ux,t in 0,1×0,1 using exact input data with N=5,15,20 are shown in Figure 3. Similarly, according to (5), the approximate solution rt of the inverse problem can be obtained by means of the approximate solution ux,t when N=20, see Figure 4. By (30), the error of inverse solution rtt∈0,1 is shown in Table 2. Moreover, the running time of the scheme in this paper is 1m48s, while the algorithm in [20] takes 7m25s.Figure 3
Reconstructions of Example2. (a) First line: exact solution of ux,t (left); approximate solution of ux,t when N=5 (right); (b) second line: approximate solutions of ux,t when N=15 (left) and N=20 (right), respectively.
(a)(b)Figure 4
Reconstructions ofrt of Example 2.Table 2
The error of solutionrtt∈0,1 of Example 2.
x0.10.20.30.40.5Error0.008 681 510.042 939 70.025 728 30.029 755 50.042 485 8x0.60.70.80.91Error0.011 920 70.049 506 70.070 640 70.024 357 30.078 507


## 5. Discussion and Conclusions

In fact, the inverse problem is frequently proposed by using the noisy datahδt satisfying hδ−hC10,T≤δ. 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 ℏ 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 ℏ on the convergence of the solution series given by the HAM, according to the ℏ-curves, it is easy to find the valid region of ℏ, 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].



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