# A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

Authors: Songshu Liu; Lixin Feng
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.


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## Body

## 1. Introduction

In many industrial applications [1] one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements. This problem leads us to consider a Cauchy problem of heat conduction equation, which can be considered as a data completion problem that means to achieve the remaining information from boundary conditions for both the solution and its normal derivative on the boundary. This sort of problem occurs in a wide range of scientific and engineering areas [1] including manufacturing process control, metallurgy, chemical, aerospace and nuclear engineering, food science, and medical diagnostics.Mathematically, the Cauchy problem of heat equation belongs to the class of problems called the ill-posed problems; i.e., small errors in the measured data can lead to large deviations in the estimated quantities. As a consequence, its solution does not satisfy the general requirement of existence, uniqueness, and stability under small changes to the input data. To overcome such difficulties, a variety of techniques for solving the Cauchy problem of heat equation have been proposed [2–10].In this paper, we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation. We remark here that the Cauchy problem of one-dimensional heat equation has been well studied in the last few decades. Due to severe ill-posedness, however, much more difficulties exist to solve the Cauchy problem of heat conduction equation in the 2D case than in the 1D case.To the knowledge of the authors, there are still few results on Cauchy problem of heat conduction equation in the 2D case. In 2007, Qian and Fu [11] applied Fourier method and modified equation method to solve problem 1 and prove some error estimates of Hölder type for the solution. In 2011, Li and Wang [12] used a simplified Tikhonov regularization method to treat it. Following the works of [11, 12], Zhao et al. [13] also treated this problem by a modified kernel method in 2015.In this article, we continue to consider the following Cauchy problem of 2D heat conduction equation.Problem 1.
(1) u t x , y , t = u x x x , y , t + u y y x , y , t ,  0 < x < 1 ,   y > 0 ,   t > 0 , (2) u 0 , y , t = g y , t ,  y ≥ 0 ,   t ≥ 0 ,with corresponding measured data function gδy,t,(3)ux0,y,t=0,y≥0,t≥0,ux,y,0=0,0≤x≤1,y≥0,ux,0,t=0,0≤x≤1,t≥0.
We wish to determine the temperatureux,y,t for 0<x≤1 from temperature measurement gδy,t. We will adopt a revised Tikhonov regularization method to deal with this problem. This method was first introduced by Carasso in [14], and then the idea of this method has been successfully used for solving various types of ill-posed problems [15–18]. Under a priori selection rule for the regularization parameter, the convergence of the revised Tikhonov regularization method will also be given.The paper is organized as follows. In Section2 we formulate a Cauchy problem of 2D heat conduction equation. Section 3 is devoted to the convergence estimate for this method. A numerical example is tested in Section 4. Finally, the paper ends with a brief conclusion in Section 5.


## 2. Mathematical Formulation of the Cauchy Problem of 2D Heat Conduction Equation

In order to use the Fourier transform technique, we extend the functionsux,y,t,gy,t,gδy,t to the whole real y,t plane by defining them to be zero everywhere in y,t,y<0,t<0. We wish to determine the temperature ux,y,t∈L2R2 for 0<x≤1 from the temperature measurement gδy,t∈L2R2.We also use the correspondingL2-norm defined as(4)g=∫R2gy,t2dydt1/2.We now could assume that the measured data function gδy,t satisfies(5)gδ-g≤δ,where the constant δ>0 represents a bound on the measurement error. Assume that there exists a constant E>0 such that the following a priori bound exists:(6)u1,·,·≤E.Let(7)g^ξ,η=12π∫R2gy,te-iξy+ηtdydt,ξ,η∈R,be the Fourier transform of a function gy,t. Then(8)gy,t=12π∫R2g^ξ,ηeiξy+ηtdξdη.Applying this transform to (1) with respect to y and t, we obtain(9)u^xxx,ξ,η=iη+ξ2u^x,ξ,η,which is a second-order ordinary differential equation for fixed ξ and η. Now using the boundary condition in the frequency domain, we can easily get(10)u^x,ξ,η=cosh⁡τxg^ξ,η,and taking the inverse Fourier transform, the solution of problem 1 is(11)ux,y,t=12π∫R2eiξy+ηtg^ξ,ηcosh⁡τxdξdη,where τ is the principal value of iη+ξ2; i.e.,(12)τ=iη+ξ2=η2+ξ4+ξ22+isignηη2+ξ4-ξ22=a+σbi,where σ=signη.We note here that, for fixed0<x≤1, the value of cosh⁡τx in (10) is unbounded as |ξ|→∞. We can see that small errors in the data can blow up and completely destroy the solution for 0<x≤1. Thus the Cauchy problem of 2D heat conduction equation is ill-posed. A feasible approach to stabilize the problem is to eliminate all high frequencies or to replace the “kernel” cosh⁡τx by a bounded approximation kernel denoted by kx,τ,α, which has the following two common properties:(I)
If the parameterα is small, then for small τ, the kernel kx,τ,α is close to cosh⁡τx.(II)
Ifα is fixed, kx,τ,α is bounded.


## 3. Revised Tikhonov Regularization Method and Error Estimates

To get a bounded approximation kernelkx,τ,α, a popular method is Tikhonov regularization, which is applied to solve the following minimization problem:(13)min⁡Fαu=min⁡Ku·,y,t-gδy,tL2R22+αu·,y,tL2R22,where α is a regularization parameter and K:u·,y,t→gy,t is a linear bounded operator.Lemma 2.
There exists a unique solution to the above minimization problem. It is given by(14)uαδx,y,t=12π∫R2cosh⁡τx1+αcosh⁡τx2g^δξ,ηeiξy+ηtdξdη.Proof.
LetI denote identity operator in L2 and let K∗ be the adjoint of K. Then, the unique solution of the minimization problem (13) is given by(15)uαδ=αI+K∗K-1K∗gδ.In order to obtain the explicit formula (14) from (15), we use Parseval’s formula; one has(16)Kxu^,v^=Kxu,v=u,K∗v=u^,K∗v^.According to Ku^=1/cosh⁡τxu^x,ξ,η, we find(17)K∗v^=1cosh⁡τx¯v^x,ξ,η.Likewise,(18)K∗Ku^=1cosh⁡τx2u^x,ξ,η.Consequently,(19)α+1cosh⁡τx2u^=K∗gδ^.Using (17) and solving for u^αδ(x,ξ,η) in (19), we obtain(20)u^αδx,ξ,η=cosh⁡τx1+αcosh⁡τx2g^δξ,η.Finally, (14) follows by an inverse Fourier transform.From Lemma2, we can see that the kernel function kx,τ,α=cosh⁡(τx)/1+αcosh⁡τx2 corresponds to the Tikhonov regularization method.In the following, we use a much better filter1/1+αe2|τ| to replace the original 1/1+αcosh⁡τx2; i.e., kx,τ,α=cosh⁡τx/1+αe2|τ|. It corresponds to the regularized solution as follows:(21)uαδx,y,t=12π∫R2eiξy+ηtcosh⁡τx1+αe2τg^δξ,ηdξdη,and is regarded as a revised Tikhonov regularization solution of u(x,y,t) of problem 1.In the following, we give the convergence estimate foruαδx,y,t-ux,y,t by using an a priori choice rule for the regularization parameter. Before proceeding to derive the main result, we recall a proposition which we will use in the proof below.Proposition 3 (see [14]).
Let0<x<2,α>0. We have the following inequality:(22)supτ≥0eτx1+αe2τ≤1αx.Theorem 4.
Supposeu is the solution of problem 1 with the exact data g and uαδ is the regularized solution with the noise data gδ, and let gδ satisfy (5) and the exact solution u at x=1 satisfy (6). If α=δ2/E2 is selected, then for fixed 0<x<1, one gets the error estimate(23)uαδx,y,t-ux,y,t≤3δ1-xEx.Proof.
Using the Parseval’s identity and Proposition3, we have(24)uαδx,y,t-uαx,y,t=u^αδx,ξ,η-u^αx,ξ,η=∫-∞∞∫-∞∞cosh⁡τx1+αe2τg^δξ,η-cosh⁡τx1+αe2τg^ξ,η2dξdη1/2≤supξ,η∈Rcosh⁡τx1+αe2τg^δ-g^≤supξ,η∈Reτx1+αe2τδ≤1αxδ,(25)uαx,y,t-ux,y,t=u^αx,ξ,η-u^x,ξ,η=∫-∞∞∫-∞∞cosh⁡τx1+αe2τg^-coshτxg^2dξdη1/2=∫-∞∞∫-∞∞αcosh⁡τxe2τ1+αe2τg^2dξdη1/2=∫-∞∞∫-∞∞αcosh⁡τxe2τ1+αe2τ1cosh⁡τu^1,ξ,η2dξdη1/2≤αsupξ,η∈Re2τ1+αe2τcosh⁡τxcosh⁡τu^1,·,·≤αsupξ,η∈Re2τ1+αe2τ2e-1-xτu^1,·,·≤2α1α1+xE.Combining (24), (25), and the condition α=δ2/E2, we have(26)uαδx,y,t-ux,y,t≤3δ1-xEx.Remark 5.
Compared with [11, 12], where the convergence estimate is of logarithmic type, the convergence estimate we get is of Hölder type.Remark 6.
In Theorem4, we do not obtain the error estimate at x=1; in order to obtain that, we should use a stronger priori assumption as follows:(27)u1,·,·p≤E,where ·p denotes the norm on Sobolev space HpR2 defined by(28)u1,·,·p=∫-∞∞∫-∞∞1+ξ2+η2pu^1,ξ,η2dξdη1/2,p>0.Theorem 7.
Supposeu1,y,t is the solution of problem 1 with the exact data g and uαδ1,y,t is the regularized solution with the noise data gδ, and let gδ satisfy (5) and the exact solution u at x=1 satisfy (27). If α=δ/E is selected, then for fixed x=1, one gets the error estimate(29)uαδ1,y,t-u1,y,t≤δ1/2E1/2+Emax⁡δE1/2,18ln⁡Eδ-p.Proof.
Similar to the proof of Theorem4, we know that(30)uαδ1,y,t-uα1,y,t≤δα,(31)uα1,y,t-u1,y,t=u^α1,ξ,η-u^1,ξ,η=∫-∞∞∫-∞∞cosh⁡τ1+αe2τg^-coshτg^2dξdη1/2=∫-∞∞∫-∞∞αcosh⁡τe2τ1+αe2τg^2dξdη1/2=∫-∞∞∫-∞∞αe2τ1+αe2τ1+ξ2+η2-p/21+ξ2+η2p/2coshτg^2dξdη1/2≤supξ,η∈Rαe2τ1+αe2τ1+ξ2+η2-p/2u^1,·,·p.Now, we distinguish two cases to estimate (31).
Case 1. For η+ξ≤1/2ln1/α, we have(32)τ=iη+ξ2≤η+ξ.Therefore, we can derive that(33)αe2τ1+αe2τ1+ξ2+η2-p/2≤αe2τ≤αe2η+ξ≤α.We can see that it goes to zero as α→0.
Case 2. For η+ξ≥1/2ln1/α, we have(34)αe2τ1+αe2τ1+ξ2+η2-p/2≤1+ξ2+η2-p/2.If η≥ξ≥0, we have(35)2η≥η+ξ≥12ln1α;i.e.,η≥1/4ln1/α. Therefore,(36)1+ξ2+η2-p/2≤ξ2+η2-p/2≤η-p=η-2p≤14ln1α-2p.If 0≤η≤ξ, we have(37)2ξ≥η+ξ≥12ln1α;i.e., ξ≥1/4ln1/α. Therefore,(38)1+ξ2+η2-p/2≤ξ2+η2-p/2≤ξ-p≤14ln1α-p.Combining (31), (33), (36), and (38), we obtain(39)uα1,y,t-u1,y,t≤Emax⁡α,14ln1α-p.Now using the triangle inequality and combining (30) and (39), we can easily get the convergence estimate(40)uαδ1,y,t-u1,y,t≤u^αδ1,ξ,η-u^α1,ξ,η+u^α1,ξ,η-u^1,ξ,η≤δ1/2E1/2+Emax⁡δE1/2,18lnEδ-p.


## 4. Numerical Aspect

In the section, we present a numerical example intended to illustrate the behaviour of the proposed method.The numerical example is constructed in the following way: first we select the initial datau0,y,t=gy,t and ux0,y,t=0. Then we added a random distributed perturbation to the data function obtaining vector gδy,t; i.e.,(41)gδy,t=gy,t+εrandnsizegy,t,where(42)gy,t=gy1,t1,gy2,t2,…,gyn,tnT,yi=-10+i-1Δy,Δy=20n-1,i=1,2,…,n,tj=-10+j-1Δt,Δt=20n-1,j=1,2,…,n.Then the total noise δ can be measured in the sense of Root Mean Square Error according to(43)δ≔gδy,t-gy,tl2=1n2∑i=1n∑j=1ngiδyi,tj-giyi,tj2.The function “randn·” generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1, and standard deviation σ=1; “randn(size (g(y,t)))” returns array of random entries that is of the same size as gy,t.Example 8.
We choose(44)gy,t=e-y2-t2.
Our tests about this example correspond to Figures1–3. The numerical results for u and uαδ with different error level ε=0.0001,0.001,0.01 at different point x=0.8,0.5 are shown in Figures 1-2. The numerical result for u and uαδ with error level ε=0.01 at different point x=0.9,0.6,0.1 is shown in Figure 3.
From Figures1-2, we can see that the smaller the ε is, the better the computed solution is. Figure 3 shows that the numerical result becomes worse when x approaches 1.Figure 1
The exact solution and its approximation solution atx=0.8: (a) the exact solution, (b) its approximation solution with ε=0.0001, (c) its approximation solution with ε=0.001, (d) its approximation solution with ε=0.01.
(a) (b) (c) (d)Figure 2
The exact solution and its approximation solution atx=0.5: (a) the exact solution, (b) its approximation solution with ε=0.0001, (c) its approximation solution with ε=0.001, (d) its approximation solution with ε=0.01.
(a) (b) (c) (d)Figure 3
The exact solution and its approximation solution: (a) the exact solution atx=0.9, (b) its approximation solution with ε=0.01 at x=0.9, (c) the exact solution at x=0.6, (d) its approximation solution with ε=0.01 at x=0.6, (e) the exact solution at x=0.1, (f) its approximation solution with ε=0.01 at x=0.1.
(a) (b) (c) (d) (e) (f)


## 5. Conclusion

In this paper, the Cauchy problem of 2D heat conduction equation is considered. We regularize it by a revised Tikhonov regularization method for overcoming its ill-posedness. The error estimate is obtained under an a priori regularization parameter choice rule. A numerical example shows that our proposed method is effective and stable.



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