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

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.


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][3][4][5][6][7][8][9][10].
In this paper, we investigate a Cauchy problem of twodimensional (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.
The paper is organized as follows.In Section 2 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.

Mathematical Formulation of the Cauchy Problem of 2D Heat Conduction Equation
In order to use the Fourier transform technique, we extend the functions (, , ), (, ),   (, ) to the whole real (, ) plane by defining them to be zero everywhere in {(, ),  < 0,  < 0}.We wish to determine the temperature (, , ) ∈  2 (R 2 ) for 0 <  ≤ 1 from the temperature measurement   (, ) ∈  2 (R 2 ).We also use the corresponding  2 -norm defined as We now could assume that the measured data function   (, ) satisfies where the constant  > 0 represents a bound on the measurement error.Assume that there exists a constant  > 0 such that the following a priori bound exists: Let (, )  −(+)  , ,  ∈ R, (7) be the Fourier transform of a function (, ).Then Applying this transform to (1) with respect to  and , we obtain û (, , ) = ( +  2 ) û (, , ) , which is a second-order ordinary differential equation for fixed  and .Now using the boundary condition in the frequency domain, we can easily get û (, , ) = cosh () ĝ (, ) , (10) and taking the inverse Fourier transform, the solution of problem 1 is where  is the principal value of √ +  2 ; i.e., where  = sign().We note here that, for fixed 0 <  ≤ 1, the value of cosh() in ( 10) is unbounded as || → ∞.We can see that small errors in the data can blow up and completely destroy the solution for 0 <  ≤ 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() by a bounded approximation kernel denoted by (, , ), which has the following two common properties: (I) If the parameter  is small, then for small , the kernel (, , ) is close to cosh().

Revised Tikhonov Regularization Method and Error Estimates
To get a bounded approximation kernel (, , ), a popular method is Tikhonov regularization, which is applied to solve the following minimization problem: where  is a regularization parameter and  : (⋅, , ) → (, ) is a linear bounded operator.
Lemma 2. There exists a unique solution to the above minimization problem.It is given by Proof.Let  denote identity operator in  2 and let  * be the adjoint of .Then, the unique solution of the minimization problem ( 13) is given by In order to obtain the explicit formula ( 14) from ( 15), we use Parseval's formula; one has According to K = (1/ cosh())û(, , ), we find Likewise, Consequently, Using (17) and solving for û  (, , ) in ( 19), we obtain Finally, (14) follows by an inverse Fourier transform.
In the following, we give the convergence estimate for ‖   (, , ) − (, , )‖ 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]).Let 0 <  < 2,  > 0. We have the following inequality: Theorem 4. Suppose  is the solution of problem 1 with the exact data  and    is the regularized solution with the noise data   , and let   satisfy (5) and the exact solution  at  = 1 satisfy (6).If  =  2 / 2 is selected, then for fixed 0 <  < 1, one gets the error estimate         (, , ) −  (, , )      ≤ 3 Combining ( 24), (25), and the condition  =  2 / 2 , we have 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 Theorem 4, we do not obtain the error estimate at  = 1; in order to obtain that, we should use a stronger priori assumption as follows: where ‖ ⋅ ‖  denotes the norm on Sobolev space   (R 2 ) defined by Therefore, we can derive that We can see that it goes to zero as  → 0.

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 data (0, , ) = (, ) and   (0, , ) = 0. Then we added a random distributed perturbation to the data function obtaining vector   (, ); i.e.,   (, ) =  (, ) +  randn (size ( (, ))) , Then the total noise  can be measured in the sense of Root Mean Square Error according to The function "randn(⋅)" generates arrays of random numbers whose elements are normally distributed with mean 0, variance  2 = 1, and standard deviation  = 1; "randn(size ((, )))" returns array of random entries that is of the same size as (, ).
Example 8. We choose  with error level  = 0.01 at different point  = 0.9, 0.6, 0.1 is shown in Figure 3.
From Figures 1-2, we can see that the smaller the  is, the better the computed solution is. Figure 3 shows that the numerical result becomes worse when  approaches 1.

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.

Figure 1 :
Figure 1: The exact solution and its approximation solution at  = 0.8: (a) the exact solution, (b) its approximation solution with  = 0.0001, (c) its approximation solution with  = 0.001, (d) its approximation solution with  = 0.01.
Suppose (1, , ) is the solution of problem 1 with the exact data  and    (1, , ) is the regularized solution with the noise data   , and let   satisfy (5) and the exact solution  at  = 1 satisfy (27).If  = / is selected, then for fixed  = 1, one gets the error estimate ⋅    û (1, , )     2  )