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.

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. with corresponding measured data function ,We wish to determine the temperature for from temperature measurement . 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 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.

2. 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 . We wish to determine the temperature for from the temperature measurement .

We also use the corresponding -norm defined asWe now could assume that the measured data function satisfieswhere the constant represents a bound on the measurement error. Assume that there exists a constant such that the following a priori bound exists:

Letbe the Fourier transform of a function . ThenApplying this transform to (1) with respect to and , we obtainwhich is a second-order ordinary differential equation for fixed and . Now using the boundary condition in the frequency domain, we can easily getand taking the inverse Fourier transform, the solution of problem 1 iswhere is the principal value of ; i.e.,where .

We note here that, for fixed , the value of in (10) is unbounded as . We can see that small errors in the data can blow up and completely destroy the solution for . 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” 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 .(II)If is fixed, is bounded.

3. 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 and let be the adjoint of . Then, the unique solution of the minimization problem (13) is given byIn order to obtain the explicit formula (14) from (15), we use Parseval’s formula; one hasAccording to , we findLikewise,Consequently,Using (17) and solving for in (19), we obtainFinally, (14) follows by an inverse Fourier transform.

From Lemma 2, we can see that the kernel function corresponds to the Tikhonov regularization method.

In the following, we use a much better filter to replace the original ; i.e., . It corresponds to the regularized solution as follows:and is regarded as a revised Tikhonov regularization solution of of problem 1.

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 . 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 satisfy (6). If is selected, then for fixed , one gets the error estimate

Proof. Using the Parseval’s identity and Proposition 3, we haveCombining (24), (25), and the condition , 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 ; in order to obtain that, we should use a stronger priori assumption as follows:where denotes the norm on Sobolev space defined by

Theorem 7. 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 satisfy (27). If is selected, then for fixed , one gets the error estimate

Proof. Similar to the proof of Theorem 4, we know thatNow, we distinguish two cases to estimate (31).
Case 1. For , we haveTherefore, we can derive thatWe can see that it goes to zero as .
Case 2. For , we haveIf , we havei.e.,. Therefore,If , we havei.e., . Therefore,Combining (31), (33), (36), and (38), we obtainNow using the triangle inequality and combining (30) and (39), we can easily get the convergence estimate