Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation

Abstract

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions. Firstly, the problem is reduced to an equivalent problem which is easy to handle using variational iteration method. Secondly, variational iteration method is used to solve the reduced problem. Using this method a rapid convergent sequence can be produced which tends to the exact solution of the problem. Furthermore, variational iteration method does not require the discretization of the problem. Two numerical examples are presented to illustrate the strength of the method.

1. Introduction

In the process of transportation, diffusion, and conduction of natural materials, the following heat equation is induced: 𝑢_𝑡−𝑎²Δ𝑢=𝑓(𝑥,𝑡;𝑢),(𝑥,𝑡)∈Ω×(0,𝑇], (1.1) where 𝑢 represents state variable, 𝑎 is the diffusion coefficient, Ω is a bounded domain in 𝑅^𝑑, and 𝑓 denotes physical laws, which means source terms here. There are many researches on such inverse problems of determining source terms from 1970s, since the characteristics of sources in practical problems are always unknown. And the inverse problems are unstable in nature from indirect observable data which contain measurement errors. The major difficulty in establishing any numerical algorithm for approximating the solution is the ill-posedness of the problem and the ill-conditioning of the resultant discretized matrix. The inverse problem of determining an unknown heat-source function in the heat conduction equation has been considered in many papers [1–8]. For 𝑓=𝑓(𝑢), the inverse source problem with additional data was studied by Cannon, Duchateau, and Fatullayev [1, 2]. In [3, 4], the source is sought as a function of both space and time but is additive or separable. However, in all the other studies the source has been sought as a function of space or time only [5–8].

In this paper, the heat source is taken to be time dependent only, and the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable; hence the problem is ill-posed. The inverse problems are formulated in Section 2. Several numerical methods have been proposed for the inverse source problem [5–12]. In this work, we extend the use of variational iteration method (VIM) to this inverse source problem. The VIM was proposed originally by He [13, 14]. This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a functional. This method gives rapidly convergent successive approximations of the exact solution if such a solution exists. For concrete problems, a few number of approximations can be used for numerical purposes with a high degree of accuracy. Furthermore, VIM does not require the discretization of the problem. Thus the variational iteration method is suitable for finding the approximation of the solution without discretization of the problem. It was successfully applied to two-point boundary value problems, partial differential equations, evolution equations, and other fields [13–26].

The rest of the paper is organized as follows. In the next section, we formulate the problem mathematically. The VIM is introduced and applied to the inverse source problem in Section 3. The numerical examples are present in Section 4. Section 5 ends this paper with a brief conclusion.

2. Formulation of the Inverse Problem

Let 𝑇>0, 𝛼∈(0,1) be fixed numbers, and let us consider first the one-dimensional time-dependent problem in which the source 𝑓(𝑥,𝑡;𝑢)=𝑓(𝑡) depends on time only.

Find the temperature 𝑢∈𝐻^{2+𝛼,1+𝛼/2}([0,1]×[0,𝑇]) and the heat source 𝑓∈𝐻^𝛼/2([0,𝑇]) which satisfy the heat-conduction equation with a time-dependent source, namely, 𝑢_𝑡=𝑎²𝑢_𝑥𝑥+𝑓(𝑡),(𝑥,𝑡)∈(0,1)×(0,𝑇], (2.1) subject to the initial and boundary conditions 𝑢(𝑥,0)=𝑢₀(𝑥)𝑢(),0≤𝑥≤1,0,𝑡=𝑔₀(𝑡),𝑢(1,𝑡)=𝑔₁(𝑡),0≤𝑡≤𝑇 (2.2)(2.3) and the overspecified condition 𝑢𝑥₀,𝑡=ℎ(𝑡),0≤𝑡≤𝑇, (2.4) where 𝑥₀∈(0,1) is the interior location of a thermocouple recording the temperature measurement, 𝑢₀(𝑥),  𝑔₀(𝑡),  𝑔₁(𝑡),  ℎ(𝑡) are given functions satisfying the compatibility conditions 𝑢₀(0)=𝑔₀(0),𝑢₀(1)=𝑔₁(0),𝑢₀𝑥₀=ℎ(0), (2.5) and the functions 𝑢(𝑥,𝑡) and 𝑓(𝑡) are unknown.

From [5], we know that, if 𝑔₀, 𝑔₁, ℎ∈𝐻^1+𝛼/2([0,𝑇]), 𝑢₀∈𝐻^2+𝛼([0,1]) and conditions (2.2)–(2.4) are consistent up to the first order, then problem (2.1)–(2.4) has a unique solution.

The model problem presented here used to describe a heat transfer process with a time-dependent source produces the temperature at a given point 𝑥₀ in the spatial domain at time 𝑡. Thus, the purpose of solving this inverse problem can be viewed as an inverse control problem to identify the source control parameter that produces at any given time a desired temperature at a given point 𝑥₀ in the spatial domain.

Although sufficient conditions for the solvability of the problem are provided [5], problem (2.1)–(2.5) is still ill-posed since small errors, inherently present in any practical measurement, give rise to unbounded and highly oscillatory solutions. We will change (2.1) to an equation with one unknown function which is easy to handle using VIM.

Using the two following transformations: 𝑟(𝑡)=^𝑡₀𝑓(𝜉)𝑑𝜉,𝑣(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑟(𝑡). (2.6) Equation (2.1) is transformed into the following equation: 𝑣_𝑡=𝑎²𝑣_𝑥𝑥,(𝑥,𝑡)∈(0,1)×(0,𝑇], (2.7) with the initial and boundary conditions 𝑣(𝑥,0)=𝑢₀(𝑥)𝑣(),0≤𝑥≤1,0,𝑡=𝑔₀(𝑡)−𝑟(𝑡),𝑣(1,𝑡)=𝑔₁(𝑡)−𝑟(𝑡),0≤𝑡≤𝑇 (2.8) and the overspecification at a point 𝑥₀: 𝑣𝑥₀,𝑡=ℎ(𝑡)−𝑟(𝑡),0≤𝑡≤𝑇. (2.9)

3. Analysis and Application of He’s Variational Iteration Method

Consider the differential equation 𝐿𝑢+𝑁𝑢=𝑔(𝑥), (3.1) where 𝐿 and 𝑁 are linear and nonlinear operators, respectively, and 𝑔(𝑥) is the source inhomogeneous term. In [9–15], the VIM was introduced by He where a correct functional for (3.1) can be written as 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀𝜆𝐿𝑢_𝑛(𝑡)+𝑁𝑢_𝑛(𝑡)−𝑔(𝑡)𝑑𝑡, (3.2) where 𝜆 is a general Lagrangian multiplier [13], which can be identified optimally via variational theory, and 𝑢_𝑛 is a restricted variation which means 𝛿𝑢_𝑛=0. By this method, it is required first to determine the Lagrangian multiplier 𝜆 that will be identified optimally. The successive approximates 𝑢_𝑛+1,𝑛≥0, of the solution 𝑢 will be readily obtained upon using the determined Lagrangian multiplier and any selective function 𝑢₀. Consequently, the solution is given by 𝑢=lim_𝑛→∞𝑢_𝑛. (3.3)

The variational iteration method has been shown to solve easily and accurately a large class of problems with approximations converging rapidly to accurate solutions.

For variational iteration method, the key is the identification of Lagrangian multiplier. For linear source, their exact solutions can be obtained by only one iteration step due to the fact that the Lagrangian multiplier can be identified exactly. For nonlinear source, the lagrange multiplier is difficult to be identified exactly. To overcome the difficulty, we apply restricted variations to nonlinear term. Due to the approximate identification of the Lagrangian multiplier, the approximate solutions converge to their exact solutions relatively slowly. It should be specially pointed out that the more accurate the identification of the multiplier, the faster the approximations converge to their exact solutions.

For (2.7), according to the VIM, we consider its correct functional in 𝑡-direction in the following form: 𝑣_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+^𝑡₀𝜆𝜕𝑣_𝑛(𝑥,𝑠)𝜕𝑠−𝑎²𝜕²𝑣_𝑛(𝑥,𝑠)𝜕𝑥²𝑑𝑠, (3.4) where 𝜆 is the general Lagrangian multiplier, which can be identified optimally via variational theory, and 𝑣_𝑛 is a restricted variation which means 𝛿𝑣_𝑛=0.

To find the optimal value of 𝜆, we have 𝛿𝑣_𝑛+1(𝑥,𝑡)=𝛿𝑣_𝑛(𝑥,𝑡)+𝛿^𝑡₀𝜆𝜕𝑣_𝑛(𝑥,𝑠)𝜕𝑠−𝑎²𝜕²𝑣_𝑛(𝑥,𝑠)𝜕𝑥²𝑑𝑠=0, (3.5) that results 𝛿𝑣_𝑛+1(𝑥,𝑡)=𝛿𝑣_𝑛(𝑥,𝑡)(1+𝜆)−^𝑡₀𝛿𝑣_𝑛(𝑥,𝑠)𝜆^′𝑑𝑠=0, (3.6) which yields 𝜆^′(𝑡)(𝑡)=0,1+𝜆=0. (3.7) Thus we have 𝜆(𝑡)=−1, (3.8) and therefore, we have the following iteration formula: 𝑣_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)−^𝑡₀𝜕𝑣_𝑛(𝑥,𝑠)𝜕𝑠−𝑎²𝜕²𝑣_𝑛(𝑥,𝑠)𝜕𝑥²𝑑𝑠. (3.9)

Now taking 𝑣₀(𝑥,𝑡)=𝑣(𝑥,0)=𝑢₀(𝑥) as an initial value and using (3.9), we can obtain the solution of (2.7) as a convergent sequence. The solution of (2.1) is obtained in the following form: 𝑓(𝑡)=𝑔₀(𝑡)−𝑣(0,𝑡)^′,𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)+𝑟(𝑡)=𝑣(𝑥,𝑡)+𝑔₀(𝑡)−𝑣(0,𝑡). (3.10)

4. Numerical Examples

In this section, we present and discuss the numerical results by employing VIM for two test examples. For these examples, we have taken 𝑎=1 and 𝑇=1. Results demonstrate present method is remarkably effective.

Example 4.1. With the input data 𝑢(𝑥,0)=𝑢₀(𝑡)(𝑥)+1=sin4𝑥⁴,𝑢(0,𝑡)=𝑔₀(𝑡)=0,𝑢(1,𝑡)=𝑔₁(𝑡)=𝑒^−𝑡(1)1sin+3𝑡+4,𝑢𝑥₀,𝑡=ℎ(𝑡)=𝑒^−𝑡1sin2+341𝑡+,64 (4.1) the inverse problem (2.1)–(2.5) has the unique solution given by 𝑢(𝑥,𝑡)=𝑒^−𝑡sin(𝑥)+3𝑡𝑥²+14𝑥⁴,𝑓(𝑡)=−6𝑡. (4.2)

Beginning with 𝑣₀()()(𝑥)+1𝑥,𝑡=𝑣𝑥,0=sin4𝑥⁴, (4.3) according to (3.9), one can obtain the successive approximations 𝑣_𝑛(𝑥,𝑡) of 𝑣(𝑥,𝑡).

From (3.10), one can obtain the 𝑛-order approximation of 𝑓(𝑥) by 𝑓_𝑛(𝑡)=𝑔₀(𝑡)−𝑣_𝑛(0,𝑡)^′ (4.4) and the 𝑛-order approximation of 𝑢(𝑥,𝑡) by 𝑢_𝑛(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+𝑔₀(𝑡)−𝑣_𝑛(0,𝑡). (4.5)

When the input dataware is exact, the maximum absolute errors of 𝑓_𝑛 and 𝑢_𝑛 are given in Table 1.

Table 1 

Maximum absolute errors of 𝑓_𝑛 and 𝑢_𝑛 with exact input data for Example 4.1.

When the input dataware is noisy, with various levels of noise 𝜎=0.1%, 𝜎=1%, 𝜎=5%, the relative errors of 𝑓₅ and 𝑢₅ are given in Figures 1 and 2.

(a)

(b)

(c)

Figure 1 

The figure of relative error |(𝑓−𝑓5)/𝑓| with noise 𝜎=0.1%, 1%, 5%, respectively, for Example 4.1.

(a)

(b)

(c)

Figure 2 

The figure of relative error |(𝑢−𝑢5)/𝑢| with noise 𝜎=0.1%, 1%, 5%, respectively, for Example 4.1.

Example 4.2. With the input data 𝑢(𝑥,0)=𝑢₀(𝑡)=𝑥²,𝑢(0,𝑡)=𝑔₀(𝑡)=2𝑡+sin(2𝜋𝑡),𝑢(1,𝑡)=𝑔₁(𝑡)(),𝑢𝑥=1+2𝑡+sin2𝜋𝑡₀(𝑡)=1,𝑡=ℎ4+2𝑡+sin(2𝜋𝑡), (4.6) the inverse problem (2.1)–(2.5) has the unique solution given by 𝑢(𝑥,𝑡)=𝑥²(),𝑓(𝑡)().+2𝑡+sin2𝜋𝑡=2𝜋cos2𝜋𝑡 (4.7)

Beginning with 𝑣₀(𝑥,𝑡)=𝑣(𝑥,0)=𝑥², (4.8) according to (3.9), one can obtain the successive approximations 𝑣_𝑛(𝑥,𝑡) of 𝑣(𝑥,𝑡).

From (3.10), one can obtain the 𝑛-order approximation of 𝑓(𝑥) by 𝑓_𝑛(𝑡)=𝑔₀(𝑡)−𝑣_𝑛(0,𝑡)^′ (4.9) and the 𝑛-order approximation of 𝑢(𝑥,𝑡) by 𝑢_𝑛(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+𝑔₀(𝑡)−𝑣_𝑛(0,𝑡). (4.10)

When the input dataware is exact, the maximum absolute errors of 𝑓_𝑛 and 𝑢_𝑛 are zero.

When the input dataware is noisy, with various levels of noise 𝜎=0.1%, 𝜎=1%, 𝜎=5%, the relative errors of 𝑓₅ and 𝑢₅ are given in Figures 3 and 4.

(a)

(b)

(c)

Figure 3 

Figures of 𝑓(𝑡) and 𝑓1(𝑡) with noise 𝜎=0.1%, 1%, 5%, respectively, for Example 4.2 (the real line is 𝑓(𝑡) and the dashed line is 𝑓1(𝑡)).

(a)

(b)

(c)

Figure 4 

The figure of relative error |(𝑢−𝑢1)/𝑢| with noise 𝜎=0.1%, 1%, 5%, respectively, for Example 4.2.

Remark 4.3. From the previous two numerical examples, it can be seen that the numerical results are quite satisfactory. When the input data are exact, the numerical solutions obtained using our method are of high degree of accuracy. Even with the noise level of input data up to 𝜎=5%, the numerical solutions are still in good agreement with the exact solutions.

5. Conclusion

In this paper, He’s variational iteration method was employed successfully for solving the inverse heat source problems. This method solves the problem without any discretization of the variables. Therefore, it is not affected by computation round-off errors, and one is not faced with necessity of large computer memory and time. The numerical results show that the VIM is an accurate and reliable numerical technique for the solution of the inverse time-dependent heat source problem.

Acknowledgments

The work of the author is supported by the Special Funds of the National Natural Science Foundation of China (Grant no. 11141003). The author would like to thank the referees for constructive suggestions and comments.