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

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.


Introduction
In the process of transportation, diffusion, and conduction of natural materials, the following heat equation is induced: 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 illposed.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.

Formulation of the Inverse Problem
Let T > 0, α ∈ 0, 1 be fixed numbers, and let us consider first the one-dimensional timedependent 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, subject to the initial and boundary conditions and the overspecified condition where x 0 ∈ 0, 1 is the interior location of a thermocouple recording the temperature measurement, u 0 x , g 0 t , g 1 t , h t are given functions satisfying the compatibility conditions and the functions u x, t and f t are unknown.
The model problem presented here used to describe a heat transfer process with a time-dependent source produces the temperature at a given point x 0 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 x 0 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: Equation 2.1 is transformed into the following equation: with the initial and boundary conditions and the overspecification at a point x 0 : 2.9

Analysis and Application of He's Variational Iteration Method
Consider the differential equation 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 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 δ u 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 u 0 .Consequently, the solution is given by 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: where λ is the general Lagrangian multiplier, which can be identified optimally via variational theory, and v n is a restricted variation which means δ v n 0.
To find the optimal value of λ, we have

3.7
Thus we have and therefore, we have the following iteration formula: 3.9 Now taking v 0 x, t v x, 0 u 0 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: 3.10

Numerical Examples
In this section, we present and discuss the numerical results by  Beginning with 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 and the n-order approximation of u x, t by u n x, t v n x, t g 0 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.
When the input dataware is noisy, with various levels of noise σ 0.1%, σ 1%, σ 5%, the relative errors of f 5 and u 5 are given in Figures 1 and 2 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 f 5 and u 5 are given in Figures 3 and 4  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.

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 timedependent heat source problem.

Table 1 :
Maximum absolute errors of f n and u n with exact input data for Example 4.1.
employing VIM for two test examples.For these examples, we have taken a 1 and T 1. Results demonstrate present method is remarkably effective.

:
. The figure of relative error | u − u 5 /u| with noise σ 0.1%, 1%, 5%, respectively, for Example 4.1.Figures of f t and f 1 t with noise σ 0.1%, 1%, 5%, respectively, for Example 4.2 the real line is f t and the dashed line is f 1 t .
c Figure 3