Advances in Mathematical Physics
Volume 2016 (2016), Article ID 9504829, 5 pages
http://dx.doi.org/10.1155/2016/9504829

Research Article

A Lipschitz Stability Estimate for the Inverse Source Problem and the Numerical Scheme

Xianzheng Jia

Shandong University of Technology, Shandong 255000, China

Received 4 May 2016; Accepted 27 June 2016

Academic Editor: Soheil Salahshour

Copyright © 2016 Xianzheng Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the inverse source problem for heat equation, where the source term has the form . We give a numerical algorithm to compute unknown source term . Also, we give a stability estimate in the case that is a piecewise constant function.

1. Introduction to the Problem

The inverse source problems in heat equation are solving the heat equation with unknown heat sources. They are well known to be ill-posed. There were many studies to identify different types of heat sources. For example, Cannon and DuChateau [1] estimated the nonlinear temperature dependent heat source. In [2, 3], the method of fundamental solutions has been presented for an inverse time-dependent heat source problem. In [4–10], several numerical schemes have been proposed to determine a space-dependent heat source. In [11–13], these numerical methods were considered to solve some two- and three-dimensional inverse heat source problems. In Savateev [14], Trong et al. [15], and Shidfar et al. [16], the authors considered the source as a function of both space and time, but it was additive or separable.

Throughout this paper, we will assume that is an open set in . We consider the following initial-boundary value problem:

Problems of the above equations include inverse problems, heat conduction processes, hydrology, material sciences, and heat transfer problems. In the context of heat conduction and diffusion, represents either temperature or concentration. In this system, is spatial density and unknown is interpreted as either a heat or a material source, respectively, in a chemical or a biochemical application.

When , it means that term is a source. We assume that is a given function and satisfies the following condition:

Our problem is to derive and a conditional stability in the determination of , from the observation

where .

2. Some Existing Results

To derive the stability estimate, we need the following inequalities, which can be found in [17]. Here, we just list the result.

Theorem 1. Let , , , and satisfy

Then,

In particular, for

and for , one has

In [17], by using the above theorem, a Hölder estimate was obtained for the case that and , where

Theorem 2. Let satisfy (2), and set to be defined as above definition.
Then, for arbitrarily given , there exists a constant such that

for any .

Corollary 3. In Theorem 2, if one replaces by

also the same result can be obtained.

But we are more interested in the case that is a piecewise constant function. In that case, can have infinity number of zeros. For that case, we obtained the similar result, and we will claim the result and the proof in the next section.

3. The Estimate

For problem (1), we can see that solution can be expressed in the following form:

where is the fundamental function for problem (1). See [18].

Also from [18], we can see that fundamental solution satisfies the following conditions:

Therefore, setting

we have

which is a Volterra integral equation of the first kind with respect to .

Here, we consider the case that .

Theorem 4 (uniqueness). For problem (1), given , then is uniquely determined except for finite point in .

Proof. Suppose that there are two functions satisfying (1). We assume that the discontinuous points of and are and , respectively, and and .
After some resorting procedure, we get points ; then, for ,

and , so we get .
For ,

therefore, .
Repeating the above procedure, we can get .
Thus, end the proof.

Suppose that changes its values at , and . Here, we assume that there exists a constant , such that .

Theorem 5. If satisfies (2), .
Then, for , there exists a constant such that

for any . Here, is the number of intervals in that .

Proof. Since is positive and bounded and (13), we can see that

Choosing norm from to , we obtain

Thus, we can see that

where is a constant.
For , we can see that

Then, we can obtain

Then

Therefore, in the case of and (20), we get

Continuing this argument until , we get the proof of this theorem.

4. Numerical Scheme

For problem (1), if is given, we need to reconstruct function .

Consider an auxiliary problem:

Actually, we can get the solution of problem (25) and suppose that is the solution. Then,

Given data , where , then, for every , we have

We suppose that ; then

Thus, we can use the above formula to get the discrete data for function .

Example 6. We choose as the following function:

Choose , , , , and outside interval . Given data . Then, we can get the numerical approximation of using the following formula:

And the numerical result is as shown in Figure 1.

Figure 1: Numerical result.

Figure 1 is the numerical result and Figure 2 is the exact function.

Figure 2: Exact solution.

Figure 3 shows the error between the numerical result and exact solution.

Figure 3: The error between numerical result and exact solution.

Example 7. Now, we will give a numerical result for the 2D case. Suppose the problem is also (1), and the domain , and

Let and , and the numerical result is as shown in Figure 4.

Figure 4: Numerical result.

Figure 4 is the numerical result and Figure 5 is the exact function.

Figure 5: Exact solution.

Figure 6 is the error between exact and the numerical result.

Figure 6: The error between numerical result and exact solution.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China Grant no. 11371231 and Grant no. 11071148 and the Natural Science Foundation of Shandong Province, China, Grant no. ZR2011AQ014.

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