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In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a priori bound of exact solution are not needed for the a posteriori choice rule. Numerical results are presented to show the stability and effectiveness of the method.

A variety of important problems in science and engineering involve the solution to the modified Helmholtz equation, e.g., in implicit marching schemes for the heat equation, in Debye-Huckel theory, and in the linearization of the Poisson-Boltzmann equation [

where

problem is called the inverse source problem. In practice, the data at

where

Inverse source problems arise in many branches of science and engineering, e.g., heat conduction, crack identification electromagnetic theory, geophysical prospecting and pollutant detection. The main difficulty of these problems is that they are ill-posed (the solution, if it exists, does not depend continuously on the data). Thus, the numerical simulation is very difficult and some special regularization is required. Many papers have presented the mathematical analysis and efficient algorithms of these problems. The uniqueness and conditional stability results for these problems can be found in [

Up to now, only a few papers for identifying the unknown source on the modified Helmholtz equation have been reported. In [

This paper is organized as follows. In Section 2, we will give the method to construct approximate solution. The choices of the regularization parameter and corresponding convergence results will be found in Section 3. Some numerical results are given in Section 4 to show the effectiveness of the new method.

Let

where

It is easy to derive a solution of problem (1) by the method of separation of variables [

where

Note that the exact data

where

where

We let

where

If we let

So we have

Which means that

Then the approximate solution can be given as

Lemma 1. For any

Lemma 2. [

Lemma 3.

where

Proof.

The proposition follows by applying (16) with b replaced by

Lemma 4.

where

Proof. With the representation

and Lemma 1, we have

In this section, we consider the choices of the regularization parameter. An a priori strategy and an a posteriori choice rule will be given. Under each choice of the regularization parameter, the convergence estimate can be obtained.

Take

we can obtain the following theorem.

Theorem 5. If (2) holds and (7) holds with

Proof. With Lemma 3, Lemma 4 and (23) we obtain

Moreover, by using Hölder inequality, we have

Formulae (8) implies that

The assertion of the Lemma follows from (25)-(27).

For any

It is apparent that the function

So we can get the following lemma

Lemma 6. Let g,

for some

In the following, we denote the unique

Lemma 7. Let g,

a)

b)

where

Proof.

a) Let

then

b)

then from Lemma 1

The rest follows from a).

Theorem 8. Suppose that the conditions (2) and (30) hold, the condition (7) hold with

Proof. By using the triangle inequality we know

So, in terms of Equations (17), (19) and (33), we have

From (26),

Combining (41) and (42), we obtain

The assertion of the theorem follows from (27).

In this section, we present some numerical tests to check the effectiveness of the method. The discretization knots are

where

in practical computing. The relative errors are measured by the weighted

All tests are computed by using Matlab and we will also compare the method (M1) with the method in [

where

Example [

In

We have proposed a new method to identify the unknown source in the modified Helmholtz equation. Theoretical analysis as well as experience from computations indicates that the proposed method works well.

N = 32 | N = 64 | N = 128 | N = 32 | N = 64 | N = 128 | N = 32 | N = 64 | N = 128 | N = 32 | N = 64 | N = 128 | |

1e−1 | 0.0765 | 0.0862 | 0.0746 | 0.0611 | 0.0423 | 0.0215 | 0.0501 | 0.0392 | 0.0128 | 0.0471 | 0.0382 | 0.0099 |

1e−2 | 0.0213 | 0.0203 | 0.0211 | 0.0078 | 0.0061 | 0.0043 | 0.0062 | 0.0040 | 0.0021 | 0.0054 | 0.0039 | 0.0011 |

1e−3 | 0.0092 | 0.0090 | 0.0072 | 0.0009 | 0.0008 | 0.0007 | 0.0008 | 0.0005 | 0.0005 | 0.0006 | 0.0004 | 0.0001 |

M1 | M2 | M1 | M2 | M1 | M2 | M1 | M2 | |

1e−1 | 0.0746 | 0.1131 | 0.0215 | 0.0375 | 0.0128 | 0.0413 | 0.0099 | 0.0578 |

1e−2 | 0.0211 | 0.0584 | 0.0043 | 0.0184 | 0.0021 | 0.0194 | 0.0011 | 0.0233 |

1e−3 | 0.0072 | 0.0612 | 0.0007 | 0.0047 | 0.0005 | 0.0082 | 0.0001 | 0.0103 |

The project is supported by the National Natural Science Foundation of China (No. 11201085).

LeiYou,ZhiLi,JuangHuang,AihuaDu, (2016) The Tikhonov Regularization Method in Hilbert Scales for Determining the Unknown Source for the Modified Helmholtz Equation. Journal of Applied Mathematics and Physics,04,140-148. doi: 10.4236/jamp.2016.41017