The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

Abstract

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

1. Introduction

Inverse source problems arise in many branches of science and engineering, for example, heat conduction, crack identification, electromagnetic theory, geophysical prospecting, and pollutant detection. For the heat source identification, there has been a large number of research results for different forms of heat source [1–6]. To the author's knowledge, there were few papers for identifying the unknown source on the modified Helmholtz equation which is pointed out in [7] by regularization method.

In this paper, we consider the following inverse problem: to find a pair of functions (𝑢(𝑥,𝑦),𝑓(𝑥)) satisfyingΔ𝑢𝑥,𝑦−𝑘²𝑢(𝑥)𝑢𝑢()𝑢|𝑥,𝑦=𝑓,0<𝑥<𝜋,0<𝑦<+∞,0,𝑦=𝑢𝜋,𝑦=0,0≤𝑦<+∞,𝑥,0=0,0≤𝑥≤𝜋,𝑥,𝑦_𝑦→∞𝑢()(𝑥)bounded,0≤𝑥≤𝜋,𝑥,1=𝑔,0≤𝑥≤𝜋, (1.1) where 𝑓(𝑥) is the unknown source depending only on one spatial variable, 𝑢(𝑥,1)=𝑔(𝑥) is the supplementary condition, and the constant 𝑘>0 is the wave number. In applications, input data 𝑔(𝑥) can only be measured, and there will be measured data function 𝑔^𝛿(𝑥) which is merely in 𝐿²(0,𝜋) and satisfies𝑔−𝑔^𝛿_{𝐿2(0,𝜋)}≤𝛿, (1.2) where the constant 𝛿>0 represents a noise level of input data.

It is easy to derive a solution of problem (1.1) by the method of separation of variables𝑢𝑥,𝑦=−^∞_𝑛=11−𝑒^{−𝑛2+𝑘2𝑦}𝑛²+𝑘²𝑓_𝑛𝑋_𝑛, (1.3) where𝑋_𝑛=2𝜋sin𝑛𝑥,(𝑛=1,2…) (1.4) is an orthogonal basis in 𝐿²(0,𝜋), and𝑓_𝑛=2𝜋^𝜋₀𝑓(𝑥)sin𝑛𝑥𝑑𝑥. (1.5) By the supplementary condition, we define the operator 𝐾∶𝑓→𝑔, then we have𝑔(𝑥)=𝐾𝑓(𝑥)=−^∞_𝑛=11−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²𝑓_𝑛𝑋_𝑛. (1.6) It is easy to see that 𝐾 is a linear compact operator and the singular values {𝜎_𝑛}^𝑛=∞_𝑛=1 of 𝐾 satisfy𝜎_𝑛=−1−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²,𝑔_𝑛=−1−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²𝑓_𝑛𝑋_𝑛,𝑋_𝑛, (1.7) that is,𝑓_𝑛=𝜎_𝑛⁻¹𝑔_𝑛, (1.8) where𝑔_𝑛=2𝜋^𝜋₀𝑔(𝑥)sin𝑛𝑥𝑑𝑥. (1.9) Therefore,𝑓(𝑥)=𝐾⁻¹𝑔(𝑥)=^∞_𝑛=11𝜎_𝑛𝑔,𝑋_𝑛𝑋_𝑛=−^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}𝑔_𝑛𝑋_𝑛. (1.10)

Note that 1/𝜎_𝑛=𝑂(𝑛²) as 𝑛→∞, thus the exact data function 𝑔(𝑥) must satisfy the property that (𝑔,𝑋_𝑛) decays rapidly as 𝑂(𝑛⁻²). As for the measured data function 𝑔^𝛿(𝑥) is only in 𝐿²(0,𝜋), we cannot expect the coefficient 𝑔^𝛿_𝑛 of 𝑔^𝛿(𝑥) has the same decay rate. Thus, the problem (1.1) is ill posed. It is impossible to gain the unknown source using classical methods. In the following sections, we will use the simplified Tikhonov method to deal with the ill posed problem. Before doing that, we impose an a priori bound on the unknown source; that is,𝑓(⋅)_{𝐻𝑝(0,𝜋)}≤𝐸,𝑝>0, (1.11) where 𝐸>0 is a constant and ‖⋅‖_𝐻^𝑝_(0,𝜋) denotes the norm in Sobolev space which is defined by [8] as follows:𝑓(⋅)_{𝐻𝑝(0,𝜋)}=^∞_𝑛=11+𝑛^2𝑝𝑓,𝑋_𝑛^21/2. (1.12)

The simplified Tikhonov regularization method was based on the Tikhonov regularization method. Skillfully simplifying the filter gained by the Tikhonov regularization, a better regularization approximation solution of the inverse problem was obtained. This idea initially came from Carasso, the author who modified the filter gained by the Tikhonov regularization method and obtained the order optimal error estimate in [9]. By this method, Fu [10] considered the inverse heat conduction problem on a general sideways parabolic equation, and Cheng et al. [11, 12] considered the spherically symmetric inverse problem.

This paper is organized as follows. Section 2 gives some auxiliary results. Section 3 gives a simplified Tikhonov regularization solution and error estimation. Section 4 gives two examples to illustrate the accuracy and efficiency of this methods. Section 5 puts an end to this paper with a brief conclusion.

2. Some Auxiliary Results

Now, we give some important Lemmas, which are very useful for our main conclusion.

Lemma 2.1. For 𝑛≥1 and 𝑘 is a positive constant, there holds 11−𝑒^{−𝑛2+𝑘2}≤2. (2.1)

Lemma 2.2. For 0<𝛼<1, there holds the following inequalities: sup_𝑛≥111−1+𝛼²𝑛⁴1+𝑛^2−𝑝/2≤max𝛼²,𝛼^𝑝/2,sup_𝑛≥1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛⁴≤2𝛼+2𝑘². (2.2)

Proof. Let 𝐺(𝑛)1∶=1−1+𝛼²𝑛⁴1+𝑛^2−𝑝/2. (2.3) The proof of (2.2) can be separated from two cases.Case 1. For large values of 𝑛, that is, 𝑛≥𝑛₀∶=1/√𝛼, we get 𝐺(𝑛)≤1+𝑛^2−𝑝/2≤𝑛^−𝑝≤𝑛₀^−𝑝=𝛼^𝑝/2. (2.4)Case 2. 1≤𝑛<𝑛₀, we obtain 𝐺(𝑛)=𝛼²𝑛⁴1+𝛼²𝑛⁴1+𝑛^2−𝑝/2≤𝛼²𝑛⁴1+𝑛^2−𝑝/2≤𝛼²𝑛^4−𝑝. (2.5) If 0<𝑝≤4, above inequality becomes into 𝐺(𝑛)≤𝛼²𝑛^4−𝑝<𝛼²𝑛₀^4−𝑝=𝛼^𝑝/2. (2.6) If 𝑝>4, we get 𝐺(𝑛)≤𝛼²𝑛^4−𝑝≤𝛼². (2.7)Combining (2.4) with (2.6) and (2.7), the first inequality equation is obtained.Let 𝐵(𝑛)𝑛∶=²+𝑘²1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛⁴=𝑛²1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛⁴+𝑘²1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛⁴∶=𝐻(𝑛)+𝐽(𝑛). (2.8) Using Lemma 1, we obtain 𝐻(𝑛)≤2𝑛²1+𝛼²𝑛⁴. (2.9) Let 𝐿(𝑛)∶=2𝑛²1+𝛼²𝑛⁴, (2.10) then 𝐿^′(𝑛)=4𝑛1−𝛼²𝑛⁴1+𝛼²𝑛⁴². (2.11) Setting 𝐿^′(𝑛)=0, we can obtain 𝑛₁=1/√𝛼. It is easy to see that 𝑛₁=1/√𝛼 is a unique maximal value point of 𝐿(𝑛). So, 𝐿(𝑛)≤2𝑛²₁1+𝛼²𝑛²₁≤2𝑛²₁=2𝛼,𝐽(𝑛)≤𝑘²1−𝑒^{−𝑛2+𝑘2}≤2𝑘². (2.12) So, we get 𝐵(𝑛)≤2𝛼+2𝑘². (2.13) This completes the proof.

3. A Simplified Tikhonov Regularization Method

Since problem (1.1) is an ill-posed problem, we give an approximate solution of 𝑓(𝑥) by a Tikhonov regularization method which minimizes the quantity𝐾𝑓^𝛿−𝑔^𝛿2+𝛼²𝑓^𝛿2. (3.1) Then, by Theorem 2.12 in [8], the unique solution of the minimization problem (3.1) is equal to solve the following normal equation:𝐾^∗𝐾𝑓^𝛿(𝑥)+𝛼²𝑓^𝛿(𝑥)=𝐾^∗𝑔^𝛿(𝑥), (3.2) that is,𝑓^𝛿(𝑥)=𝐾^∗𝐾+𝛼²𝐼⁻¹𝐾^∗𝑔^𝛿(𝑥). (3.3) Because 𝐾 is a linear self-adjoint compact operator, that is, 𝐾^∗=𝐾, we have the equivalent form𝑓^𝛿(𝑥)=𝐾²+𝛼²𝐼⁻¹𝐾𝑔^𝛿(𝑥). (3.4) We define function of a compact self-adjoint operator 𝐾 by the spectral mapping theorem in the following way.

Definition 3.1 (see[13]). If 𝑓(𝑥) is a real-valued continuous function on the spectrum 𝜎(𝐾), we define 𝑓(𝐾) by𝑓(𝐾)𝑥=_𝑛𝑓𝜆_𝑛𝑥,𝜔_𝑛𝜔_𝑛, (3.5) where 𝐾 is a compact self-adjoint, 𝜆_𝑛∈𝜎(𝐾), and 𝜔_𝑛 are the corresponding orthogonal eigenvectors.

So, we obtain𝑓^𝛿(𝑥)=−^∞_𝑛=11−𝑒^{−𝑛2+𝑘2}/𝑛²+𝑘²𝛼²+1−𝑒^{−𝑛2+𝑘2}/𝑛²+𝑘²²𝑔^𝛿(𝑥),𝑋_𝑛𝑋_𝑛=−^∞_𝑛=1𝑛²+𝑘²/1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛²+𝑘²/1−𝑒^{−𝑛2+𝑘22}𝑔^𝛿(𝑥),𝑋_𝑛𝑋_𝑛=−^∞_𝑛=1𝑛²+𝑘²/1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛²+𝑘²/1−𝑒^{−𝑛2+𝑘22}𝑔^𝛿_𝑛𝑋_𝑛. (3.6)

Comparing (1.10) with (3.6), we can find that the procedure consists in replacing the unknown 𝑔(𝑥) with an appropriately filtered noised data 𝑔^𝛿(𝑥). The filter in (3.6) attenuates the coefficient 𝑔^𝛿_𝑛 of 𝑔^𝛿(𝑥) in a manner consistent with the goal of minimizing quantity (3.1). By this idea, we can use a much better filter 1/(1+𝛼²𝑛⁴) to replace the filter 1/(1+𝛼²((𝑛²+𝑘²)/(1−𝑒^{−𝑛2+𝑘2}))²) and give another approximation 𝑓^𝛿_𝛼(𝑥) of the solution 𝑓(𝑥).

We define a regularization approximate solution of problem (1.1) for noisy data 𝑔^𝛿(𝑥) which is called the simplified Tikhonov regularized solution of problem (1.1) as follows:𝑓^𝛿_𝛼(𝑥)∶=−^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}1+𝛼²𝑛⁴𝑔^𝛿,𝑋_𝑛𝑋_𝑛. (3.7)

Theorem 3.2. Let 𝑓^𝛿_𝛼(𝑥) be the simplified Tikhonov approximation of the solution 𝑓(𝑥) of problem (1.1). Let 𝑔^𝛿(𝑥) be measured data at 𝑦=1 satisfying (1.2), and let priori condition (1.11) hold for 𝑝>0. If one selects 𝛿𝛼=𝐸^2/(𝑝+2), (3.8) then the following estimate holds: 𝑓(⋅)−𝑓^𝛿_𝛼(⋅)_{𝐿2(0,𝜋)}≤2𝛿^{𝑝/(𝑝+2)}𝐸^2/(𝑝+2)11+2𝛿max1,𝐸^{(4−𝑝)/(𝑝+2)}+2𝑘²𝛿. (3.9)

Proof. Due to the triangle inequality, we have 𝑓−𝑓^𝛿_{𝛼𝐿2(0,𝜋)}=−^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_𝑛−−^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}=^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_𝑛−^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}≤^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_𝑛−^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}+^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_𝑛−^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}=^∞_𝑛=1𝑛²+𝑘²1−𝑒^{−𝑛2+𝑘2}𝑔,𝑋_𝑛𝑋_𝑛11−1+𝛼²𝑛⁴_{𝐿2(0,𝜋)}+^∞_𝑛=1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝑔−𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}≤^∞_𝑛=111−1+𝛼²𝑛⁴𝑓,𝑋_𝑛𝑋_𝑛1+𝑛^2𝑝/21+𝑛^2−𝑝/2_{𝐿2(0,𝜋)}+sup_𝑛≥1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2∞}_𝑛=1𝑔−𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}≤sup_𝑛≥111−1+𝛼²𝑛⁴1+𝑛^2−𝑝/2𝑓,𝑋_𝑛𝑋_𝑛1+𝑛^2𝑝/2_{𝐿2(0,𝜋)}+sup_𝑛≥1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2∞}_𝑛=1𝑔−𝑔^𝛿,𝑋_𝑛𝑋_{𝑛𝐿2(0,𝜋)}≤sup_𝑛≥111−1+𝛼²𝑛⁴1+𝑛^2−𝑝/2𝐸+sup_𝑛≥1𝑛²+𝑘²1+𝛼²𝑛⁴1−𝑒^{−𝑛2+𝑘2}𝛿≤max𝛼²,𝛼^(𝑝/2)2𝐸+𝛼𝛿+2𝑘²𝛿𝛿=max𝐸^4/(𝑝+2),𝛿𝐸^𝑝/𝑝+2𝛿𝐸+2𝐸^{−2/(𝑝+2)}𝛿+2𝑘²𝛿=2𝛿^{𝑝/(𝑝+2)}𝐸^2/(𝑝+2)11+2𝛿max1,𝐸^{4−𝑝/(𝑝+2)}+2𝑘²𝛿. (3.10) The proof is complete.

Remark 3.3. If 0<𝑃≤4, 𝑓(⋅)−𝑓^𝛿_𝛼(⋅)_{𝐿2(0,𝜋)}≤3𝛿^{𝑝/(𝑝+2)}𝐸^2/(𝑝+2)+2𝑘²𝛿⟶0,as𝛿⟶0. (3.11) If 𝑝>4, 𝑓(⋅)−𝑓^𝛿_𝛼(⋅)_{𝐿2(0,𝜋)}≤2𝛿^{𝑝/(𝑝+2)}𝐸^2/(𝑝+2)+𝛿^4/(𝑝+2)𝐸^{(𝑝−2)/(𝑝+2)}+2𝑘²𝛿⟶0,as𝛿⟶0. (3.12) Hence, 𝑓^𝛿_𝛼(𝑥) can be viewed as the approximation of the exact solution 𝑓(𝑥).

4. Numerical Example

From (1.10), we know that𝐾𝑓(𝑥)=^∞_𝑛=11−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²𝑓,𝑋_𝑛𝑋_𝑛=^𝜋₀2𝜋^∞_𝑛=11−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²𝑓(𝑠)sin(𝑛𝑠)sin(𝑛𝑥)𝑑𝑠=𝑔(𝑥). (4.1)

We use trapezoid's rule to approach the integral and do an approximate truncation for the series by choosing the sum of the front 𝑀+1 terms. After considering an equidistant grid 0=𝑥₁<⋯<𝑥_𝑀+1=𝜋,(𝑥_𝑖=(𝑖−1)/𝑀𝜋,𝑖=1,…,𝑀+1), we get2𝜋^𝑀+1𝑁_𝑖=1𝑛=11−𝑒^{−𝑛2+𝑘2}𝑛²+𝑘²𝑓𝑥_𝑖sin𝑛𝑥_𝑖sin𝑛𝑥_𝑗ℎ=𝑔𝑥_𝑗, (4.2) where𝜋ℎ=𝑀. (4.3)

Example 4.1. It is easy to see that the function 𝑢(𝑥,𝑦)=(1−𝑒^{−√2𝑘𝑦})sin𝑘𝑥 and the function 𝑓(𝑥)=−2𝑘²sin𝑘𝑥 are the exact solutions of the problem (1.1). Consequently, the data function is 𝑔(𝑥)=(1−𝑒^−√2𝑘)sin𝑘𝑥, and 𝑓(⋅)_{𝐻𝑝(0,𝜋)}=^∞_𝑛=11+𝑛^2𝑝𝑓,𝑋_𝑛^21/2=1+𝑘^2𝑝/2𝑘²√2𝜋. (4.4) Adding a random distributed perturbation to each data function, we obtain vector 𝑔^𝛿; that is, 𝑔^𝛿=𝑔+𝜀randnsize𝑔. (4.5) The function “randn(⋅)” generates arrays of random numbers whose elements are normally distributed with mean 0, variance 𝜎²=1, and standard deviation 𝜎=1. “Randn(size(g))” returns an array of random entries that is the same size as 𝑔. The total noise level 𝛿 can be measured in the sense of root mean square error (RMSE) according to 𝛿=𝑔^𝛿−𝑔₂=1𝑀+1^𝑀+1_𝑛=1𝑔_𝑛−𝑔^𝛿_𝑛^21/2. (4.6)

Using 𝑔^𝛿 as data function, we obtain the computed approximation 𝑓^𝛿_𝛼(𝑥) by using (3.7). The relative error is given as follows:𝑓𝑓rerr∶=^𝛿𝛼−𝑓₂𝑓₂, (4.7) where ‖⋅‖₂ is defined by (4.6).

Tables 1-2 show 𝑀 and 𝑁 have small influence on the relative error when they become larger. So, we will always take 𝑀=100 and 𝑛=7 in the following examination.

Table 1 

Relative errors rerr(𝑓) with 𝜀=0.01, 𝑘=2, 𝑁=10 and 𝑝=2 for different 𝑀.

Table 2 

Relative errors rerr(𝑓) with 𝜀=0.01, 𝑘=2, 𝑀=100 and 𝑝=2 for different 𝑁.

Test 1
We choose 𝑝=1/2, 𝑝=1, 𝑝=2 and 𝑝=3 in Tables 3, 4, 5, and 6 to compute the parameters 𝛿, 𝛼, and rerr(𝑓), respectively.

Table 3 

𝛿, 𝛼 and rerr(𝑓) with 𝑝=1/2, 𝑘=2 and 𝐸=14.9931.

Table 4 

𝛿, 𝛼 and rerr(𝑓) with 𝑝=1, 𝑘=2 and 𝐸=22.4200.

Table 5 

𝛿, 𝛼 and rerr(𝑓) with 𝑝=2, 𝑘=2 and 𝐸=50.1326.

Table 6 

𝛿, 𝛼, and rerr(𝑓) with 𝑝=3, 𝑘=2 and 𝐸=112.0998.

These tables indicate that parameter 𝛿, 𝛼, and rerr(𝑓) all depend on the perturbation 𝜀. 𝛿, 𝛼, and rerr(𝑓) decrease with the decrease of 𝜀. These are consistent with our regularization methods. In addition, rerr(𝑓) decreases with the increase of 𝑝 at first, but it ceases decreasing when 𝑝 reaches to some extent. This means that rerr(𝑓) does not decrease for stronger “smoothness” assumptions on the exact solution 𝑓(𝑥).

Test 2
Figure 1 shows the comparison between the exact solution 𝑓(𝑥) and the regularization solution 𝑓^𝛿_𝛼(𝑥) for 𝑘=1, 𝑘=2, 𝑘=3, and 𝑘=4 with the perturbation 𝜀=0.1, 𝜀=0.01, and 𝜀=0.001 with Example 4.1.

(a)

(b)

(c)

(d)

Figure 1 

The exact solution 𝑓(𝑥) and its approximation 𝑓𝛿𝛼(𝑥): (a) 𝑘=1, (b) 𝑘=2, (c) 𝑘=3, and (d) 𝑘=4.

Figure 1 indicates these regularized solutions approximate to the exact solution, as the amount of 𝜀 decreases, while the numerical results are not so good as the parameter 𝑘 becomes larger.

Test 3
The unknown sources 𝑓(𝑥) is given. The numerical test was constructed in the following way. First, we selected the solution 𝑓(𝑥) and obtained the exact data function 𝑔(𝑥) using (1.6). Then, we added a normally distributed perturbation to each data function giving vectors 𝑔_𝛿(𝑥). Finally, we obtained the regularization solutions using (3.7).

Example 4.2. Consider a piecewise smooth source: 𝑓(𝑥)=𝜋0,0≤𝑥≤4,4𝜋𝜋𝑥−1,4𝜋<𝑥≤2,43−𝜋𝜋𝑥,2<𝑥≤3𝜋4,0,3𝜋4<𝑥≤𝜋. (4.8)

In Example 4.2, since the direct problem with the heat source 𝑓(𝑥) does not have an analytical solution, the data 𝑔(𝑥) is obtained by solving the direct problem. Figures 2-3 show the comparisons between the exact solution and its computed approximation with different noise level for Example 4.2. It can be seen that as the amount of noise 𝜀 decreases, the regularized solutions approximate better the exact solution. In addition, even when 𝑘=10, the regularized solutions still approximate the exact solution. From Figures 2-3, it can be seen that the numerical solution is less than that of Example 4.1. It is not difficult to see that the well-known Gibbs phenomenon and the recovered data near the nonsmooth points are not accurate. Taking into consideration the ill posedness of the problem, the results presented in Figures 2-3 are reasonable.

(a)

(b)

Figure 2 

The exact solution 𝑓(𝑥) and its approximation 𝑓𝛿𝛼(𝑥) for 𝑘=10: (a) 𝑝=2, and (b) 𝑝=3.

(a)

(b)

Figure 3 

The exact solution 𝑓(𝑥) and its approximation 𝑓𝛿𝛼(𝑥) for 𝑝=2: (a) 𝑘=3, and (b) 𝑘=10.

5. Conclusions

In this paper, we considered the inverse problem of determining the unknown source using the simplified Tikhonov regularization method for the modified Helmholtz equation. It was shown that with a certain choice of the parameter, a stability estimate was obtained. Meanwhile, the numerical example verified the efficiency and accuracy of this method.

Acknowledgment

The project is supported by the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015).