#### Abstract

The Quasi-Reversibility Regularization Method (Q-RRM) provides stable approximate solution of the Cauchy problem of the Helmholtz equation in the Hilbert space by providing either additional information in the Laplace-type operator in the Helmholtz equation or the imposed Cauchy boundary conditions on the Helmholtz equation. To help bridge this gap in the literature, a Modified Quasi-reversibility Regularization Method (MQ-RRM) is introduced to provide additional information in both the Laplace-type operator occurring in the Helmholtz equation and the imposed Cauchy boundary conditions on the Helmholtz equation, resulting in a strong stable solution and faster convergence of the solution of the Helmholtz equation than the regularized solutions provided by Q-RRM and its variants methods.

#### 1. Introduction

In the 20^{th} century, the term ill-posed problem was coined by Hadamard (1902); for example, the authors in [1] outlined a number of mathematical equations that do not fully satisfy the criteria of well-posedness. This poorly formulated equation loses important information because of the oversimplified assumptions of the problem, the incorrect functional space that contains the solution of the problem, and the inevitably noisy data function emanating from the field data. As a result, the mathematical equation defining such an occurrence can actually explain a completely other phenomenon. Although Hadamard gave conditions for detecting ill-posed equation, he did not provide any means for correcting such ill-posed equation. In 1944, Tikhonov introduced so-called the Tikhonov Regularization Method (TRM), in the form of a quadratic functional, which minimizes the disparity between the exact values and the observed values, giving rise to the problem of the correctness of the ill-posed equation, for instance, see [2]. One of the auxiliary conditions imposed on the Helmholtz equation which has interested many researchers across the globe is the Cauchy boundary conditions. The unique solution is observed when Cauchy boundary conditions are imposed on the Helmholtz equation; however, this unique solution is not stable to small perturbations in the data function in the Cauchy boundary conditions.

Recent studies have shown that the Q-RRM is an effective method for regularizing the solution of the Cauchy problem of the Helmholtz equation in Hilbert space. This method of restoring the stable solution of the Cauchy problem of the Helmholtz equation is in two folds: one of the methods is to add extra information to the Laplace-type operator that appears in the Helmholtz equation; see [3], for more information. The other method is to provide extra information in the Cauchy boundary conditions that are placed on the Helmholtz equation; see [4]. This extra information, whether it be in the Helmholtz equation or the imposed Cauchy boundary conditions, suppresses the growth of the cosine function in the solution of the Cauchy problem of the Helmholtz equation, stabilizing the solution in Hilbert space. The authors in [5] truncated the high frequencies arising from the Fourier series coefficients in the regularized solution of the Cauchy problem of the Helmholtz equation after subtracting the inner product of square of the regularization parameter and the dependent variable with respect to independent variables in using the Q-RRM due to authors in [3]. Thus, the resultant infinite series solution is reduced into a regularized finite series solution by cutting the high frequencies of the Fourier series coefficients, which requires discarding part of the terms. Owing to the applications of the Q-RRM due to author in [4], the authors in [6] observed the generalized way of regularizing the unstable solution of the Helmholtz equation with imposed Cauchy boundary conditions. Similar to this method, authors in [7–9] have used the to obtain regularized solution to the Helmholtz equation with various stability estimations in functional spaces. However, other techniques including the Divergence Regularization Method (DRM) and the Lavrentiev Regularization Method (LRM) developed by the authors in [10] and [11], respectively, have successfully produced regularized solutions of the Helmholtz equation with imposed Cauchy boundary conditions. The numerical schemes for restoring the stable solution of the Helmholtz equation have been observed by researchers across the globe. For examples, the authors in [12] observed the Conjugate Gradient Method (CGM), the authors in [13] formulated the iterative version of the TRM, the authors in [14] introduced the Krylov Subspace Method (KSM), the authors in [15] applied the Q-RRM together with Haar wavelet, the authors in [16] applied the Legendre Wavelet Collocation Method (LWCM), and the authors in [17] developed the Radial Basis Functions (LRMFs) with scale-3 Haar wavelets. All of these methods for regaining the stability of the unique solution of the Helmholtz equation with imposed Cauchy boundary conditions give extra information about either the Laplace-type operator given in the Helmholtz equation or the imposed Cauchy boundary conditions, but not about both of them. In addition to this vacuum in the research, neither the stability estimates nor the convergence rates of these regularization methods are very impressive. Despite this, because of their inevitable errors, numerical methods frequently produce undesirable results.

In this paper, the Q-RRM is improved to recover the stable solution of the Cauchy problem of the Helmholtz equation by adding more information to the Laplace-type operator in the Helmholtz equation and extra information in the imposed Cauchy boundary conditions. Thus, the Helmholtz equation is modified by subtracting the inner product of the regularization parameter and fourth-order partial derivatives of the dependent variable with respect to independent variables, as well as by adding extra information, a partial derivative of the dependent variable with respect to an independent variable, to the Dirichlet boundary condition in the Cauchy boundary conditions, which results in a stable solution of the Helmholtz equation in the Hilbert space. Unquestionably, when the MQ-RRM is compared to the Q-RRM and its modified versions by researchers throughout the globe, our method, a RM, produces strong, stable estimates and has a fast rate of convergence of solution.

This paper is organized as follows. The section 1 contains the introduction and the proof of the unique solution of the Cauchy problem of the Helmholtz equation, as well as unstable solution as a result of the small perturbations in the data function of the Cauchy boundary conditions leads to large change in the solution as seen in the section 2. In Section 3, a MQ-RRM is introduced for restoring the stable solution of the Cauchy problem of the Helmholtz equation. The error estimation of the regularized solution of the Helmholtz equation is observed in section 4. Convergence estimates and some numerical results including relative root mean square errors of the regularized solution are given in Section 5. The comparison of the regularized solutions by the MQ-RRM with other methods is presented in Section 6, and the conclusion of the findings in this paper is in Section 7.

*Definition 1. *Let
where is the sought solution, is the given right-hand side, and are some Hilbert spaces, and is a Laplace-type operator. Equation (1) presents a well-posed problem according to Hadamard, if the following three conditions are satisfied:
(1)For any , there exist an element such that ; that is, the range of equation (1) is closed(2)The solution is uniquely determined by the element (3)The function depends on continuously; that is, the inverse Laplace-type operator is continuous, see [18].

#### 2. Preliminary Results

In this section, we consider the Helmholtz equation with Cauchy boundary conditions. The Cauchy problem for the Helmholtz equation is a mixture of the Dirichlet boundary condition and the Neumann boundary condition as below.

We can see that the boundary conditions imposed on the Helmholtz equation in equation (2) are the Dirichlet boundary condition and the Neumann boundary condition (Cauchy boundary conditions). Thus, equation (4) is the homogeneous Neumann boundary condition which is equal to the zero on the right hand side of equation (2). This implies that equation (2) together with the boundary conditions (3)–(5) has a solution in the Hilbert space. By the method of separation of variables, the solution of the Helmholtz equation with the boundary conditions (2)–(5) has a solution given below: where

To establish that the function in equation (6) is the only solution of the system of equations (2)–(5), by contradiction, suppose that and are two different smooth solutions of system of equations (2)–(5) in the Hilbert space. Also, set be the solution (2)–(5) in the Hilbert space. The supremum of the function is used throughout this paper to establish the results.

Multiplying the expressions on both sides of equation (2) by and applying the divergence theorem, we have: where is the Laplace type operator in two dimensions.

We can see that:

This implies that the integrand in each term must be equal to the zero. Thus,

Hence, the function in (6) is the only solution of the system of equations (2)–(5).

We now show that the function that appears in (6) is unstable in Hilbert space. To show this, we start by setting the data function , in equation (3) of the system of equations (2)–(5).

The change in the boundary condition is as follows:

We can see that the change in the data function in the Cauchy boundary condition approaches unity as goes to infinity. Thus, a small perturbation in the data function is bounded.

The corresponding change in the solution is as follows:

Since the corresponding change in the solution grow unboundedly as increases, it implies that the solution in (6) is unstable to the small perturbations in the data function.

#### 3. Main Results

In this paper, the MQ-RRM is introduced to regularize the Cauchy problem of the Helmholtz equation in the Hilbert space. This method of regularization provides additional information on the Helmholtz equation and extra information on the imposed Cauchy boundary conditions on the Helmholtz equation, which in turn, yields a strong stability estimate as well as the convergence of the solution in the Hilbert space. Thus, the MQ-RRM subtracts from the Laplace-type operator occurring in the Helmholtz equation, and also, add to in the Dirichlet condition in the imposed Cauchy boundary condition. The regularized Helmholtz equation together with the regularized Cauchy boundary condition is given below.

Theorem 2. *Let Cauchy boundary conditions to be imposed on Helmholtz equation where the function of boundary deflection is zero over the boundary; then, the Cauchy problem of the Helmholtz equation is given by
where which has stable unique solution in the Hilbert space.*

*Proof. *Multiplying each term in equation (12) by and integrating the result over the yields
Applying the divergence theorem, we have:
Since,
We can see that which in turn gives is zero
where This completes the proof.

##### 3.1. Existence of Solution of the Modified Quasi-reversibility Regularization Method for the Cauchy Problem of the Helmholtz Equation

We can observed that the Neumann boundary condition is homogeneous and the integral of the inhomogeneous function over is zero. In particular, such function is defined on . Since the integral of each of the function at the right hand of equations (13) and (14) is zero with implies that the equation (12) together with the boundary conditions (13)–(15) has a solution in the Hilbert space. By the method of separation of variables, the solution of the Helmholtz equation with imposed regularized Cauchy boundary conditions (12)–(15) is as follows: where .

##### 3.2. The uniqueness of Solution of the Modified Quasi-reversibility Regularization Method for the Cauchy Problem of the Helmholtz Equation

To establish that the function in equation (20) is the only solution of the system of equations (12)–(15), by contradiction, suppose that and are two different smooth solutions of equations (12)–(15) in the Hilbert space. Set be a solution of system of equations (12)–(15) in the Hilbert space.

Multiplying the expressions on both sides of equation (12) by , integrating the results over the and finally applying the divergence theorem yields where has usual meaning.

We observed that

This implies that each integral term must be equal to zero. We can see that the first integral on the left hand side of (22) is zero, if

Furthermore, the second integral term in the equation must also equal to zero. Thus,

Lastly, the third term integral in the equation must be equal to zero. Thus,

Hence, the function in (20) is the only solution of the system of equations (12)–(15).

##### 3.3. Stable Solution of the Modified Quasi-reversibility Regularization Method for the Cauchy Problem of the Helmholtz Equation

In this subsection, we show that the solution in equation (20) is a stable solution. To start with, the following inequalities are observed. Setting . The function is defined as . At the maximum point, . For a unique minimize , we have:

Also, we can see that,

Setting the data function . The change in the boundary condition is as follows:

The corresponding change in the solution is as follows:

#### 4. Error Estimate of the Regularized Solution of the Cauchy Problem of the Helmholtz Equation

We can see that: then , ,

When is odd, we have:

#### 5. Convergence Estimate of the Regularized Solution of the Cauchy Problem of the Helmholtz Equation

To restore the stability of the solution at , we assume that there is a priori bounded solution

Then, the following convergence result is obtained:

The case is odd:

#### 6. Comparison of the Solution by a MQ-RRM with Some Existing Methods

This section compares the regularized solution using the MQ-RRM in equation (20) with the exact solution in equation (6) and also, compares the regularized solution by the MQ-RRM and some existing methods. Setting .

Figures 1–4 show the numerical results of the regularized solution using the MQ-RRM for different values of and . We observed that the regularized solution by the MQ-RRM becomes stable as both the values of and the wave number increases.

Figures 5–8 are the corresponding two-dimensional plots of the Figures 1–4.

Figures 5–8 show the comparison between the exact solution and the regularized solution by the MQ-RRM in equation (20). We observed that there is a blow-up in the exact solution as compared to the MQ-RRM.

The authors in [6] applied the Q-RRM to regularize the system of equations (2)–(5) as follows: which yields a solution

In comparing our result, the MQ-RRM in equation (20). Set and to be the solutions of the system of equations (12)–(15) and the method by [6]. If the ; then we have

This inequality implies that the regularized solution by the MQ-RRM in equation (20) is more stable than the regularized solution using the MQ-RRM by the authors in [6].

Figures 9 and 10 compare the modified solution and the regularized solution by the authors in [6] with We note that the result of the modified solution is more stable than that of the solution of [6] since the points in the domain are more consistent.

[19] Q-RRM is as follows: which yields a regularized solution

Supposed the solution of the system of eqautions (12)–(15) is equal to the solution of the method by the authors in [19]; then we have: since

We can see that the regularized solution of the Cauchy problem for the Helmholtz equation in equation (20) changes slowly compared to the regularized solution of the [19]. The corresponding numerical result of this analysis is shown in Figures 11 and 12. This implies that the regularized solution of the MQ-RRM for the Cauchy problem of the Helmholtz equation is more stable than that of the regularized solution due to the authors in [19].

Figure 11 shows the comparison between the modified solution and regularized solution [19] with , and whereas Figure 12 shows the comparison between the modified solution and regularized solution [19] with , and . We can see that irrespective of the parameter value, the modified solution yields better result than the solution by [19] as shown in Figures 11 and 12.

The Q-RRM due to the authors in [5] is given below.

By the method of separation of variables, we have

The regularized solution of the system of equations (12)–(15) is equal to the solution of the method due to the authors in [5] as given below.