Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation

The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term ð1 + α2mÞem, where α > 0 is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.


Introduction
The Helmholtz equation, Δw + k 2 w = 0, where k is a constant, i.e., the wave number, models reduced wave equation which yields a time-harmonic solution or monofrequency. This time-harmonic Helmholtz equation has numerous applications in seismology, sonar technology, and noise scattering.
In order to make the Helmholtz equation useful in life, some boundary conditions (the values of the solutions) at the end points of the domains must be provided.
The computation of a Helmholtz equation's solution in a suitable functional space is a direct problem. Imposing boundary constraints on a Helmholtz equation may not have a solution in any functional space in the majority of cases. The existence and uniqueness conditions of wellposedness of a Helmholtz equation are all influenced to a great extent by the number of auxiliary conditions put on the equation. These auxiliary criteria can be solely boundary conditions or a combination of boundary and initial conditions. Usually, evolutionary challenges entail a mixture of factors. Despite this, the boundary requirements that were set on the Dirichlet problem, a Neumann issue, or a Robin problem could be the Helmholtz equation [1]. If the Helmholtz equation is subjected to these auxiliary conditions, they must be enough to guarantee the existence of solutions. The uniqueness and stability of the solution cannot be emphasized in this ongoing discussion so as to make the Helmholtz equation well-posed. Furthermore, proper boundary or starting data aids in the comprehension of the stationary process. Otherwise, there will be no solution to the Helmholtz equation if these auxiliary conditions are too numerous. If the auxiliary requirements are insufficient, the solution will exist but will not be unique [2]. A typical example of an ill-posed problem for the Helmholtz equation that has received a lot of attention in the scientific community dating back to the 20th century is the Cauchy problem. Cauchy data is simply a combination of Dirichlet and Neumann data on some part of the boundary domain of the Helmholtz equation. The Cauchy problem for the Helmholtz equation is highly ill-posed in the sense of Hadamand. The Cauchy condition specifies data from an unknown field as well as its derivative. This auxiliary condition specifies data information only for a portion of the unknown function's domain, not for the complete domain. Imposing such mixed boundary conditions on the Helmholtz equation as a result does not result in a solution.
The ill-posed Helmholtz equation, according to Hadamard, has no practical use or is physically nonsensical. The current trends in inverse problems, however, have refuted this assertion. The vibrating membrane system and laser beam models, for example, are an ill-posed [3].
Some regularization approaches have been suggested for solving the Cauchy issue of the Helmholtz equation, based on the assumption of the existence of a unique solution. The Tikhonov regularization method (TRM) is based on the existence of a linear bounded operator A that connects one Hilbert space X to another Hilbert space Y, with x, y ∈ DðAÞ. The TRM makes the Laplace-type operator in the Helmholtz equation regular. The (precise) solution and variations of the data function in the Helmholtz equation are both constrained in this method to prevent the data function from blowing up due to its emitted faults [4]. The quasireversibility regularization method (Q-RRM) was established by Lattes and Lions [5], which assumes that a linear Laplace-type operator in the Helmholtz equation is linear, but the inverse Laplace-type operator is not continuous from its range into a domain. Only the Helmholtz equation is regularized by the Q-RRM by subtracting a product of a square of a regularization parameter α and a mixed fourth-order partial derivative from the Laplace-type operator in the Helmholtz equation, which is of the form: Khoa and Nhan [6] introduced the stabilized operator to complement the effort of the Laplace-like operator occurring in the Helmholtz equations. The introduction of the stabilized operator rather increased the complexity of the problem of regularizing Helmholtz equation, whose weak solution is sought in the Sobolev spaces.
The Cauchy issue of the Helmholtz equation has been solved using wavelet methods. Vani and Avudainayagam [7] solved the problem in the (Meyer) wavelet domain and demonstrated that the regularized solution converges as the Cauchy data perturbations approach zero. Solutions are occasionally sought in a more convenient function space rather than regular Hilbert space. Dou and Fu [8] regularized the problem in a Sobolev space, and when the number of disturbances in the Cauchy data increased, the numerical solution remained stable. Other less effective methods include the energy regularization method introduced by Han et al. [9] for solving the Cauchy issue of the Helmholtz equation. For the identification of a source data for regularizing the Helmholtz equation with Dirichlet boundary conditions, Zhao et al. [10] compared the efficiency and convergence rate of the mollification method, the modified Tikhonov regularization method, and the Fourier regularization method. All of these regularization methods fail to ensure not only the existence of a unique solution to the Helmholtz equation but also the stability of the solution, especially when the inhomogeneous Cauchy boundary deflection is imposed on the equation.
In [11], the authors employed Fourier truncation regularization to solve the Helmholtz equation with an altered wave number. The regularized solution's error estimation and the exact solution obtained as a regularization parameter are both varied. However, the regularization method has been observed to avoid singularity in the equation. The singular boundary approach, for example, changes the Helmholtz equation into a boundary integral equation for the determination of singularities in the equation and finally produces the regularized solution to the Helmholtz equation, as described in [12]. All of the strategies mentioned in this work fail to guarantee a solution to the Helmholtz equation with the Cauchy boundary conditions but instead restore the problem's stability.
The third condition of well-posedness can be ensured by the bounded inverse theorem. We note that the Helmholtz equation has a solution if the smoothness requirement is satisfied together with the data compatibility conditions, which we give below.

Preliminary Result
In this section of the paper, the lemma and theorem that will be used in achieving the main result in the next section are provided. Also, throughout this paper, the supremum norm is applied in establishing results.

Lemma 1 (Compatibility of data in Neumann problem).
Let Ω denote a bounded region in R 2 having a smooth boundary ∂Ω. The Neumann problem for the Helmholtz equation A necessary and sufficient condition for the existence of a solution to the Neumann problem for the Helmholtz (homogeneous) equation is [1] ð Proof. Considering the vector identity Δu = ∇⋅ ∇u, applying Green's first identity to the operator equation above, we have ð Theorem 2 (Bounded inverse theorem). Let A be a bounded linear Laplace-type operator in the Helmholtz equation from a subspace Ω in a Hilbert space H into a Hilbert space H.

Abstract and Applied Analysis
Then. A has a continuous inverse operator A −1 from its range RðAÞ into Ω. Conversely, if there is a continuous inverse operator then there is a positive constant C such that [13] A In some cases, the Helmholtz equation with the abovementioned boundary conditions does not produce a solution in any functional space. As a result, none of the three well-posedness conditions are satisfied, rendering the Cauchy problem of the Helmholtz equation ill-posed in Hadamard's meaning.
The following is the Helmholtz equation with Cauchy boundary conditions.
We prove that in Hilbert space, the aforementioned Helmholtz equation with Cauchy boundary conditions has no solution.
Furthermore, we can observe from the Helmholtz equation's initial deflection condition that Thus, If n = 1, we obtain Thus, The data compatibility requirement is not satisfied by the equation. This means that there is no solution to the Helmholtz equation with Cauchy boundary conditions. As a result, in Hadamard's understanding, the Helmholtz equation with Cauchy boundary conditions is ill-posed.
How can there be a stable solution without a solution to the Helmholtz equation? Assuming heuristically, this is not unusual among the scientific community. How can there be an estimation of error between the regularized solution and an exact solution when the Helmholtz equation has no solution in any functional space? It takes more information in its construction, the right number of boundary conditions, and appropriate restrictions to make the inhomogeneous boundary deflection in the Cauchy problem of the Helmholtz equation well-posed. Numerical stability is unquestionably possible when the inhomogeneous Cauchy problem of the Helmholtz equation is well-posed, whether addressed with finite precision or with data errors.
The DRM is used to regularize both imposed Cauchy boundary conditions on the Helmholtz equation and equations in a Hilbert space. This method uses a positive integer scalar η to homogenize the inhomogeneous boundary deflection in the Cauchy data of the Helmholtz equation. As a result, the Helmholtz equation has a solution thanks to the positive integer scalar. The DRM employs a regularization term ð1 + α 2m Þe m , which restores the stability of solution of the Helmholtz equation. The approach uses Green's first identity to the Laplace-type operator of ð1 + α 2m Þe m and wðx, yÞ appearing in the Helmholtz equation to determine the uniqueness of the solution. This results in a piecewise smooth boundary of two disjoint complementary sections ∂Ω 1 and ∂Ω 2 with wðx , 0Þ and ∂wðx, lÞ/∂y on ∂Ω 1 and wðl, yÞ and ∂wð0, yÞ/∂x on ∂Ω 2 , respectively. The uniqueness of the solution of the regularized Helmholtz equation with regularized Cauchy boundary conditions is proven by contradiction with the help of DRM. Using the DRM, we demonstrated that the operator R ðAÞ has a closed range, the null space of the operator NðAÞ is trivial, and the inverse operator has a continuous range.

Main Result
We can see that equation (3) is ill-posed, and that Cauchy data is provided on the arc of the boundary ∂Ω instead of the whole boundary of the domain Ω. We first extend the arc of the boundary (hypersurface) ∂Ω to the full boundary of the domain Ω using the DRM to restore the well-posedness of the Helmholtz equation with Cauchy boundary conditions where the boundary deflection is inhomogeneous. As a result, the domain of Cauchy data's boundary is quadratured to a piecewise smooth boundary comprising two disjoint complementary parts.

Theorem 3 (Divergence regularization method). Let Cauchy boundary conditions be imposed on Helmholtz equation
where the boundary deflection is inhomogeneous; then, the Cauchy problem of the Helmholtz equation is given by 3 Abstract and Applied Analysis where The corresponding regularized Cauchy problem of the Helmholtz equation is where ð hðxÞ ≠ 0, hðyÞ ≠ 0, α ∈ ð−∞,−1Þ ∪ ð1,∞Þ is the regularization parameter, m ∈ Z + is a positive integer, k is the wave number, ½0, l is the square domain with l is a radian number, and η is any even positive integer.
This Theorem 3 ensures the existence, uniqueness, and stability of the Cauchy problem for the Helmholtz equation.
Proof. We write the homogeneous Helmholtz equation as Applying the dot product, product rule, and integrating both sides over Ω, we obtain ð ð The x − coordinate of the unknown function wðx, yÞ in equation (22) is then scaled by a factor η. The trigonometric function of the Helmholtz equation at inhomogeneous boundary deflection becomes zero as a result of this scalar, and when integrated over the domain boundary, we get Substituting v = ð1 + α 2m Þe m into equation (3), the right hand side becomes ð ð Applying Green's first identity to the first term of above equation yields ð ð A regularized Helmholtz equation with regularized Cauchy boundary conditions is obtained as shown above.
We show that our regularized Helmholtz equation, when combined with Cauchy boundary conditions, meets all three well-posedness requirements. We begin by demonstrating that equations (15)-(19) has a solution in the Hibert space.

Well-Posedness of Regularized Helmholtz Equation with
Regularized Cauchy Boundary Conditions. We now show that the regularized Helmholtz equation with regularized Cauchy boundary conditions is well-posed.
and inhomogeneous ∂w n,0 x, l ð Þ ∂y It is clear that the homogeneous boundary deflection satisfies the data compatibility condition. We demonstrate that the inhomogeneous boundary deflection also satisfies the data compatibility condition as follows: Since hðηxÞ is a periodic function such as sin ðxÞ or cos ðxÞ the integral of the periodic function of a scalar multiplier of the spatial variable over the boundary always yields zero, then This result implies that the regularized Helmholtz equation with regularized Cauchy boundary conditions has a solution in the Hilbert space.
We prove that the DRM provides a unique solution of regularized Helmholtz equation together with regularized Cauchy boundary conditions as follows. .

Abstract and Applied Analysis
Also, we observe that h ηx ð Þ = 0 on ∂Ω 1 , Thus, w η,0 ðx, yÞ is smooth and zero in the domain Ω and its boundary ∂Ω. This implies that Hence, equation (22) has only one solution in Hilbert Space.
Theorem 6 (Stability). In demonstrating that the regularized Helmholtz equation is stable to the small changes in the regularized Cauchy boundary conditions, the spatial variable y is perturbed from ε to δ, where δ > ε. Then, and the corresponding solutions are ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi The change in the regularized boundary condition is lim m,n⟶∞ on the grounds that hðηxÞ ≤ 1 and hðδxÞ ≤ 1 are a periodic function which is sin ðηxÞ or cos ðηxÞ. This implies that there is a small change in the boundary condition. Moreover, we observe the corresponding change in the solution wðx, yÞ as lim m,n⟶∞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Thus, lim m,n⟶∞ We can see that This implies that a small change in the regularized Cauchy boundary condition from x 1 = ε to x 2 = δ results in a small change in solution Thus, the regularized Helmholtz equation is stable. Hence, the regularized Cauchy problem for the regularized Helmholtz equation is well-posed.

Applications
The DRM is used in this part to find solutions to the Helmholtz equation with Cauchy boundary conditions in the upper half-plane and the Helmholtz equation with Neumann boundary conditions in the lower half-plane.

Helmholtz Equation with
Cauchy Boundary Conditions where the Boundary Deflection Is Inhomogeneous. In this subsection of the paper, the DRM is applied to regularize 6 Abstract and Applied Analysis the Cauchy problem of the Helmholtz equation in a domain ½0, ðπ/2Þ × ½0, 2π. The problem is as follows: We show that when Cauchy boundary conditions are applied on a homogeneous Helmholtz equation with a nonzero boundary deflection, we get the following result using the method of separation of variables For the above function wðx, yÞ in Equation (52) to be called a solution to Equation (52) together with Cauchy boundary conditions, it must satisfy the smoothness requirement condition as well as the data compatibility condition. The integral of the boundary deflection ∂wðx, 0Þ/∂y over ½0 , ðπ/2Þ is Now, is not a solution of equations (49), (50), (51). Hence, the equation is ill-posed in the sense of Hadamard.
To regularize this equation, we choose η = 4 in the Theorem 3, and we obtain By the method of separation of variables, we obtain It can easily be shown that the regularized equations (56), (57), (58), (59), (60) together with the regularized boundary conditions is well-posed since Applying the uniqueness of the DRM in Theorem 3, the solution in equation (61) is unique.