Some remarks on a modified Helmholtz equation with inhomogeneous source

Nguyen, Huy Tuan; Viet, Tran Quoc; Nguyen, Van Thinh

doi:10.1016/j.apm.2012.03.014

Cited by 21 publications

(5 citation statements)

References 12 publications

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“…Interestingly, all linear and nonlinear cases presented in Section well. We are also aware of the possibilities of solving the modified Helmholtz equation [12] and the elliptic sine-Gordon equation with Cauchy data in the present paper.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…Currently, there have been many other fields of study where the Helmholtz-type equations can be greatly used, such as the influence of the frequency on the stability of Cauchy problems [7], finding the shape of a part of a boundary in [2], regularization of the modified Helmholtz equation in [12] and the problem of identifying source functions in [1,10].…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of the electric field of the Helmholtz equation in 3D

Nguyen¹,

Khoa²,

Minh³

et al. 2014

Preprint

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In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Regińska and Regiński [14].Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in L 2 -norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reconstruction of the electric field of the Helmholtz equation in 3D

Nguyen¹,

Khoa²,

Minh³

et al. 2014

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Helmholtz equation appears in numerous physical applications involving electromagnetic wave propagation or acoustics. It is applied in many technical fields, such as biomedical imaging and geophysical imaging measurement, see [31,32], and references contained therein.…”

Section: Introductionmentioning

confidence: 99%

Adaptive anadromic regularization method for the Cauchy problem of the Helmholtz equation

Omri,

Jday

2023

Inverse Problems

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In this research work, we investigate the Cauchy problem for the Helmholtz equation. Considering the completion data problem in a bounded cylindrical domain with Neumann and Dirichlet conditions on a part of the boundary. An immediate approximation of missing boundary data is obtained using a method that factorizes the boundary value problem. This factorization uses the Neumann to Dirichlet (NtD) or Dirichlet to Neumann (DtN) operators that satisfy the Riccati equation. Some singularities appear in the solution of the Riccati equation for a particular length of the waveguide of the Helmholtz equation. We elaborate a new numerical method called "adaptive anadromic regularization method" that can solve these operators beyond the singularity. In addition, we introduce a scaling matrix technique to the linear matrix equations associated with the Riccati equations to generate normalized solutions. Our numerical procedure not only approximates missing boundary data, but also provides an error estimate that allows efficient time stepping. Numerical tests proved to confirm the theory even in the presence of high noise levels.

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“…The solution of the modified Helmholtz equation plays a crucial role in the science and engineering fields, such as for boundary detection problems [1], water wave problems [2], Cauchy problems [3,4], diffusion equations [5], topological sensitivity analyses [6], and boundary value problems [7]. Over the past 10 years, conventional approaches have been widely adopted for solving the modified Helmholtz equation [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

Xiao

Yeih

et al. 2019

Mathematics

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This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). The multiple source collocation scheme in the MSMA stems from the MFS and the basis functions are formulated using the CTM. The solution of the modified Helmholtz equation is therefore approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain. To deal with the two-dimensional modified Helmholtz equation in layered materials, the domain decomposition method was adopted. Numerical examples were carried out to validate the method. The results illustrate that the MSMA is relatively simple because it avoids a complicated procedure for finding the appropriate position of the sources. Additionally, the MSMA for solving the modified Helmholtz equation is advantageous because the source points can be collocated on or within the domain boundary and the results are not sensitive to the location of source points. Finally, compared with other methods, highly accurate solutions can be obtained using the proposed method.

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Some remarks on a modified Helmholtz equation with inhomogeneous source

Cited by 21 publications

References 12 publications

Reconstruction of the electric field of the Helmholtz equation in 3D

Reconstruction of the electric field of the Helmholtz equation in 3D

Adaptive anadromic regularization method for the Cauchy problem of the Helmholtz equation

On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

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