Relevant bibliographies by topics / Ill-posed Helmholtz equation

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Ill-posed Helmholtz equation'

Author: Grafiati
Published: 3 June 2025
Last updated: 31 July 2025

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Journal articles on the topic "Ill-posed Helmholtz equation"

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Benedict, Barnes, O. Boateng F., K. Amponsah S., and Osei-Frimpong E. "On the Notes of Quasi-Boundary Value Method for Solving both Cauchy-Dirichlet Problem of the Helmholtz Equation." British Journal of Mathematics & Computer Science 22, no. 2 (2017): 1–10. https://doi.org/10.9734/BJMCS/2017/32727.

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The Cauchy-Dirichlet problem of the Helmholtz equation yields unstable solution, which when solved with the Quasi-Boundary Value Method (Q-BVM) for a regularization parameter = 0. At this point of regularization parameter, the solution of the Helmholtz equation with both Cauchy and Dirichlet boundary conditions is unstable when solved with the Q-BVM. Thus, the quasi-boundary value method is insufficient and inefficient for regularizing ill-posed Helmholtz equation with both Cauchy and Dirichlet boundary conditions. In this paper, we introduce an expression 1 (1+ 2) ; ∈ R, where is the regulari
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Kabanikhin, Sergey Igorevich, M. A. Shishlenin, D. B. Nurseitov, A. T. Nurseitova, and S. E. Kasenov. "Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/786326.

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We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.
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Barnes, Benedict, Anthony Y. Aidoo, and Joseph Ackora-Prah. "Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation." Abstract and Applied Analysis 2022 (March 29, 2022): 1–10. http://dx.doi.org/10.1155/2022/4628634.

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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 1 + α 2 m e m , where α > 0 is the regularization parameter. As a result, DRM re
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Chen, Yong-Gang, Fan Yang, and Qian Ding. "The Landweber Iterative Regularization Method for Solving the Cauchy Problem of the Modified Helmholtz Equation." Symmetry 14, no. 6 (2022): 1209. http://dx.doi.org/10.3390/sym14061209.

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In this manuscript, the Cauchy problem of the modified Helmholtz equation is researched. This inverse problem is a serious ill-posed problem. The classical Landweber iterative regularization method is designed to find the regularized solution of this inverse problem. The error estimations between the exact solution and the regularization solution are all obtained under the a priori and the a posteriori regularization parameter selection rule. The Landweber iterative regularization method can also be applied to solve the Cauchy problem of the modified Helmholtz equation on the spherically symme
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Kashirin, A. A., and S. I. Smagin. "On the solvability on the spectrum of Fredholm boundary integral equations of the first kind for the three-dimensional transmission problem." Дифференциальные уравнения 60, no. 2 (2024): 211–23. http://dx.doi.org/10.31857/s0374064124020054.

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The paper considers two weakly singular Fredholm boundary integral equations of the first kind, to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on spectra, where they are ill-posed. For the first equation, it is shown that if its solution exists on the spectrum, it allows us to find a solution to the transmission problem. The second equation in this case always has infinitely many solutions, only one of which gives a solution to the transmission problem. The interpolation method for finding approximate solution
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Juraev, Davron Aslonqulovich, and Samad Noeiaghdam. "Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane." Axioms 10, no. 2 (2021): 82. http://dx.doi.org/10.3390/axioms10020082.

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In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).
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Dou, Fang-Fang, and Chu-Li Fu. "A Wavelet Method for the Cauchy Problem for the Helmholtz Equation." ISRN Applied Mathematics 2012 (January 4, 2012): 1–18. http://dx.doi.org/10.5402/2012/435468.

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We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.
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He, Shangqin, and Xiufang Feng. "A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 05 (2019): 1950029. http://dx.doi.org/10.1142/s0219691319500292.

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In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.
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Yang, Fan, Ping Fan, and Xiao-Xiao Li. "Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number." Mathematics 7, no. 8 (2019): 705. http://dx.doi.org/10.3390/math7080705.

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In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.
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Li, Xiao-Xiao, Fan Yang, Jie Liu, and Lan Wang. "The Quasireversibility Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/245963.

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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 quasireversibility 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.
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Dissertations / Theses on the topic "Ill-posed Helmholtz equation"

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Watson, Francis Maurice. "Better imaging for landmine detection : an exploration of 3D full-wave inversion for ground-penetrating radar." Thesis, University of Manchester, 2016. https://www.research.manchester.ac.uk/portal/en/theses/better-imaging-for-landmine-detection-an-exploration-of-3d-fullwave-inversion-for-groundpenetrating-radar(720bab5f-03a7-4531-9a56-7121609b3ef0).html.

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Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging f
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Book chapters on the topic "Ill-posed Helmholtz equation"

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"Cauchy problem for the Helmholtz equation." In Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis. De Gruyter, 2003. http://dx.doi.org/10.1515/9783110936520.143.

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Conference papers on the topic "Ill-posed Helmholtz equation"

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DeLillo, Thomas K., Tomasz Hrycak, and Nicolas Valdivia. "Iterative Regularization Methods for Inverse Problems in Acoustics." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32730.

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We consider the use of conjugate-gradient-like iterative methods for the solution of integral equations arising from an inverse problem in acoustics in a bounded three dimensional region. The inverse problem is the computation of the normal velocities on the boundary of a region from pressure measurements on an interior surface. The pressure satisfies the Helmholtz equation in the region. Two formulations are considered: one based on the representation of pressures by a single layer potential and the other based on the Helmholtz-Kirchhoff integral equation. Both formulations can be used to app
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Lopez de Bertodano, Martin A., and William D. Fullmer. "Two Equation Two-Fluid Model Analysis for Stratified Flow Under Kinematic and Dynamic Instabilities." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-66743.

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The unstable one-dimensional incompressible two-fluid model including a hydrostatic force is reduced to a two equation model in terms of the liquid volume fraction and the liquid velocity. For small density ratios the model may be simplified to a formulation that is equivalent tothe Shallow Water Theory (SWT) equations [Whitham, 1975] with a source term corresponding to the two-fluid model constitutive relations for wall and interfacial shear and to a void gradient term that contains the Kelvin-Helmholtz mechanism. Linear stability of the SWT equations shows that the model is made well-posed s
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Journal articles
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