Wavelet moment method for the Cauchy problem for the Helmholtz equation
type: Item
instance of: scholarly article
title: Wavelet moment method for the Cauchy problem for the Helmholtz equation
zbMATH Open document ID: 1165.65069
author: Teresa Regińska
author: Andrzej Wakulicz
published in: Journal of Computational and Applied Mathematics
publication date: 2008-11-20T00:00:00Z
review text: \textit{J. Cheng, Y. C. Hon, T. Wei} and \textit{M. Yamamoto} [Z. Angew. Math. Mech. 81, No.~10, 665--674 (2001; Zbl 0999.65100)] investigated a Cauchy problem of the Laplace equation in two space dimensions, where mixed Dirichlet-Neumann conditions are given on an open subset of the boundary. This formulation represents an inverse problem and thus it is ill-posed. The equations are transformed into an equivalent moment problem, i.e., a weak form of a boundary integral equation. Accordingly, Galerkin type methods with polynomials as basis functions can be applied. In the paper at hand, the authors consider this Cauchy problem of the Helmholtz equation. A corresponding moment problem is formulated in the general case of three space dimensions. The authors focus on the moment approach for a model problem in two space dimensions. The Meyer wavelet generates an orthonormal basis of \(L^2(\mathbb{R})\). A subset of this basis is applied to construct a numerical approximation of the solution of the boundary integral equation for possibly perturbed data. It follows that a regularisation parameter can be chosen in dependence on a perturbation such that the approximations converge to the exact solution in case of perturbations tending to zero. Accordingly, a regularised formulation of the Helmholtz equation is achieved, where Neumann conditions are given on the whole boundary. Methods for the computation of the defined numerical approximation or simulation results are not within the scope of the article. Instead of that, the authors prove the properties of the numerical approximation using the wavelet projection and corresponding techniques.
reviewed by: Roland Pulch
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 65N38
zbMATH DE Number: 5370029
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Neumann problem
zbMATH Keywords: moment problem
zbMATH Keywords: ill-posed problem
zbMATH Keywords: regularisation
zbMATH Keywords: Meyer wavelet
zbMATH Keywords: wavelet projection
MaRDI profile type: Publication
full work available at URL: j.cam.2008.01.005
OpenAlex ID: W1986257886
cites work: Numerical solution of a Cauchy problem for the Laplace equation
cites work: Numerical computation of a Cauchy problem for Laplace's equation
cites work: Q4434436
cites work: Ten Lectures on Wavelets
cites work: Q3993273
cites work: The detection of surface vibrations from interior acoustical pressure
cites work: A stability estimate for a Cauchy problem for an elliptic partial differential equation
cites work: Q4895893
cites work: Increased stability in the continuation of solutions to the Helmholtz equation
cites work: Inverse problems for partial differential equations
cites work: On Neumann's method for the exterior Neumann problem for the Helmholtz equation
cites work: Approximate solution of a Cauchy problem for the Helmholtz equation
cites work: Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation
cites work: Q4850052
cites work: Q4762985
DOI: 10.1016/J.CAM.2008.01.005
description: scientific article; zbMATH DE number 5370029