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**Wavelet moment method for the Cauchy problem for the Helmholtz equation.**
*(English)*
Zbl 1165.65069

J. Cheng, Y. C. Hon, T. Wei and 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.

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.

Reviewer: Roland Pulch (Wuppertal)

### MSC:

65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

44A60 | Moment problems |

65N38 | Boundary element methods for boundary value problems involving PDEs |

### Keywords:

Helmholtz equation; Cauchy problem; Neumann problem; moment problem; ill-posed problem; regularisation; Meyer wavelet; wavelet projection### Citations:

Zbl 0999.65100
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\textit{T. Regińska} and \textit{A. Wakulicz}, J. Comput. Appl. Math. 223, No. 1, 218--229 (2009; Zbl 1165.65069)

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### References:

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