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The Fourier regularization for solving the Cauchy problem for the Helmholtz equation. (English) Zbl 1169.65333

Summary: The Cauchy problem for the Helmholtz equation in an infinite “strip” is considered. The Cauchy data are at the boundary \(x=0\) given in an approximate manner and the solution is sought in the region \(\{(x,y)|0<x\leqslant 1,y\in \mathbb R^n, n\geqslant 1\} \). This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. In this paper we use the Fourier regularization method to solve the problem. The method is independent of the interval length and wave number. Some sharp error estimates between the exact solution and its regularization approximation are given and numerical examples show that the method works effectively.

MSC:

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R25 Ill-posed problems for PDEs
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