Qiu, Chun-Yu; Fu, Chu-Li Wavelets and regularization of the Cauchy problem for the Laplace equation. (English) Zbl 1135.35093 J. Math. Anal. Appl. 338, No. 2, 1440-1447 (2008). Summary: The Cauchy problem for the two-dimensional Laplace equation in the strip \(0<x \leqslant \)1 is considered. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data; a small perturbation in the data can cause a dramatically large error in the solution for \(0<x\leqslant \)1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in \(H^r(\mathbb R)\)-norm is also provided. Cited in 21 Documents MSC: 35R25 Ill-posed problems for PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35B35 Stability in context of PDEs 65T60 Numerical methods for wavelets Keywords:Cauchy problem; Laplace equation; Meyer wavelet; regularization PDFBibTeX XMLCite \textit{C.-Y. Qiu} and \textit{C.-L. Fu}, J. Math. Anal. Appl. 338, No. 2, 1440--1447 (2008; Zbl 1135.35093) Full Text: DOI References: [1] Vani, C.; Avudainayagam, A., Regularized solution of the Cauchy problem for the Laplace equation using Meyer wavelets, Math. Comput. Modelling, 36, 1151-1159 (2002) · Zbl 1045.35103 [2] Regińska, T., Sideways heat equation and wavelets, J. Comput. Appl. Math., 63, 209-214 (1995) · Zbl 0858.65099 [3] Fu, C. L.; Qiu, C. Y.; Zhu, Y. B., A note on “Sideways heat equation and wavelets” and constant \(e^\ast \), Comput. Math. Appl., 43, 1125-1134 (2002) · Zbl 1051.65090 [4] Daubechies, I., Ten Lectures on Wavelets (1992), SIAM: SIAM Philadelphia · Zbl 0776.42018 [5] Nho Hào, Dinh; Schneider, A.; Reinhardt, H.-J., Regularization of a non-characteristic Cauchy problem for a parabolic equation, Inverse Problems, 11, 1247-1263 (1995) · Zbl 0845.35130 [6] Tautenhahn, U., Optimal stable solution of Cauchy problem for elliptic equation, J. Anal. Appl., 15, 4, 961-984 (1996) · Zbl 0865.65076 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.