Shi, R.; Wei, T.; Qin, H. H. A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation. (English, Russian) Zbl 1212.65355 Numer. Math., Theory Methods Appl. 2, No. 3, 326-340 (2009). Summary: This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain \(0<x \leqslant 1\), \(y\in \mathbb R\). The Cauchy data at \(x = 0\) is given and the solution is then sought for the interval \(0<x \leqslant 1\). This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under suitable choices of the regularization parameters and the a priori assumption on the bounds of the exact solution. A numerical implementation is considered and numerical examples show that the proposed method is effective and stable. Cited in 9 Documents 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 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs Keywords:Cauchy problem; Helmholtz equation; ill-posed problem; fourth-order modified method; convergence; regularization; numerical implementation PDFBibTeX XMLCite \textit{R. Shi} et al., Numer. Math., Theory Methods Appl. 2, No. 3, 326--340 (2009; Zbl 1212.65355) Full Text: DOI