A Modified Method for a Cauchy Problem of the Helmholtz Equation
- 160 Downloads
In this paper, a Cauchy problem for the Helmholtz equation is investigated. It is well known that this problem is severely ill-posed in the sense that the solution (if it exists) does not depend continuously on the given Cauchy data. To overcome such difficulties, we propose a modified regularization method to approximate the solution of this problem, and then analyze the stability and convergence of the proposed regularization method based on the conditional stability estimates. Finally, we present two numerical examples to illustrate that the proposed regularization method works well.
KeywordsCauchy problem Helmholtz equation Ill-posed problem Regularization Error estimate
Mathematics Subject Classification35R30 65N12 65N20
The authors would like to thank the reviewers’ valuable comments and suggestions that have improved our manuscript. The work described in this paper was supported in part by the Fundamental Research Funds for the Central Universities (2015QNA49).
- 10.Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems, V. H. Winston & Sons, Washington, DC: John Wiley & Sons, New York (1977)Google Scholar
- 36.Showalter, R.E.: Cauchy problem for hyper-parabolic partial differential equations. In: Lakshmikantham, V. (ed.) Trends in the Theory and Practice of Non-Linear Analysis. Elsevier, North-Holland (1985)Google Scholar