Optimal error bound and a quasi-boundary value regularization method for a Cauchy problem of the modified Helmholtz equation

Abstract

In this paper, the Cauchy problem for the modified Helmholtz equation is investigated in a rectangle, where the Cauchy data is given for $y=0$ and boundary data for $x=0$ and $x=\pi$. The solution is sought in the interval $0<y\le 1$. We propose a quasi-boundary value regularization method to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates. In addition, we also carry out numerical experiments and compare numerical results of our method with Qin's methods [Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation, Math. Comput. Simulation 80 (2009), pp. 352–366] and Tuan's methods [Regularization and new error estimates for a modified Helmholtz equation, An. Stiint Univ. ‘Ovidius' Constanta Ser. Mat. 18(2) (2010), pp. 267–280]. It shows that our quasi-boundary value method give a better results than quasi-reversibility method of Qin and modified regularization method of Tuan.

Keywords:

• ill-posed
• the Cauchy problem of modified Helmholtz equation
• quasi-boundary value method
• error estimate
• stability estimate
• regularization

2000 AMS Subject Classification:

• 35R40

Acknowledgements

The authors would like to thank the reviewers for their very careful reading and for pointing out several mistakes as well as for their useful comments and suggestions. We would also like to thank Mrs Chen for all her kindness and help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by Natural Science Foundation of Hubei Province of China (2015CFB708), Educational Commission of Hubei Province of China (D20152801) and the Fundamental Research Funds for the Central Universities No. SWJTU11BR078