A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation

Abstract

In this paper, we study a fractional Tikhonov regularization method (FTRM) for solving a Cauchy problem of Helmholtz equation in the frequency domain. On the one hand, the FTRM retains the advantage of classical Tikhonov method. On the other hand, our method can prevent the effect of oversmoothing of classical Tikhonov method and conveniently control the amount of damping. The convergence error estimates between the exact solution and its regularization approximation are constructed. Several interesting numerical examples are provided, which validate the effectiveness of the proposed method.

Keywords:

• Inverse problem
• ill-posed problem
• filtering idea
• fractional Tikhonov method
• Cauchy problem for Helmholtz equation

AMS Subject Classifications:

• 65N20
• 65N21
• 65N12

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11271187], [grant number 11671199], [grant number 11401456]; Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Natural Science Basic Research Plan in Shaanxi Province of China [No. 2015JQ1016].