238
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Numerical method for a Cauchy problem for multi-dimensional Helmholtz equation

ORCID Icon & ORCID Icon
Article: 2321450 | Received 20 Jul 2023, Accepted 15 Feb 2024, Published online: 11 Mar 2024

References

  • Klibanov MV. Two classes of inverse problems for partial differential equation. Ann New York Acad Sci. 1992;661(1):93–111. doi: 10.1111/j.1749-6632.1992.tb26036.x
  • Tadi M, Nandakumaran AK, Sritharan SS. An inverse problem for Helmholtz equation. Inverse Probl Sci Eng. 2011;19(6):839–854. doi: 10.1080/17415977.2011.556705
  • Jin B, Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Engrg. 2006;195(19-22):2270–2288. doi: 10.1016/j.cma.2005.05.013
  • Jin B, Rundell W. A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl. 2015;31(3):035003. doi: 10.1088/0266-5611/31/3/035003
  • Bao G, Li P, Zhao Y. Stability in the inverse source problem for elastic and electromagnetic waves with multi-frequencies, 2017. Preprint, arXiv: 1703.03890.
  • EI Badia A, EI Hajj A. Stability estimates for an inverse source problem of Helmholtz equation from single Cauchy data at a fixed frequency. Inverse Probl. 2013;29(12):125008. doi: 10.1088/0266-5611/29/12/125008
  • Bao G, Lin J, Triki F. Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data. Contemp Math. 2011;548:45–60. doi: 10.1090/conm/548
  • Zhang D, Guo Y. Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation. Inverse Probl. 2015;31(3):035007. doi: 10.1088/0266-5611/31/3/035007
  • Wang X, Guo Y, Zhang D, et al. Fourier method for recovering acoustic sources from multi-frequency far-field data. Inverse Probl. 2017;33(3):035001. doi: 10.1088/1361-6420/aa573c
  • Wang G, Ma F, Guo Y, et al. Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, 2017. Preprint, arXiv: 1708.00673.
  • Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory[M]. Berlin: Springer; 1998.
  • Guo Y, Monk P, Colton D. Toward a time domain approach to the linear sampling method. Inverse Probl. 2013;29(9):095016. doi: 10.1088/0266-5611/29/9/095016
  • Marin L, Elliot L, Heggs PJ, et al. BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method. Eng Anal Bound Elem. 2004;28:1025–1034. doi: 10.1016/j.enganabound.2004.03.001
  • Kirsch A. An introduction to the mathematical theory of inverse problems. 2nd ed. New York: Springer; 2011. (Mathematical sciences).
  • Manselli P, Miller K. Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side. Ann Mat Pura Appl. 1980;123(1):161–183. doi: 10.1007/BF01796543
  • Murio DA. The mollification method and the numerical solution of ill-Posed problems. John Wiley and Sons Inc; 2011.
  • Ha`o DN. A mollification method for ill-posed problems. Numer Math. 1994;68:469–506. doi: 10.1007/s002110050073
  • Manselli P, Miller K. Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side. Ann Mat Pura Appl. 1980;123(1):161–183. doi: 10.1007/BF01796543
  • Murio DA. Numerical method for inverse transient heat conduction problems. Rev Unión Mat Argent. 1981;30:25–46.
  • Murio DA. On the estimation of the boundary temperature on a sphere from measurements at its center. J Comput Appl Math. 1982;8:111–119. doi: 10.1016/0771-050X(82)90064-X
  • He SQ, Feng XF. A regularization method to solve a Cauchy problem for the two-dimensional modified Helmholtz equation. Mathematics. 2019;7:360. doi: 10.3390/math7040360
  • Kirsch A. An introduction to the mathematical theory of inverse problems. 2nd ed. Springer; 2011. (Applied mathematical sciences).