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Mathematical Problems in Engineering
Volume 2014, Article ID 917972, 9 pages
http://dx.doi.org/10.1155/2014/917972
Research Article

An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

School of Science, Jiangnan University, Jiangsu Province, Wuxi 214122, China

Received 12 January 2014; Accepted 19 February 2014; Published 25 March 2014

Academic Editor: Rolf Stenberg

Copyright © 2014 Hao Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. E. Beskos, “Boundary element methods in dynamic analysis: part II (1986–1996),” Applied Mechanics Reviews, vol. 50, no. 3, pp. 149–197, 1997. View at Google Scholar · View at Scopus
  2. J. T. Chen and F. C. Wong, “Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition,” Journal of Sound and Vibration, vol. 217, no. 1, pp. 75–95, 1998. View at Google Scholar · View at Scopus
  3. I. Harari, P. E. Barbone, M. Slavutin, and R. Shalom, “Boundary infinite elements for the Helmholtz equation in exterior domains,” International Journal for Numerical Methods in Engineering, vol. 41, no. 6, pp. 1105–1131, 1998. View at Google Scholar · View at Scopus
  4. W. S. Hall and X. Q. Mao, “A boundary element investigation of irregular frequencies in electromagnetic scattering,” Engineering Analysis with Boundary Elements, vol. 16, no. 3, pp. 245–252, 1995. View at Google Scholar · View at Scopus
  5. L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 5-6, pp. 709–722, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. L. Marin and D. Lesnic, “The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations,” Computers and Structures, vol. 83, no. 4-5, pp. 267–278, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation,” International Journal for Numerical Methods in Engineering, vol. 60, no. 11, pp. 1933–1947, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method,” Engineering Analysis with Boundary Elements, vol. 28, no. 9, pp. 1025–1034, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. T. Regińska and A. Wakulicz, “Wavelet moment method for the Cauchy problem for the Helmholtz equation,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 218–229, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. H.-H. Qin and T. Wei, “Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation,” Mathematics and Computers in Simulation, vol. 80, no. 2, pp. 352–366, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. X. L. Feng, C. L. Fu, and H. Cheng, “A modified Tikhonov regularization for solving the Cauchy problem for the Helmholtz equation,” http://www.mai.liu.se/xifen.
  12. H. H. Qin, T. Wei, and R. Shi, “Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 39–53, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. C.-L. Fu, X.-L. Feng, and Z. Qian, “The Fourier regularization for solving the Cauchy problem for the Helmholtz equation,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2625–2640, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. T. Regińska and K. Regiński, “Approximate solution of a Cauchy problem for the Helmholtz equation,” Inverse Problems, vol. 22, no. 3, pp. 975–989, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. T. Regińska and U. Tautenhahn, “Conditional stability estimates and regularization with applications to cauchy problems for the helmholtz equation,” Numerical Functional Analysis and Optimization, vol. 30, no. 9-10, pp. 1065–1097, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. H. Cheng and C.-L. Fu, “An iteration regularization for a time-fractional inverse diffusion problem,” Applied Mathematical Modelling, vol. 36, no. 11, pp. 5642–5649, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  18. M. T. Nair, E. Schock, and U. Tautenhahn, “Morozov's discrepancy principle under general source conditions,” Zeitschrift für Analysis und ihre Anwendungen, vol. 22, no. 1, pp. 199–214, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus