A Modified Method for a Cauchy Problem of the Helmholtz Equation

Article

Abstract

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.

Keywords

Cauchy problem Helmholtz equation Ill-posed problem Regularization Error estimate 

Mathematics Subject Classification

35R30 65N12 65N20 

Notes

Acknowledgments

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).

References

  1. 1.
    Beskos, D.E.: Boundary element methods in dynamic analysis: part II (1986–1996). Appl. Mech. Rev. 50, 149–197 (1997)CrossRefGoogle Scholar
  2. 2.
    Chen, J.T., Wong, F.C.: Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J. Sound. Vib. 217, 75–95 (1998)CrossRefGoogle Scholar
  3. 3.
    Hall, W.S., Mao, X.Q.: Boundary element investigation of irregular frequencies in electromagnetic scattering. Eng. Anal. Bound. Elem. 16, 245–252 (1995)CrossRefGoogle Scholar
  4. 4.
    Harari, I., Barbone, P.E., Slavutin, M., Shalom, R.: Boundary infinite elements for the Helmholtz equation in exterior domains. Int. J. Numer. Methods Eng. 41, 1105–1131 (1998)CrossRefMATHGoogle Scholar
  5. 5.
    Arendt, W., Regińska, T.: An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains. J. Inverse Ill-Posed Probl. 17, 703–711 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, G., Zhou, J.: Boundary Element Methods. Computational Mathematics and Applications. Academic Press, London (1992)MATHGoogle Scholar
  7. 7.
    Regińska, T., Regiński, K.: Approximate solution of a Cauchy problem for the Helmholtz equation. Inverse Probl. 22, 975–989 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)MATHGoogle Scholar
  9. 9.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences, vol. 127. Springer, New York (1998)MATHGoogle Scholar
  10. 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
  11. 11.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Kluwer Academic, Dordrecht (1996)CrossRefMATHGoogle Scholar
  12. 12.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol. 120. Springer, New York (2011)MATHGoogle Scholar
  13. 13.
    Cheng, H., Fu, C.L., Feng, X.L.: An optimal filtering method for the Cauchy problem of the Helmholtz equation. Appl. Math. Lett. 24, 958–964 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Feng, X.L., Fu, C.L., Cheng, H.: A regularization method for solving the Cauchy problem for the Helmholtz equation. Appl. Math. Model. 35, 3301–3315 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fu, C.L., Feng, X.L., Qian, Z.: The Fourier regularization for solving the Cauchy problem for the Helmholtz equation. Appl. Numer. Math. 59, 2625–2640 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jin, B.T., Marin, L.: The plane wave method for inverse problems associated with Helmholtz-type equations. Eng. Anal. Bound. Elem. 32, 223–240 (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Jin, B.T., Zheng, Y.: Boundary knot method for some inverse problems associated with the Helmholtz equation. Int. J. Numer. Methods Eng. 62, 1636–1651 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Marin, L., Elliott, L., Heggs, P.J., Ingham, D.B., Lesnic, D., Wen, X.: Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations. Comput. Mech. 31, 367–377 (2003)MathSciNetMATHGoogle Scholar
  19. 19.
    Marin, L., Elliott, L., Heggs, P.J., Ingham, D.B., Lesnic, D., Wen, X.: BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method. Eng. Anal. Bound. Elem. 28, 1025–1034 (2004)CrossRefMATHGoogle Scholar
  20. 20.
    Marin, L., Elliott, L., Heggs, P.J., Ingham, D.B., Lesnic, D., Wen, X.: Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation. Int. J. Numer. Methods Eng. 60, 1933–1947 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput. Struct. 83, 267–278 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Qin, H.H., Wei, T.: Modified regularization method for the Cauchy problem of the Helmholtz equation. Appl. Math. Model. 33, 2334–2348 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Regińska, T., Wakulicz, A.: Wavelet moment method for the Cauchy problem for the Helmholtz equation. J. Comput. Appl. Math. 223, 218–229 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sun, Y., Zhang, D.Y., Ma, F.M.: A potential function method for the Cauchy problem of elliptic operators. J. Math. Anal. Appl. 395, 164–174 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wei, T., Hon, Y.C., Ling, L.: Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng. Anal. Bound. Elem. 31, 373–385 (2007)CrossRefMATHGoogle Scholar
  26. 26.
    Xiong, X.T., Zhao, X.C., Wang, J.X.: Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation. Numer. Algorithms 63, 691–711 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zhang, H.W., Qin, H.H., Wei, T.: A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation. Int. J. Comput. Math. 88, 839–850 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Xiong, X.T.: A regularization method for a Cauchy problem of the Helmholtz equation. J. Comput. Appl. Math. 233, 1723–1732 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Han, H., Reinhardt, H.J.: Some stability estimates for Cauchy problems for elliptic equations. J. Inverse Ill-Posed Probl. 5, 437–454 (1997)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tautenhahn, U.: Optimal stable solution of Cauchy problems for elliptic equations. J. Anal. Appl. 15, 961–984 (1996)MathSciNetMATHGoogle Scholar
  31. 31.
    Cheng, J., Yamamoto, M.: One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Probl. 16, L31–L38 (2000)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kabanikhin, S.I., Schieck, M.: Impact of conditional stability: convergence rates for general linear regularization methods. J. Inverse Ill-Posed Probl. 16, 267–282 (2008)MathSciNetMATHGoogle Scholar
  33. 33.
    Hào, D.N., Van Duc, N., Lesnic, D.: A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Probl. 25, 055002 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wei, T., Qin, H.H., Zhang, H.W.: Convergence estimates for some regularization methods to solve a Cauchy problem of the Laplace equation. Numer. Math. Theor. Methods Appl. 4, 459–477 (2011)MathSciNetMATHGoogle Scholar
  35. 35.
    Qin, H.H., Wei, T.: Some filter regularization methods for a backward heat conduction problem. Appl. Math. Comput. 217, 10317–10327 (2011)MathSciNetMATHGoogle Scholar
  36. 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
  37. 37.
    Ames, K.A., Clark, G.W., Epperson, J.F., Oppenheimer, S.F.: A comparison of regularizations for an ill-posed problem. Math. Comput. 67, 1451–1471 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and TechnologyXuzhouChina

Personalised recommendations