Advertisement

On the Cauchy Problem for the Wave Equation with Data on the Boundary

  • M. N. DemchenkoEmail author
Article

The Cauchy problem for the wave equation in Ω × ℝ with data given on some part of the boundary ∂Ω × ℝ is considered. A reconstruction algorithm for this problem based on analytic expressions is given. This result is applicable to the problem of determining a nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Applied Mathematical Sciences, 127, Springer (2006).Google Scholar
  2. 2.
    M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, AMS Transl. of Math., Monographs, 64 (1986).Google Scholar
  3. 3.
    D. Tataru, “Unique Continuation for Solutions to PDE’s; between Hörmander’s Theorem and Holmgren’s Theorem,” Communications in Partial Differential Equations, 20, Nos. 5–6, 855–884 (1995).MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. I. Kabanikhin, D. B. Nurseitov, M. A. Shishlenin, and B. B. Sholpanbaev, “Inverse problems for the ground penetrating radar,” J. Inverse Ill-Posed Probl., 21, 885–892 (2013).MathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Natterer, “Photo-acoustic inversion in convex domains,” Inverse Probl. Imaging, 6, No. 2, 315–320 (2012).MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. A. Kruger, P. Liu, Y. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Medical Physics, 22, 1605–1609 (1995).CrossRefGoogle Scholar
  7. 7.
    T. A. Voronina, V. A. Tcheverda, and V. V. Voronin, “Some properties of the inverse operator for a tsunami source recovery,” Siberian Electr. Math. Reports, 11, 532–547 (2014).MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. I. Belishev, “Recent progress in the boundary control method,” Inverse Problems, 23, No. 5, R1–R67 (2007).MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. N. Demchenko, “The dynamical 3-dimensional inverse problem for the Maxwell system,” St. Petersburg Math. J., 23, No. 6, 943–975 (2012).MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. I. Belishev and M. N. Demchenko, “Elements of noncommutative geometry in inverse problems on manifolds,” J. Geom. Phys., 78, 29–47 (2014).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. S. Blagoveshchensky and F. N. Podymaka, “On a Cauchy problem for the wave equation with data on a time-like hyperplane,” in: Proceedings of the International Conference Days on Diffraction (2016), pp. 31–34.Google Scholar
  12. 12.
    E. T. Quinto, A. Rieder, and T. Schuster, “Local inversion of the sonar transform regularized by the approximate inverse,” Inverse Problems, 27, No. 3, 035006 (2011).MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Finch, S. K. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal., 35, No. 5, 1213–1240 (2004).MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math., 68, 392–412 (2007).MathSciNetCrossRefGoogle Scholar
  15. 15.
    R. Courant and D. Hilbert, Methods of Mathematical Physics: Volume II, Partial Differential Equations, Wiley Classics Edition (1989).Google Scholar
  16. 16.
    V. P. Palamodov, “Reconstruction from Limited Data of Arc Means,” J. Fourier Anal. Appl., 6, No. 1, 25–42 (2000).MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. N. Demchenko, “Reconstruction of a solution to the wave equation from Cauchy data on the boundary,” in: Proceedings of the International Conference Days on Diffraction (2018), pp. 66–70.Google Scholar
  18. 18.
    L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin (2009).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

Personalised recommendations