Skip to main content
SpringerLink
Log in

A Method for Studying Singular Integral Equations

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We examine a singular integral equation of the first kind on a bounded open set of an n-dimensional space. Open subsets with a common (contact) (n — 1)-dimensional piecewise smooth part of boundaries are selected. The underdetermined case is treated in which the unknown part of the integrand depends on 2n independent variables whereas a given integral depends only on n variables. In this situation we pose the problem of finding the contact part of the boundaries and prove unique solvability of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mikhlin S. G., Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford, London, Edinburgh, New York, Paris, and Frankfurt (1965).

    MATH  Google Scholar 

  2. Stefan J., “Sitzungsber,” Wien. Akad. Math.-Naturwiss., Bd 98a, 473–484 (1890).

    Google Scholar 

  3. Anikonov D. S., “The inverse problem of determination of a body for a transport equation,” Differentsial’nye Uravneniya, 12, No. 1, 567–570 (1976).

    MathSciNet  Google Scholar 

  4. Anikonov D. S., “On the boundedness of a singular integral operator in the space C6h,” Math. USSR-Sb., 33, No. 4, 447–464 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  5. Anikonov D. S., “Stefan-type problem for a transport equation,” Dokl. Phys., 39, No. 9, 599–602 (1994).

    MathSciNet  Google Scholar 

  6. Anikonov D. S., “Integro-differentiation indicator of non-homogeneity in a tomography problem,” J. Inverse Ill-Posed Probl., 7, No. 1, 17–59 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  7. Anikonov D. S., Kovtanyuk A. E., and Prokhorov I. V., Application of the Transport Equation in Tomography [in Russian], Logos, Moscow (2000).

    Google Scholar 

  8. Anikonov D. S., Kovtanyuk A. E., and Prokhorov I. V., Transport Equation and Tomography, VSP, Utrecht (2002).

    MATH  Google Scholar 

  9. Anikonov D. S., “A special problem of integral geometry,” Dokl. Math., 76, No. 1, 483–485 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  10. Anikonov D. S., “The indicator of contact boundaries for an integral geometry problem,” Siberian Math. J., 49, 587–600 (2008).

    Article  MathSciNet  Google Scholar 

  11. Tikhonov A. N. and Samarskii A. A., Equations of Mathematical Physics, Pergamon Press, Oxford etc. (1963).

    MATH  Google Scholar 

  12. Faridani A., Keinert F., Ritman T. L., and Smith K. T., “Local and global tomography,” in: Signal Processing, Springer-Verlag, New York, 1990, pp. 241–245 (IMA Vol. Math. Appl.; 23).

    Google Scholar 

  13. Louis A. K. and Maass P., “Contour reconstruction in 3D X-Ray CT,” IEEE Trans. Med. Imaging, 12, No. 4, 109–115 (1993).

    Article  Google Scholar 

  14. Katsevich A. I. and Ramm A. G., “New methods for finding values of the jumps of a function from its local tomographic data,” Inverse Probl., 11, No. 5, 1005–1023 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  15. Lavrent’ev M. M., “Mathematical problems of tomography and hyperbolic mappings,” Siberian Math. J., 42, No. 5, 916–925 (2001).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    D. S. Anikonov

Authors
  1. D. S. Anikonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. S. Anikonov.

Additional information

Original Russian Text Copyright © 2010 Anikonov D. S.

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 5, pp. 961-973, September-October, 2010

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anikonov, D.S. A Method for Studying Singular Integral Equations. Sib Math J 51, 765–775 (2010). https://doi.org/10.1007/s11202-010-0077-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-010-0077-5

Keywords

Search

Navigation