Siberian Mathematical Journal

, Volume 44, Issue 4, pp 659–670 | Cite as

A Stability Estimate for a Solution to a Two-Dimensional Inverse Problem of Electrodynamics

  • V. G. Romanov


We consider the problem of finding the three coefficients c(x), σ(x), and q(x) in a hyperbolic equation. Here c(x) is the coefficient at the Laplace operator, σ(x) is the coefficient of the first time derivative, and q(x) is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients c(x)-1, σ(x), and q(x) are small in some norm and their supports are contained in some disk B. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function δ(t)δ(x·ν) localized on the set t=0, x·ν=0. Here ν is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside B is measured at points of the boundary of the domain B on some time interval of a fixed length T counted from the moment of arrival of the signal from the source for three different values of the parameter ν. It is proven that, for a sufficiently large T, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.

inverse problem equation of electrodynamics hyperbolic equation stability uniqueness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Romanov V. G. and Kabanikhin S. I., Inverse Problems for Geoelectrics [in Russian], Nauka, Moscow (1991).Google Scholar
  2. 2.
    Ola P., Päivärinta L., and Somersalo E., “An inverse boundary value problem in electrodynamics,” Duke Math. J., 70, 617–653 (1993).Google Scholar
  3. 3.
    Ola P. and Somersalo E., “Electromagnetic inverse problems and generalized Sommerfeld potential,” SIAM J. Appl. Math., 560, 1129–1145 (1996).Google Scholar
  4. 4.
    Yakhno V. G., “Multidimensional inverse problems in ray formulation for hyperbolic equations,” J. Inverse Ill-Posed Probl., 6, No. 4, 373–386 (1998).Google Scholar
  5. 5.
    Belishev M. I. and Glasman A. K., “A dynamical inverse problem for Maxwell's system: recovery of velocity in a regular zone,” Algebra i Analiz, 12, No. 2, 131–187 (2000).Google Scholar
  6. 6.
    Romanov V. G., “Inverse problems for electrodynamics,” Dokl. Akad. Nauk, 386, No. 3, 304–309 (2002).Google Scholar
  7. 7.
    Romanov V. G., “On a stability estimate for a solution to an inverse problem for a hyperbolic equation,” Sibirsk. Mat. Zh., 39, No. 2, 436–449 (1998).Google Scholar
  8. 8.
    Romanov V. G. and Yamamoto M., “Multidimensional inverse hyperbolic problem with impulse input and single boundary measurement,” J. Inverse Ill-Posed Probl., 7, No. 6, 573–588 (1999).Google Scholar
  9. 9.
    Romanov V. G., “Stability estimation in the inverse problem of determining the speed of sound,” Sibirsk. Mat. Zh., 40, No. 6, 1323–1338 (1999).Google Scholar
  10. 10.
    Glushkova D. I., “A stability estimate for a solution to the inverse problem of determination of an absorption coefficient,” Differentsial'nye Uravneniya, 37, No. 9, 1203–1211 (2001).Google Scholar
  11. 11.
    Glushkova D. I. and Romanov V. G., “A stability estimate for a solution to the problem of determination of two coefficients of a hyperbolic equation,” Sibirsk. Mat. Zh., 44, No. 2, 311–321 (2003).Google Scholar
  12. 12.
    Romanov V. G., Investigation Methods for Inverse Problems, VSP, Utrecht (2002).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • V. G. Romanov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirsk

Personalised recommendations