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A two-dimensional inverse problem for the viscoelasticity equation

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Abstract

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ℝ2 | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.

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References

  1. Lorenzi A., Messina F., and Romanov V. G., “Recovering a Lamé kernel in a viscoelastic system,” Appl. Anal., 86, No. 11, 1375–1395 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  2. Romanov V. G. and Yamamoto M., “Recovering a Lamé kernel in a viscoelastic equation by a single boundary measurement,” Appl. Anal., 89, No. 3, 377–390 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  3. Romanov V. G., “A stability estimate for a solution to the problem of determining the kernel of the viscoelastic equation,” in: Nonclassical Equations of Mathematical Physics [in Russian], Izdat. Inst. Mat., Novosibirsk, 2010, pp. 246–253.

    Google Scholar 

  4. Romanov V. G., “A two-dimensional inverse problem of viscoelasticity,” Dokl. Math., 84, No. 2, 649–652 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  5. Romanov V. G., “Stability estimates for the solution in the problem of determining the kernel of the viscoelasticity equation,” Sibirsk. Zh. Industr. Mat., 15, No. 1, 86–98 (2012).

    Google Scholar 

  6. Lorenzi A. and Romanov V. G., “Recovering two Lamé kernels in a viscoelastic system,” Inverse Probl. Imaging, 5, No. 2, 431–464 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  7. Romanov V. G., “A three-dimensional inverse problem of viscoelasticity,” Dokl. Math., 84, No. 3, 833–836 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  8. Mukhometov R. G., “The problem of recovery of a two-dimensional Riemannian metric and integral geometry,” Soviet Math., Dokl., 18, 27–31 (1977).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. G. Romanov

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  1. V. G. Romanov
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Correspondence to V. G. Romanov.

Additional information

Original Russian Text Copyright © 2012 Romanov V.G.

The author was supported by the Russian Foundation for Basic Research (Grant 11-01-00105-a), the Siberian Division of the Russian Academy of Sciences (Integration Project No. 14), and the Visby Program.

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Translated from Sibirskiı Matematicheskiı Zhurnal, Vol. 53, No. 6, pp. 1401–1412, November–December, 2012.

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Romanov, V.G. A two-dimensional inverse problem for the viscoelasticity equation. Sib Math J 53, 1128–1138 (2012). https://doi.org/10.1134/S0037446612060171

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  • DOI: https://doi.org/10.1134/S0037446612060171

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