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Iterative Process for Numerical Recovering the Lowest Order Space-Wise Coefficient in Parabolic Equations

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Finite Difference Methods. Theory and Applications (FDM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11386))

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Abstract

In this work we suggest an iterative process for coefficient inverse problem. A parabolic equation in a bounded area supplied with initial condition and monotonic nondecreasing on time Dirichlet condition on a boundary is considered. We state a problem to recover the lowest order coefficient that depends only on spatial variables under an additional information as the observation of a solution taken at the final point of time. For numerical recovering of the coefficient we build the iterative process, at each iteration we perform finite-element approximation in space and fully implicit two-level discretization in time. For capabilities of given iterative process we present computational test for a model problem.

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References

  1. Alifanov, O.M.: Inverse Heat Transfer Problems. Springer, Heidelberg (2012)

    Google Scholar 

  2. Alifanov, O.M., Artiukhin, E.A., Rumiantsev, S.V.: Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems. Begell House, New York (1995)

    Google Scholar 

  3. Burman, E., Ern, A.: Discrete maximum principle for galerkin approximations of the laplace operator on arbitrary meshes. Comptes Rendus Mathematique 338(8), 641–646 (2004). https://doi.org/10.1016/j.crma.2004.02.010

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciarlet, P., Raviart, P.A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2(1), 17–31 (1973). https://doi.org/10.1016/0045-7825(73)90019-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, W.: Discrete maximum principle and a delaunay-type mesh condition for linear finite element approximations of two-dimensional anisotropic diffusion problems. Numer. Math.: Theory Methods Appl. 4(3), 319–334 (2011). https://doi.org/10.1017/S1004897900000374

    Article  MathSciNet  MATH  Google Scholar 

  6. Isakov, V.: Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math. 44(2), 185–209 (1991)

    Article  MathSciNet  Google Scholar 

  7. Isakov, V.: Inverse Problems for Partial Differential Equations, vol. 127, 3rd edn. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-51658-5

    Book  MATH  Google Scholar 

  8. Kostin, A.B.: Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation condition. Differ. Equ. 51(5), 605–619 (2015)

    Article  MathSciNet  Google Scholar 

  9. Prilepko, A.I., Solov’ev, V.V.: On the solvability of inverse boundary value problems for the determination of the coefficient preceding the lower derivative in a parabolic equation. Differentsial’nye Uravneniya 23, 136–143 (1987)

    MathSciNet  Google Scholar 

  10. Prilepko, A.I., Kostin, A.B.: Inverse problems of the determination of the coefficient in parabolic equations. I. Siberian Math. J. 33(3), 489–496 (1992)

    Article  Google Scholar 

  11. Prilepko, A.I., Kostin, A.B.: On inverse problems of determining a coefficient in a parabolic equation. II. Siberian Math. J. 34(5), 923–937 (1993)

    Article  MathSciNet  Google Scholar 

  12. Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)

    Book  Google Scholar 

  13. Vabishchevich, P.N.: Iterative computational identification of a space-wise dependent source in parabolic equation. Inverse Probl. Sci. Eng. 25(8), 1168–1190 (2017). https://doi.org/10.1080/17415977.2016.1230611

    Article  MathSciNet  MATH  Google Scholar 

  14. Vasil’ev, F.P.: Solution Methods for Incorrect Extremal Problems. Nauka, Moscow (1981). In Russian

    MATH  Google Scholar 

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Funding

The work for the first author was supported by the mega-grant of the Russian Federation Government (N 14.Y26.31.0013), and for the second by the Russian Foundation for Basic Research (project 17-01-00689).

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Authors and Affiliations

  1. Institute of Mathematics and Information Science, North-Eastern Federal University, 58, Belinskogo, Yakutsk, Russia

    D. Kh. Ivanov & P. N. Vabishchevich

  2. Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, Moscow, Russia

    P. N. Vabishchevich

Authors
  1. D. Kh. Ivanov
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  2. P. N. Vabishchevich
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Corresponding author

Correspondence to D. Kh. Ivanov .

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Editors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

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Ivanov, D.K., Vabishchevich, P.N. (2019). Iterative Process for Numerical Recovering the Lowest Order Space-Wise Coefficient in Parabolic Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_32

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  • DOI: https://doi.org/10.1007/978-3-030-11539-5_32

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11538-8

  • Online ISBN: 978-3-030-11539-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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