Numerical Determination of Source from Point Observation in a Time-Fractional Boundary-Value Problem on Disjoint Intervals | SpringerLink
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Numerical Determination of Source from Point Observation in a Time-Fractional Boundary-Value Problem on Disjoint Intervals

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Large-Scale Scientific Computations (LSSC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13952))

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Abstract

In the present work, we solve numerically an inverse problem for identification of sources from point observations in a time-fractional diffusion-reaction problem defined on disjoint intervals. The fractional derivative is in Caputo sense with different fractional order on each of the subintervals. We propose algorithms, based on decomposition with respect to the sources on time adaptive mesh. Numerical tests illustrate the efficiency of the proposed approach.

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References

  1. Amosov, A.A.: Global solvability of a nonlinear nonstationary problem with a nonlocal boundary condition of radiation heat transfer type. Differ. Equ. 41(1), 96–109 (2005)

    Article  MathSciNet  Google Scholar 

  2. Caputo, M.: Vibrations of infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56(3), 897–904 (1974)

    Article  Google Scholar 

  3. Datta, A.K.: Biological and Bioenvironmental Heat and Mass Transfer. Marcel Dekker, New York (2002)

    Book  Google Scholar 

  4. Dib, F., Kirane, M.: An inverse source problem for a two terms time-fractional diffusion equation. Bol. Soc. Paran. Mat. 40, 1–15 (2022)

    MathSciNet  Google Scholar 

  5. Duc, N.V., Thang, N.V., Thanh, N.T.: The quasi-reversability method for an inverse sourse problem for time-space fractional parabolic equations. J. Differ. Equ. 344, 102–130 (2023)

    Article  Google Scholar 

  6. Hasanoglu, A., Romanov, V.G.: Introduction to Inverse Problems for Differential Equations, 1st edn, 261 p. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62797-7

  7. Hussein, M., Lesnic, D., Kamynin, V., Kostin, A.: Direct and inverse problems for degenerate parabolic equations. J. Inverse Ill-Posed Probl. 28(3), 425–448 (2020)

    Article  MathSciNet  Google Scholar 

  8. Jovanovic, B.S., Delic, A., Vulkov, L.G.: About some boundaery value problems for fractional PDE and their numerical solution. Proc. Appl. Math. Mech. 13, 445–446 (2013)

    Article  Google Scholar 

  9. Jovanovic, B.S., Vulkov, L.G., Delic, A.: Boundary value problems for fractional PDE and their numerical approximation. In: Dimov, I., et al. (eds.) NAA 2012. LNCS, vol. 8236, pp. 38–49. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41515-9_4

    Chapter  Google Scholar 

  10. Klibas, A.A., Sriv, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  11. Koleva, M.N., Milovanović Jeknić, Z. D., Vulkov, L.G.: Determination of external boundary conditions of a stationary nonlinear problem on disjoint intervals at point observation. Study in Computational Intelligence. Springer, Cham (accepted)

    Google Scholar 

  12. Lesnic, D.: Inverse Problems with Applications in Science and Engineering, p. 349. CRC Press, Abingdon (2021)

    Book  Google Scholar 

  13. Liu, Y., Li, Z., Yamamoto, M.: Inverse problem of determining sources of the fractional partial differential equations. Fract. Diff. Equat. 2, 411–430 (2019)

    Article  MathSciNet  Google Scholar 

  14. Milovanovic, Z.: Finite difference scheme for a parabolic transmission problem in disjoint domains. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 403–410. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41515-9_45

    Chapter  Google Scholar 

  15. Podlubny, I.: Fractional Differential Rquations, Academic Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (1998)

    Google Scholar 

  16. Qatanani, N., Barham, A., Heeh, Q.: Existence and uniqueness of the solution of the coupled conduction radiation energy transfer on diffuse gray surfaces. Surv. Math. Appl. 2, 43–58 (2007)

    MathSciNet  Google Scholar 

  17. Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems in Mathematical Physics, 438 p. de Gruyter, Berlin (2007)

    Google Scholar 

  18. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)

    Article  MathSciNet  Google Scholar 

  19. Zhang, Y., Sun, Z., Liao, H.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)

    Article  MathSciNet  Google Scholar 

  20. Zhuo, L., Lesnic, D., Meng, S.: Reconstruction of the heat transfer coefficient at the interface of a bi-material. Inverse Probl. Sci. Eng. 28, 374–401 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is supported by the Bulgarian National Science Fund under Project KP-06-N 62/3 from 2022 and partly by FNSE-03.

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Authors and Affiliations

  1. University of Ruse, 8 Studentska Street, 7017, Ruse, Bulgaria

    Miglena N. Koleva & Lubin G. Vulkov

Authors
  1. Miglena N. Koleva
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  2. Lubin G. Vulkov
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Correspondence to Miglena N. Koleva .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Sofia, Bulgaria

    Ivan Lirkov

  2. Institute of Information and Communication Technologies, Sofia, Bulgaria

    Svetozar Margenov

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Koleva, M.N., Vulkov, L.G. (2024). Numerical Determination of Source from Point Observation in a Time-Fractional Boundary-Value Problem on Disjoint Intervals. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computations. LSSC 2023. Lecture Notes in Computer Science, vol 13952. Springer, Cham. https://doi.org/10.1007/978-3-031-56208-2_13

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  • DOI: https://doi.org/10.1007/978-3-031-56208-2_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-56207-5

  • Online ISBN: 978-3-031-56208-2

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