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Iterative Identification of the Diffusion Coefficient in an Initial Boundary Value Problem for the Subdiffusion Equation

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Abstract

We propose an iterative solution method for an implicit finite-difference analog of the inverse problem of identifying the diffusion coefficient in an initial boundary value problem for the subdiffusion equation with the fractional Caputo time derivative. We consider the two different ways of setting the overdetermination condition at the final time point: the value of the solution at some given point and a weighted integral of the solution. The results of numerical implementation of the iterative method are presented on model problems with exact solutions. These results confirm the sufficiently high accuracy of the method.

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REFERENCES

  1. I. Podlubny, Fractional Differential Equations (Acad. Press, San Diego, 1999) [Ser. Mathematics in Science and Engineering].

  2. A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 2000).

    MATH  Google Scholar 

  3. A. M. Nakhushev, Elements of Fractional Calculus and Their Application (Kabardino-Balkar Sci. Center, Nalchik, 2000; Fizmatgiz, Moscow, 2003) [in Russian].

  4. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).

    MATH  Google Scholar 

  5. K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Berlin, 2010).

    Book  Google Scholar 

  6. M. Caputo, “Linear Model of Dissipation Whose Q Is Almost Frequency Independent. Part II,” Geophys. J. Astronom. Soc. 13, 529–539 (1967).

    Article  Google Scholar 

  7. O. P. Agrawal, “Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain,” Nonlinear Dynamics 29, 145–155 (2002).

    Article  MathSciNet  Google Scholar 

  8. P. Zhuang and F. Liu, “Implicit Difference Approximation for the Time Fractional Diffusion Equation,” J. Appl. Math. Comput. 22 (3), 87–99 (2006).

    Article  MathSciNet  Google Scholar 

  9. Y. Lin and C. Xu, “Finite Difference/Spectral Approximations for the Time-Fractional Diffusion Equation,” J. Comput. Phys. 225, 1533–1552 (2007).

    Article  MathSciNet  Google Scholar 

  10. D. A. Murio, “Time Fractional IHCP with Caputo Fractional Derivatives,” Comput. Math. Appl. 56, 2371–2381 (2008).

    Article  MathSciNet  Google Scholar 

  11. A. A. Alikhanov, “A New Difference Scheme for the Time Fractional Diffusion Equation,” J. Comput. Phys. 280, 424–438 (2015).

    Article  MathSciNet  Google Scholar 

  12. A. A. Alikhanov, “Stability and Convergence of Difference Schemes for the Boundary Value Problems of Fractional Diffusion Equations,” Zh. Vychisl. Mat. Mat. Fiz. 56 (4), 572–586 (2016) [Comput. Math. Math. Phys. 56 (4), 561–575 (2016)].

    Article  MathSciNet  Google Scholar 

  13. A. K. Bazzaev and I. D. Tsopanov, “Difference Schemes for Partial Differential Equations of Fractional Order,” Ufim. Mat. Zh. 11 (2), 19–35 (2019).

    Article  MathSciNet  Google Scholar 

  14. V. M. Goloviznin and I. A. Korotkin, “Methods for Numerical Solution of Some One-Dimensional Equations with Fractional Derivatives,” Differentsial’nye Uravneniya 42 (7), 907–913 (2006) [Differential Equations 42, 967–973 (2006)].

    Article  MathSciNet  Google Scholar 

  15. M. Zecova and J. Terpak, “Heat Conduction Modeling by Using Fractional-Order Derivatives,” Appl. Math. Comput. 257, 365–373 (2015).

    MathSciNet  MATH  Google Scholar 

  16. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Springer, Netherlands, 1995).

    Book  Google Scholar 

  17. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics (Walter de Gruyter, Berlin, 2008) [Ser. Inverse and Ill-Posed Problems, Vol. 52].

  18. S. I. Kabanikhin, Inverse and Ill-posed Problems: Theory and Applications (Walter de Gruyter, Berlin, 2011) [Ser. Inverse and Ill-Posed Problems, Vol. 55].

  19. J. Janno, “Determination of the Order of Fractional Derivative and a Kernel in an Inverse Problem for a Generalized Time Fractional Diffusion Equation,” Electron. J. Differ. Equat. 199, 1–28 (2016).

    MathSciNet  MATH  Google Scholar 

  20. M. D’Ovidio, P. Loreti, A. Momenzadeh, and S. Ahrabi, “Determination of Order in Linear Fractional Differential Equations,” J. Fract. Calculus Appl. Anal. 21 (4), 937–948 (2018).

    Article  MathSciNet  Google Scholar 

  21. Z. Li and M. Yamamoto, “Uniqueness for Inverse Problems of Determining Orders of Multi-Term Time-Fractional Derivatives of Diffusion Equation,” Appl. Anal. 94, 570–579 (2015).

    Article  MathSciNet  Google Scholar 

  22. A. N. Bondarenko and D. S. Ivaschenko, “Methods for the Numerical Solution of Boundary Value Problems in Theory of Anomalous Diffusion,” Sibir. Electron. Mat. Izv. 5, 581–594 (2008).

    MATH  Google Scholar 

  23. G. Li, D. Zhang, X. Jia, and M. Yamamoto, “Simultaneous Inversion for the Space-Dependent Diffusion Coefficient and the Fractional Order in the Time-Fractional Diffusion Equation,” Inverse Probl. 29, 065014 (2013).

    Article  MathSciNet  Google Scholar 

  24. L. Sun, X. Yan, and T. Wei, “Identification of Time-Dependent Convection Coefficient in a Time-Fractional Diffusion Equation,” J. Comput. Appl. Math. 346, 505–517 (2019).

    Article  MathSciNet  Google Scholar 

  25. A. Kardashevsky, “Fractional Derivative Order with Respect to Time for a Diffusion Equation: An Iterative Method of Determination,” J. Phys.: Conf. Series 1715, 012035 (2021).

    Google Scholar 

  26. V. I. Vasiliev and A. M. Kardashevsky, “Numerical Identification of Order of the Fractional Time Derivative in a Subdiffusion Model,” Mat. Zametki Sev.-Vost. Fed. Univ. 27 (4), 60–69 (2020).

    Google Scholar 

  27. A. I. Kozhanov, “The Heat Transfer Equation with an Unknown Heat Capacity Coefficient,” Sibir. Zh. Ind. Mat. 23 (1), 93–106 (2020) [J. Appl. Ind. Math. 14 (1), 104–114 (2020)].

    Article  MathSciNet  Google Scholar 

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Funding

The authors were supported by the Ministry of Education and Science of the Russian Federation (supplementary agreement no. 075–02-=2020–1543/1).

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Authors and Affiliations

  1. Ammosov North-Eastern Federal University, Yakutsk, 677000, Russia

    V. I. Vasil’ev & A. M. Kardashevsky

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  1. V. I. Vasil’ev
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  2. A. M. Kardashevsky
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Correspondence to V. I. Vasil’ev or A. M. Kardashevsky.

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Translated by L.B. Vertgeim

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Vasil’ev, V.I., Kardashevsky, A.M. Iterative Identification of the Diffusion Coefficient in an Initial Boundary Value Problem for the Subdiffusion Equation. J. Appl. Ind. Math. 15, 343–354 (2021). https://doi.org/10.1134/S1990478921020162

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  • DOI: https://doi.org/10.1134/S1990478921020162

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