Iterative identification of the diffusion coefficient in an initial boundary value problem for the subdiffusion equation

V. I. Vasil'ev ; A. M. Kardashevsky

Sibirskij žurnal industrialʹnoj matematiki, Tome 24 (2021) no. 2, pp. 23-37.

Résumé

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.

Détail

Keywords: Caputo fractional time derivative, finite-difference method, iterative secant method.
Mots-clés : subdiffusion equation, inverse problem, identification of the diffusion coefficient

Bibliographie

[1] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Acad. Press, San Diego, 1999 | MR | Zbl

[2] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Dekker, Marcel, 2000 | MR | Zbl

[3] A. M. Nakhushev, Elements of Fractional Calculus and Their Application, Fizmatgiz, M., 2003 (in Russian)

[4] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Sci, Amsterdam, 2006 | MR | Zbl

[5] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verl, Berlin, 2010 | MR | Zbl

[6] M. Caputo, “Lineal model of dissipation whose Q is almost frequancy independent. Part II”, Geophys. J. Astronom. Soc., 13 (1967), 529–539 | DOI

[7] O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynamics, 29 (2002), 145–155 | DOI | MR | Zbl

[8] P. Zhuang, F. Liu, “Implicit difference approximation for the time fractional diffusion equation”, J. Appl. Math. Comput., 22:3 (2006), 87–99 | DOI | MR | Zbl

[9] Y. Lin, C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation”, J. Comput. Phys., 225 (2007), 1533–1552 | DOI | MR | Zbl

[10] D. A. Murio, “Time fractional IHCP with Caputo fractional derivatives”, Comput. Math. Appl., 56 (2008), 2371–2381 | DOI | MR | Zbl

[11] A. A. Alikhanov, “A new difference scheme for the time fractional diffusion equation”, J. Comput. Phys., 280 (2015), 424–438 | DOI | MR | Zbl

[12] A. A. Alikhanov, “Stability and convergence of difference schemes for the boundary value problems of fractional diffusion equations”, Comput. Math. Math. Phys., 56:4 (2016), 561–575 | DOI | DOI | MR | Zbl

[13] A. K. Bazzaev, I. D. Tsopanov, “Difference schemes for partial differential equations of fractional order”, Ufim. Mat. Zh., 11:2 (2019), 19–35 (in Russian) | DOI | MR | Zbl

[14] V. M. Goloviznin, I. A. Korotkin, “Methods for numerical solution of some one-dimensional equations with fractional derivatives”, Differential Equations, 42:7 (2006), 967–973 | DOI | MR | Zbl

[15] M. Zecova, J. Terpak, “Heat conduction modeling by using fractional-order derivatives”, Appl. Math. Comput., 257 (2015), 365–373 | DOI | MR | Zbl

[16] A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and Its Applications, Springer-Verl., Netherlands, 1995 | MR

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

[18] S. I. Kabanikhin, Inverse and Ill-posed Problems Theory and Applications, Inverse and Ill-Posed Problems. Ser., 55, Walter de Gruyter, Berlin, 2011 | DOI | MR

[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. Equ., 199 (2016), 1–28 ; http://ejde.math.txstate.eduhttp://ejde.math.unt.edu | MR | Zbl

[20] M. D'Ovidio, P. Loreti, A. Momenzadeh, S. Ahrabi, “Determination of order in linear fractional differential equations”, J. Fract. Calculus Appl. Anal., 21:4 (2018), 937–948 | DOI | MR

[21] Z. Li, M. Yamamoto, “Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation”, Appl. Anal., 94 (2015), 570–579 | DOI | MR | Zbl

[22] A. N. Bondarenko, D. S. Ivashchenko, “Methods for the numerical solution of boundary value problems in theory of anomalous diffusion”, Sib. Electron. Mat. Izv., 5 (2008), 581–594 (in Russian) | Zbl

[23] G. Li, D. Zhang, X. Jia, M. Yamamoto, “Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation”, Inverse Probl, 29 (2013), 065014 | DOI | MR | Zbl

[24] L. Sun, X. Yan, T. Wei, “Identification of time-dependent convection coefficient in a time-fractional diffusion equation”, J. Comput. Appl. Math, 346 (2019), 505–517 | DOI | MR | Zbl

[25] A. Kardashevsky, “Fractional derivative order with respect to time for diffusion equation: an iterative method of determination”, J. Phys.: Conf. Series, 1715 (2021), 012035 | DOI

[26] V. I. Vasil'ev, A. M. Kardashevskii, “Numerical identification of order of the fractional time derivative in a subdiffusion model”, Mat. Zametki Sev. Vost. Fed. Univ., 27:4 (2020), 60–69 (in Russian) | DOI

[27] A. I. Kozhanov, “The heat transfer equation with an unknown heat capacity coefficient”, J. Appl. Indust. Math., 23:1 (2020), 104–114 | DOI | DOI | MR