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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2016, Volume 9, Issue 3, Pages 105–118 (Mi vyuru333)

Programming & Computer Software

Coefficients identification in fractional diffusion models by the method of time integral characteristics

S. Yu. Lukashchuk

Ufa State Aviation Technical University, Ufa, Russian Federation

Abstract: Inverse problems of identification of the fractional diffusivity and the order of fractional differentiation are considered for linear fractional anomalous diffusion equations with the Riemann–Liouville and Caputo fractional derivatives. As an additional information about the anomalous diffusion process, the concentration functions are assumed to be known at several arbitrary inner points of calculation domain. Numerically-analytical algorithms are constructed for identification of two required parameters of the fractional diffusion equations by approximately known initial data. These algorithms are based on the method of time integral characteristics and use the Laplace transform in time. The Laplace variable can be considered as a regularization parameter in these algorithms. It is shown that the inverse problems under consideration are reduced to the identification problem for a new single parameter which is formed by the fractional diffusivity, the order of fractional differentiation and the Laplace variable. Estimations of the upper error bound for this parameter are derived. A technique of optimal Laplace variable determination based on minimization of these estimations is described. The proposed algorithms are implemented in the AD-TIC package for the Maple software. A brief discussion of this package is also presented.

Keywords: anomalous diffusion; fractional derivatives; inverse coefficient problem; identification algorithm; software package.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 11.G34.31.0042 This work was supported by the grant of the Ministry of Education and Science of the Russian Federation (contract No. 11.G34.31.0042 with Ufa State Aviation Technical University and leading scientist Professor N.H. Ibragimov).

DOI: https://doi.org/10.14529/mmp160309

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UDC: 517.9
MSC: 65M32
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Citation: S. Yu. Lukashchuk, “Coefficients identification in fractional diffusion models by the method of time integral characteristics”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 9:3 (2016), 105–118

Citation in format AMSBIB
\Bibitem{Luk16} \by S.~Yu.~Lukashchuk \paper Coefficients identification in fractional diffusion models by the method of time integral characteristics \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2016 \vol 9 \issue 3 \pages 105--118 \mathnet{http://mi.mathnet.ru/vyuru333} \crossref{https://doi.org/10.14529/mmp160309} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000390881400009} \elib{http://elibrary.ru/item.asp?id=25717239}