# Coefficients Identification in Fractional Diffusion Models by the Method of Time Integral Characteristics

S.Yu. LukashchukInverse 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.Full text

- Keywords
- anomalous diffusion; fractional derivatives; inverse coefficient problem; identification algorithm; software package.
- References
- 1. Samko S., Kilbas A., Marichev O. Fractional Integrals and Derivatives. Theory and Applications. Amsterdam, Gordon & Breach Sci. Publishers, 1993. 1006 p.

2. Podlubny I. Fractional Differential Equations. San Diego, Academic press, 1999. 340 p.

3. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier, 2006. 523 p.

4. Uchaikin V.V. Metod drobnykh proizvodnykh [Method of Fractional Derivatives]. Ul'yanovsk, Artishok Publ., 2008. 512 p.

5. Anomalous Transport: Foundations and Applications. Berlin, Willey-VCH, 2008. 584 p.

6. Fractional Dynamics: Recent Advances. Singapore, World Scientific, 2011. 532 p.

7. Metzler R., Klafter J. The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamic Approach. Physics Reports, 2000, vol. 339, pp. 1-77. DOI: 10.1016/S0370-1573(00)00070-3

8. Uchaikin V.V. Self-Similar Anomalous Diffusion and Levy-Stable Laws. Physics-Uspekhi, 2003, vol. 46, no. 8, pp. 821-849. DOI: 10.1070/PU2003v046n08ABEH001324

9. Pskhu A.V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Partial Differential Equations of Fractional Order]. Moskow, Nauka, 2005. 199 p.

10. Luchko Yu. Anomalous Diffusion: Models, Their Analysis, and Interpretation. Advances in Applied Analysis. Boston, Birkhauser Verlag, 2012.

11. Vlasov V.V., Shatalov S.Yu. et all. Teplofizicheskie izmereniya: spravochnoe posobie [Thermophysical Measurements: Reference Manual]. Tambov, VNIIRTMash Publ., 1975. 254 p.

12. Vlasov V.V., Seregina V.G., Shatalov Yu.S. Integral Characteristics in the Determination of Coefficients of Parabolic Systems and Equations. Journal of Engineering Physics and Thermophysics, 1977, vol. 32, no. 4, pp. 453-458. DOI: 10.1007/BF00867038

13. Shatalov Yu.S. Integral'nye predstavleniya posoyannikh koeffitsientov teploperenosa [Integral Representation of Constant Heat Transfer Coefficients]. Ufa, Publ. of Ufa Aviation Institute, 1992. 82 p.

14. Vlasov V.V., Shatalov Yu.S., Zotov E.N., Churikov A.A., Filin N.A. Methods and Equipment for Nondestructive Control of the Thermophysical Properties of the Materials of Massive Solids. Measurement Techniques, 1980, vol. 23, no. 6, pp. 524-528. DOI: 10.1007/BF00825973

15. Shatalov Yu.S., Lukashchuk S.Yu., Rikachev Yu.Yu. The Problem of Coefficients Identification in the Mathematical Model of the Ion Implantation Diffusion Process. Inverse Problems in Engineering, 1999, vol. 7, pp. 267-290. DOI: 10.1080/174159799088027697

16. Lukashchuk S.Yu. [Solving of Inverse Coefficients Problems for Equations of Parabolic Type by the Method of Integral Characteristic]. Vestnik UGATU, 2003, vol. 4, no. 2, pp. 67-71. (in Russian)

17. Lukashchuk S.Yu. Estimation of Parameters in Fractional Subdiffusion Equations by the Time Integral Characteristics Method. Computers and Mathematics with Applications, 2011, vol. 62, no. 3, pp. 834-844. DOI: 10.1016/j.camwa.2011.03.058

18. Krylov V.I. Priblizhennoe vichislenie integralov [Approximate Calculation of Integrals]. Moscow, Nauka, 1967. 500 p.

19. Lukashchuk S.Yu. AD-TIC: Identification of Parameters of Anomalous Diffusion Equation by the Method of Time Integral Characteristics. Certificate of State Registration for the Computer Program, No. 2016610761, January 19, 2016.