Determination of a minor coefficient in a time fractional diffusion equation

H. P. Lopushanska

doi:10.15330/ms.45.1.57-66

For a time fractional diffusion equation on bounded cylindrical domain the inverse problem is studied. It consists of the determination of a pair of functions: a classical solution of the second boundary value problem for such an equation and unknown, depending on all variables, minor coefficient in the equation under some integral type over-determination condition. Conditions of the existence and uniqueness of a solution are found.

1. Caputo M. Linear model of dissipation whose Q is almost friequency independent, II// Geofis. J. R. Astr. Soc. - 1967. - V.13. - P. 529-539.

2. Djrbashian M.M., Nersessyan A.B. Fractional derivatives and Cauchy problem for differentials of fractional order// Izv. AN Arm. SSR. Matematika. - 1968. - V.3. - P. 3-29. (in Russian)

3. Luchko Yu. Maximum principle for the generalized time-fractional diffusion equation// J. Math. Anal. Appl. - 2009. - V.351. - P. 409-422.

4. Meerschaert M.M., Nane E., Vallaisamy P. Fractional Cauchy problems on bounded domains// Ann. Probab. - 2009. - V.37. - P. 979-1007.

5. Kochubei A.N. The Cauchy problem for evolutionary equation of fractional order// Differential Equations. - 1989. - V.25, №8. - P. 1359-1368. (in Russian)

6. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. - Basel-Boston-Berlin, Birkhauser Verlag, 2004.

7. Voroshylov A.A., Kilbas A.A. Conditions of the existence of classical solution of the Cauchy problem for diffusion-wave equation with Caputo partial derivative// Dokl. Ak. Nauk. - 2007. - V.414, №4. - P. 1-4. (in Russian)

8. Cheng J., Nakagawa J., Yamamoto M., Yamazaki T. Uniqueness in an inverse problem for a onedimentional fractional diffusion equation// Inverse Problems. - 2000. - V.25. - P. 1-16.

9. El-Borai Mahmoud M. On the solvability of an inverse fractional abstract Cauchy problem// LJRRAS. - 2010. - V.4. - P. 411-415.

10. Nakagawa J., Sakamoto K., Yamamoto M. Overview to mathematical analysis for fractional diffusion equation - new mathematical aspects motivated by industrial collaboration// Journal of Math-for- Industry. - 2010. - V.2A. - P. 99-108.

11. Zhang Y., Xu X. Inverse source problem for a fractional diffusion equation// Inverse Problems. - 2011. - V.27. - P. 1-12.

12. Rundell W., Xu X., Zuo L. The determination of an unknown boundary condition in fractional diffusion equation// Applicable Analysis. - 2013. - V.92, №7. - P. 1511-1526.

13. Hatano Y., Nakagawa J., Wang Sh., Yamamoto M. Determination of order in fractional diffusion equation// Journal of Math-for-Industry. - 2013. - V.5A. - P. 51-57.

14. Lopushanskyj A. O. The solvability of the inverse boundary value problem for equation with fractional derivarive// Visnyk of Lviv. Un-ty, Ser. mech.-mat. - 2014. - V.79. - P. 97-110. (in Ukrainian)

15. Lopushanskyj A. O., Lopushanska H.P. One inverse boundary value problem for diffusion-wave equation with fractional derivarive// Ukr. math. J. - 2014. - V.66, №5. - P. 655-667. (in Ukrainian)

16. Jim B., Rundell W. A turorial on inverse problems for anomalous diffusion processes// Inverse Problems. - 2015. - V.31. - doi:10.1088/0266-5611/31/3/035003.

17. Ivanchov M. Inverse problems for equations of parabolic type. - Math. Studies: Monograph Ser., Lviv: VNTL Publ., V.10, 2003.

18. Snitko G. Inverse problem for parabolic equation with unknown young coefficient on domain with free boundary// Visnyk of Lviv. Un-ty, Ser. mech.-mat. - 2008. - V.68. - P. 231-245. (in Ukrainian)

19. Lopushanska H., Lopushanskyj A., Pasichnyk E. The Cauchy problem in a space of generalized functions for the equations possessing the fractional time derivarive// Sib. Math. J. - 2011. - V.52, №6. - P. 1288-1299.

20. Friedman A. Partial differential equations of parabolic type. - Prentice-Hall, Englewood Cliffs, NJ, 1964.

21. Kilbas A.A., Sajgo M. H-Transforms: Theory and Applications. - Boca-Raton, Chapman and Hall/CRC, 2004. - 401 p.

	Determination of a minor coefficient in a time fractional diffusion equation
Author	H. P. Lopushanska lhp@ukr.net Ivan Franko National University of Lviv
Abstract	For a time fractional diffusion equation on bounded cylindrical domain the inverse problem is studied. It consists of the determination of a pair of functions: a classical solution of the second boundary value problem for such an equation and unknown, depending on all variables, minor coefficient in the equation under some integral type over-determination condition. Conditions of the existence and uniqueness of a solution are found.
Keywords	fractional derivative; inverse boundary value problem; Green vector-function; Volterra integral equation
DOI	doi:10.15330/ms.45.1.57-66
Reference	1. Caputo M. Linear model of dissipation whose Q is almost friequency independent, II// Geofis. J. R. Astr. Soc. - 1967. - V.13. - P. 529-539. 2. Djrbashian M.M., Nersessyan A.B. Fractional derivatives and Cauchy problem for differentials of fractional order// Izv. AN Arm. SSR. Matematika. - 1968. - V.3. - P. 3-29. (in Russian) 3. Luchko Yu. Maximum principle for the generalized time-fractional diffusion equation// J. Math. Anal. Appl. - 2009. - V.351. - P. 409-422. 4. Meerschaert M.M., Nane E., Vallaisamy P. Fractional Cauchy problems on bounded domains// Ann. Probab. - 2009. - V.37. - P. 979-1007. 5. Kochubei A.N. The Cauchy problem for evolutionary equation of fractional order// Differential Equations. - 1989. - V.25, №8. - P. 1359-1368. (in Russian) 6. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. - Basel-Boston-Berlin, Birkhauser Verlag, 2004. 7. Voroshylov A.A., Kilbas A.A. Conditions of the existence of classical solution of the Cauchy problem for diffusion-wave equation with Caputo partial derivative// Dokl. Ak. Nauk. - 2007. - V.414, №4. - P. 1-4. (in Russian) 8. Cheng J., Nakagawa J., Yamamoto M., Yamazaki T. Uniqueness in an inverse problem for a onedimentional fractional diffusion equation// Inverse Problems. - 2000. - V.25. - P. 1-16. 9. El-Borai Mahmoud M. On the solvability of an inverse fractional abstract Cauchy problem// LJRRAS. - 2010. - V.4. - P. 411-415. 10. Nakagawa J., Sakamoto K., Yamamoto M. Overview to mathematical analysis for fractional diffusion equation - new mathematical aspects motivated by industrial collaboration// Journal of Math-for- Industry. - 2010. - V.2A. - P. 99-108. 11. Zhang Y., Xu X. Inverse source problem for a fractional diffusion equation// Inverse Problems. - 2011. - V.27. - P. 1-12. 12. Rundell W., Xu X., Zuo L. The determination of an unknown boundary condition in fractional diffusion equation// Applicable Analysis. - 2013. - V.92, №7. - P. 1511-1526. 13. Hatano Y., Nakagawa J., Wang Sh., Yamamoto M. Determination of order in fractional diffusion equation// Journal of Math-for-Industry. - 2013. - V.5A. - P. 51-57. 14. Lopushanskyj A. O. The solvability of the inverse boundary value problem for equation with fractional derivarive// Visnyk of Lviv. Un-ty, Ser. mech.-mat. - 2014. - V.79. - P. 97-110. (in Ukrainian) 15. Lopushanskyj A. O., Lopushanska H.P. One inverse boundary value problem for diffusion-wave equation with fractional derivarive// Ukr. math. J. - 2014. - V.66, №5. - P. 655-667. (in Ukrainian) 16. Jim B., Rundell W. A turorial on inverse problems for anomalous diffusion processes// Inverse Problems. - 2015. - V.31. - doi:10.1088/0266-5611/31/3/035003. 17. Ivanchov M. Inverse problems for equations of parabolic type. - Math. Studies: Monograph Ser., Lviv: VNTL Publ., V.10, 2003. 18. Snitko G. Inverse problem for parabolic equation with unknown young coefficient on domain with free boundary// Visnyk of Lviv. Un-ty, Ser. mech.-mat. - 2008. - V.68. - P. 231-245. (in Ukrainian) 19. Lopushanska H., Lopushanskyj A., Pasichnyk E. The Cauchy problem in a space of generalized functions for the equations possessing the fractional time derivarive// Sib. Math. J. - 2011. - V.52, №6. - P. 1288-1299. 20. Friedman A. Partial differential equations of parabolic type. - Prentice-Hall, Englewood Cliffs, NJ, 1964. 21. Kilbas A.A., Sajgo M. H-Transforms: Theory and Applications. - Boca-Raton, Chapman and Hall/CRC, 2004. - 401 p.
Pages	57-66
Volume	45
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html