Classical solution of the inverse problem
for fractional diffusion equation under time-integrated over-determination condition

H. P. Lopushanska; A. O. Lopushansky; O. M. Myaus

doi:10.15330/ms.44.2.215-220

We prove the correctness of the inverse problem on determination of a~pare of functions: a~classical solution

$u$ of the first boundary value problem for linear diffusion equation

$D^\alpha_t u-u_{xx}=F_0(x), (x,t)\in (0,l)\times (0,T]$ with regularized fractional derivative of order

$\alpha\in (0,1)$ with respect to time and function

$F_0(x)$ under integral by time over-determination condition.

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

2. Kochubei A.N. Cauchy problem for the evolutionary equation fractional order// Diff. Uravn. – 1989. – V.25, №8. – P. 1359–1368. (in Russian)

3. Povstenko Y. Linear fractional diffusion-wave equation for scientists and engeneers. – New-York, Birkhauser, 2015. – 460 p.

4. Aleroev T. S., Kirane M., Malik S. A. Determination of a source term for a time fractional diffusion equation with an integral type over-determination condition// Electronic J. of Differential Equations. – 2013.– V.270. – P. 1–16.

5. Cheng J., Nakagawa J., Yamamoto M. and Yamazaki T. Uniqueness in an inverse problem for a onedimentional fractional diffusion equation// Inverse Problems. – 2009. – V. 25. – P. 1–16.

6. Jim B., Rundell W. A turorial on inverse problems for anomalous diffusion processes // Inverse Problems. – 2015. – V. 31.

7. Lopushanska H., Rapita V. Inverse coefficient problem for semi-linear fractional telegraph equation// Electronic J. of Differential Equations. – 2015. – V.153. – P. 1–13.

8. 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.

9. Rundell W., Xu X., Zuo L. The determination of an unknown boundary condition in fractional diffusion equation// Applicable Analysis. – 2012. – V.1. – P. 1–16.

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

11. Kuz A.M. The problem with integral conditions for factorized parabolic operator with variable coefficients// Visnyk Nat. Univ. “Lviv Polytechnic”. Phys.-Mat. Sci. – 2012. – №740. – P. 24–34. (in Ukrainian)

12. Ptashnyk B.Y., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M. Nonlocal boundary value problems for equations with partial derivatives. – Kiev, Naukova dumka, 2002. – 416 p. (in Ukrainian)

13. Pulkina L.S. Nonlocal problem for equation of heat-conducting// Nonclassical problems of mathematical physics, IM SO A, Novosibirsk. – 2005. – P. 231–239. (in Russian)

14. Pukalsky I.D. Parabolic Nonlocal boundary value problem and the optimal control problem for linear equations with degeneration// Prykl. Problems. Mechanics and Mathematics. – 2012. – V.10. – P. 102– 114. (in Ukrainian)

15. Isaryuk I.M., Pukalsky I.D. Boundary value problems for parabolic equations with nonlocal conditions and degenerations// Ukr. Mat. Visnyk. – 2014. – V.11, №4. – P. 480–496. (in Russian)

16. Vladimirov V.S. Generelized functions in mathematical physycs. – 4th Edition, Nauka, Moskow, 1981. – 512 p. (in Russian)

17. Dzhrbashyan M.M. Integral transforms and representations of functions in the complex domain. – Nauka, Moskow, 1966. – 671 p. (in Russian)

18. Pollard H. The completely monotonic character of the Mittag-Leffler function

$E_\alpha(-x)$ // Bull. Amer. Math. Soc. – 1948. – V.68, №5. – P. 602–613.

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

	Classical solution of the inverse problem for fractional diffusion equation under time-integrated over-determination condition (in Ukrainian)
Author	H. P. Lopushanska, A. O. Lopushansky, O. M. Myaus lhp@ukr.net, alopushanskyj@gmail.com, myausolya@mail.ru Ivan Franko National University of Lviv, University of Zheshiv, National University Lviv Polytechnic
Abstract	We prove the correctness of the inverse problem on determination of a~pare of functions: a~classical solution $u$ of the first boundary value problem for linear diffusion equation $D^\alpha_t u-u_{xx}=F_0(x), (x,t)\in (0,l)\times (0,T]$ with regularized fractional derivative of order $\alpha\in (0,1)$ with respect to time and function $F_0(x)$ under integral by time over-determination condition.
Keywords	fractional derivative; inverse problem; Mittag-Leffler function, diffusion equation
DOI	doi:10.15330/ms.44.2.215-220
Reference	1. Caputo M. Linear model of dissipation whose Q is almost frequency independent// II. Geofis. J. R. Astr. Soc. – 1967. – V.13. – P. 529–539. 2. Kochubei A.N. Cauchy problem for the evolutionary equation fractional order// Diff. Uravn. – 1989. – V.25, №8. – P. 1359–1368. (in Russian) 3. Povstenko Y. Linear fractional diffusion-wave equation for scientists and engeneers. – New-York, Birkhauser, 2015. – 460 p. 4. Aleroev T. S., Kirane M., Malik S. A. Determination of a source term for a time fractional diffusion equation with an integral type over-determination condition// Electronic J. of Differential Equations. – 2013.– V.270. – P. 1–16. 5. Cheng J., Nakagawa J., Yamamoto M. and Yamazaki T. Uniqueness in an inverse problem for a onedimentional fractional diffusion equation// Inverse Problems. – 2009. – V. 25. – P. 1–16. 6. Jim B., Rundell W. A turorial on inverse problems for anomalous diffusion processes // Inverse Problems. – 2015. – V. 31. 7. Lopushanska H., Rapita V. Inverse coefficient problem for semi-linear fractional telegraph equation// Electronic J. of Differential Equations. – 2015. – V.153. – P. 1–13. 8. 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. 9. Rundell W., Xu X., Zuo L. The determination of an unknown boundary condition in fractional diffusion equation// Applicable Analysis. – 2012. – V.1. – P. 1–16. 10. Zhang Y., Xu X. Inverse source problem for a fractional diffusion equation// Inverse Problems. – 2011. – V.27. – P. 1–12. 11. Kuz A.M. The problem with integral conditions for factorized parabolic operator with variable coefficients// Visnyk Nat. Univ. “Lviv Polytechnic”. Phys.-Mat. Sci. – 2012. – №740. – P. 24–34. (in Ukrainian) 12. Ptashnyk B.Y., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M. Nonlocal boundary value problems for equations with partial derivatives. – Kiev, Naukova dumka, 2002. – 416 p. (in Ukrainian) 13. Pulkina L.S. Nonlocal problem for equation of heat-conducting// Nonclassical problems of mathematical physics, IM SO A, Novosibirsk. – 2005. – P. 231–239. (in Russian) 14. Pukalsky I.D. Parabolic Nonlocal boundary value problem and the optimal control problem for linear equations with degeneration// Prykl. Problems. Mechanics and Mathematics. – 2012. – V.10. – P. 102– 114. (in Ukrainian) 15. Isaryuk I.M., Pukalsky I.D. Boundary value problems for parabolic equations with nonlocal conditions and degenerations// Ukr. Mat. Visnyk. – 2014. – V.11, №4. – P. 480–496. (in Russian) 16. Vladimirov V.S. Generelized functions in mathematical physycs. – 4th Edition, Nauka, Moskow, 1981. – 512 p. (in Russian) 17. Dzhrbashyan M.M. Integral transforms and representations of functions in the complex domain. – Nauka, Moskow, 1966. – 671 p. (in Russian) 18. Pollard H. The completely monotonic character of the Mittag-Leffler function $E_\alpha(-x)$ // Bull. Amer. Math. Soc. – 1948. – V.68, №5. – P. 602–613. 19. Kilbas A.A., Sajgo M. H-Transforms: Theory and Applications. – Boca-Raton: Chapman and Hall/CRC. – 2004. – 401 p.
Pages	215-220
Volume	44
Issue	2
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Classical solution of the inverse problem for fractional diffusion equation under time-integrated over-determination condition (in Ukrainian)