Identification of the unknown parameters in the parabolic equation in a free
boundary domain

N. M. Huzyk

doi:10.15330/ms.51.1.168-182

In a free boundary domain there are established conditions of the existence and uniqueness of the classical solution to the inverse problem of identification of the time-dependent both major and minor coefficients in the parabolic equation with degeneration. The case of arbitrary weak degeneration is investigated.

1. Jones B.F., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), №5, 907-918.

2. Cannon J.R., Recovering a time-dependent coefficient in a parabolic differential equation, J. Math. Anal. Appl., 160 (1991), 572-582.

3. Azari H., Li C., Nie Y., Shang S., Determination of an unknown coefficient in a parabolic inverse problem, Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Math. Analysis, 11 (2004), 665-674.

4. Oussaeif Taki-Eddine, Bouziani Abdelfatah, An inverse coefficient problem for parabolic equation under nonlocal boundary and integral overdetermination conditions, International Journal of Partial Differential Equation and Applications, 2 (2014), №3, 38-43.

5. Mansur I. Ismailov, Bulent O.gur, An inverse diffusion problem with nonlocal boundary conditions, Numerical Methods for Partial Differential Equations, 32, 2 (2016), 564-590.

6. Cannon J.R., Peres-Esteva S., Determination of the coefficient of $u_x$ in a linear parabolic equation, Inverse Problems, 10 (1993), №3, 521-531.

7. Hong-Ming Yin, Global solvability for some parabolic inverse problems, J. Math. Anal. and Appl., 162 (1991), 392-403.

8. Trong D.D., Ang D.D., Coefficient identification for a parabolic equation, Inverse Problems, 10 (1994), №3, 733-752.

9. Dehghan M., Parameter identification in a partial differential equation from overspecified data, Math. and Computer Modelling, 41 (2005), 197-213.

10. Onyejekwe O.N., Determination of two time-dependent coefficients in a parabolic partial differential equation by homotopy analysis method, International Journal of Applied Math. Research, 3 (2014), №2, 161–167.

11. Hussein M., Lesnic D., Ivanchov M., Simultaneous determination of time-dependent coefficients in the heat equation, Computers & Mathematics with Applications, 67 (2014), №5, 1065–1091.

12. BaiyuWang, Anping Liao,Wei Liu, Simultaneous determination of unknown two parameters in parabolic equation, International Journal of Applied Math. and Computation, 4 (2012), №3, 332–336.

13. Ebru Ozbilge, Ali Demir, Identification of the unknown coefficients in a linear parabolic equation via semigroup approach, Advances in Diff. Equations, 47 (2014), 1–8.

14. Barans’ka I., Inverse problem with free boundary for a parabolic equation, Mat. Stud., 27 (2007), №1, 85–94. (in Ukrainian)

15. Barans’ka I., Ivanchov M., An inverse problem for two-dimentional heat equation in a free boundary domain, Ukrainian Mathematical Bulletin, 4 (2007), №4, 457–484. (in Ukrainian)

16. Snitko H., Inverse problem for a parabolic equation with free boundary, Matematychni Metody ta Fizyko- Mekhanichni Polya, 50 (2007), №4, 7–18. (in Ukrainian)

17. Snitko H., Determination of unknown multiplier in the coefficient at the first derivative in a parabolic equation in a free boundary domain, Visnyk Lviv. Univ. Ser. Mech.-Math., 67 (2007), 233–247. (in Ukrainian)

18. Hussein M., Lesnic D., Ivanchov M., Snitko H., Multiple time-dependent coefficient identification thermal problems with a free boundary, Applied Numerical Mathematics, 99 (2016), 24–50.

19. Snitko H., On a coefficient inverse problem for a parabolic equation in a domain with free boundary, J. of Math. Sciences, 200 (2014), №3, 374–388.

20. Zui-Cha Deng, Liu Yang, An inverse problem of identifying the coefficient of first order in a degenerate parabolic equation, J. of Computational and Applied Math., 235 (2011), 4407–4417.

21. Zui-Cha Deng, Liu Yang, An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation, Chin. Ann. Math., 35B (2014), №3, 355–382.

22. Tort J., An inverse diffusion problem in a degenerate parabolic equation, Monografias de la real academia de ciencias de Zaragoza, 38 (2012), 137–145.

23. Saldina N., Inverse problem for a degenerate parabolic equation, Visnyk Lviv Univ., Ser. Mech-Math., 64 (2005), 245–257. (in Ukrainian)

24. Ivanchov M., Saldina N., An inverse problem for strongly degenerate heat equation, J. Inv. Ill-Posed Problems, 14 (2006), №5, 465–480.

25. Hryntsiv N., Solvability of inverse problem for a degenerate parabolic equation in a free boundary domain, Scientific Bulletin of Chernivtsi University, 314–315, Mathematics (2006), 40–49. (in Ukrainian)

26. Hryntsiv N., Ivanchov M., An inverse problem for a strongly degenerate heat equation in a free boundary domain, Ukrain. Math. J., 61 (2009), 28–43. (in Ukrainian)

27. Huzyk N., Inverse problem of determining the coefficients in a degenerate parabolic equation, Electron. J. Diff. Eq., 172 (2014), 1–11.

28. Ivanchov M., Inverse problems for equations of parabolic type, VNTL Publishers, Lviv, 2003.

29. Saldina N., An inverse problem for a generally degenerate heat equation, J. of Lviv Polytechnic National University, “Physical and Mathematical sciences”, 566 (2006), 59–67.

30. Ladyzhenskaya O.A., Uralcevav N.N., Solonnikov V.A., Linear and quasilinear equations of parabolic type, Moscow, Nauka, (1973). (in Russian)

	Identification of the unknown parameters in the parabolic equation in a free boundary domain (in Ukrainian)
Author	N. M. Huzyk hryntsiv@ukr.net Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	In a free boundary domain there are established conditions of the existence and uniqueness of the classical solution to the inverse problem of identification of the time-dependent both major and minor coefficients in the parabolic equation with degeneration. The case of arbitrary weak degeneration is investigated.
Keywords	coefficient inverse problem; parabolic equation; arbitrary weak degeneration
DOI	doi:10.15330/ms.51.2.168-182
Reference	1. Jones B.F., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), №5, 907-918. 2. Cannon J.R., Recovering a time-dependent coefficient in a parabolic differential equation, J. Math. Anal. Appl., 160 (1991), 572-582. 3. Azari H., Li C., Nie Y., Shang S., Determination of an unknown coefficient in a parabolic inverse problem, Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Math. Analysis, 11 (2004), 665-674. 4. Oussaeif Taki-Eddine, Bouziani Abdelfatah, An inverse coefficient problem for parabolic equation under nonlocal boundary and integral overdetermination conditions, International Journal of Partial Differential Equation and Applications, 2 (2014), №3, 38-43. 5. Mansur I. Ismailov, Bulent O.gur, An inverse diffusion problem with nonlocal boundary conditions, Numerical Methods for Partial Differential Equations, 32, 2 (2016), 564-590. 6. Cannon J.R., Peres-Esteva S., Determination of the coefficient of $u_x$ in a linear parabolic equation, Inverse Problems, 10 (1993), №3, 521-531. 7. Hong-Ming Yin, Global solvability for some parabolic inverse problems, J. Math. Anal. and Appl., 162 (1991), 392-403. 8. Trong D.D., Ang D.D., Coefficient identification for a parabolic equation, Inverse Problems, 10 (1994), №3, 733-752. 9. Dehghan M., Parameter identification in a partial differential equation from overspecified data, Math. and Computer Modelling, 41 (2005), 197-213. 10. Onyejekwe O.N., Determination of two time-dependent coefficients in a parabolic partial differential equation by homotopy analysis method, International Journal of Applied Math. Research, 3 (2014), №2, 161–167. 11. Hussein M., Lesnic D., Ivanchov M., Simultaneous determination of time-dependent coefficients in the heat equation, Computers & Mathematics with Applications, 67 (2014), №5, 1065–1091. 12. BaiyuWang, Anping Liao,Wei Liu, Simultaneous determination of unknown two parameters in parabolic equation, International Journal of Applied Math. and Computation, 4 (2012), №3, 332–336. 13. Ebru Ozbilge, Ali Demir, Identification of the unknown coefficients in a linear parabolic equation via semigroup approach, Advances in Diff. Equations, 47 (2014), 1–8. 14. Barans’ka I., Inverse problem with free boundary for a parabolic equation, Mat. Stud., 27 (2007), №1, 85–94. (in Ukrainian) 15. Barans’ka I., Ivanchov M., An inverse problem for two-dimentional heat equation in a free boundary domain, Ukrainian Mathematical Bulletin, 4 (2007), №4, 457–484. (in Ukrainian) 16. Snitko H., Inverse problem for a parabolic equation with free boundary, Matematychni Metody ta Fizyko- Mekhanichni Polya, 50 (2007), №4, 7–18. (in Ukrainian) 17. Snitko H., Determination of unknown multiplier in the coefficient at the first derivative in a parabolic equation in a free boundary domain, Visnyk Lviv. Univ. Ser. Mech.-Math., 67 (2007), 233–247. (in Ukrainian) 18. Hussein M., Lesnic D., Ivanchov M., Snitko H., Multiple time-dependent coefficient identification thermal problems with a free boundary, Applied Numerical Mathematics, 99 (2016), 24–50. 19. Snitko H., On a coefficient inverse problem for a parabolic equation in a domain with free boundary, J. of Math. Sciences, 200 (2014), №3, 374–388. 20. Zui-Cha Deng, Liu Yang, An inverse problem of identifying the coefficient of first order in a degenerate parabolic equation, J. of Computational and Applied Math., 235 (2011), 4407–4417. 21. Zui-Cha Deng, Liu Yang, An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation, Chin. Ann. Math., 35B (2014), №3, 355–382. 22. Tort J., An inverse diffusion problem in a degenerate parabolic equation, Monografias de la real academia de ciencias de Zaragoza, 38 (2012), 137–145. 23. Saldina N., Inverse problem for a degenerate parabolic equation, Visnyk Lviv Univ., Ser. Mech-Math., 64 (2005), 245–257. (in Ukrainian) 24. Ivanchov M., Saldina N., An inverse problem for strongly degenerate heat equation, J. Inv. Ill-Posed Problems, 14 (2006), №5, 465–480. 25. Hryntsiv N., Solvability of inverse problem for a degenerate parabolic equation in a free boundary domain, Scientific Bulletin of Chernivtsi University, 314–315, Mathematics (2006), 40–49. (in Ukrainian) 26. Hryntsiv N., Ivanchov M., An inverse problem for a strongly degenerate heat equation in a free boundary domain, Ukrain. Math. J., 61 (2009), 28–43. (in Ukrainian) 27. Huzyk N., Inverse problem of determining the coefficients in a degenerate parabolic equation, Electron. J. Diff. Eq., 172 (2014), 1–11. 28. Ivanchov M., Inverse problems for equations of parabolic type, VNTL Publishers, Lviv, 2003. 29. Saldina N., An inverse problem for a generally degenerate heat equation, J. of Lviv Polytechnic National University, “Physical and Mathematical sciences”, 566 (2006), 59–67. 30. Ladyzhenskaya O.A., Uralcevav N.N., Solonnikov V.A., Linear and quasilinear equations of parabolic type, Moscow, Nauka, (1973). (in Russian)
Pages	168-182
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Identification of the unknown parameters in the parabolic equation in a free boundary domain (in Ukrainian)