Volume 9, no. 2Pages 75 - 89

Inverse Problems for Some Sobolev-Type Mathematical Models

S.G. Pyatkov, S.N. Shergin
The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
Full text
the Sobolev-type model; Sobolev equation; existence and uniqueness theorems; inverse problem; boundary value problem; plasma waves; rotating fluid; the Boussinesq - Love model.
1. Sobolev S.L. [On a New Problem of Mathematical Physics]. Izvestiya: Mathematics, 1954, vol. 18, pp. 3-50. (in Russian)
2. Boussinesq J.V. Essai sur la theorie des eaux courantes. Mem. Pesentes Divers Savants Acad. Sci. Inst. France, 1877, vol. 23, pp. 1-680.
3. Love A.E.H. A Treatise on the Mathematical Theory of Elasticity. New York, Dover Publications, 1944.
4. Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner U.D. Linear and Non-Linear Equations Sobolev's Type. Moscow, Fizmatlit, 2007. 736 p. (in Russian)
5. Ikezi H. Experimental Study of Solitons in Plasma. In Solitons in Action, N.Y., Academic Press, 1978, pp. 163-184.
6. Lyubanova A.Sh. Identification of a Coefficient in the Leading Term of a Pseudoparabolic Equation of Filtration. Journal of Applied and Industrial Mathematics, 2013, vol. 54, no. 6, pp. 1046-1058. DOI: 10.1134/s0037446613060116
7. Lyubanova A.Sh., Tani A. On Inverse Problems for Pseudoparabolic and Parabolic Equations of Filtration. Inverse Problems in Science and Engineering, 2011, vol. 19, no. 7, pp. 1023-1042. DOI: 10.1080/17415977.2011.569712
8. Kozhanov A.I. [On Solvability of Inverse Problems of Recovering the Coefficients in Composite Type Equations]. Vestnik Novosibirskogo gosudarstvennogo universiteta. Seriya Matematika, Mekhanika, Informatika, 2008, vol. 8, no. 3, pp. 81-99. (in Russian)
9. Kozhanov A.I. Nonlinear Loaded Equations and Inverse Problems. Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 4, pp. 657-675.
10. Kozhanov A.I. [On Solvability of Inverse Coefficient for Some Sobolev-Type Equations]. Belgorod State University Scientific Bulletin. Mathematics. Physics. 2010, vol. 18, no. 5, pp. 88-98.
11. Fedorov V.E., Urazaeva A.V. An Inverse Problem for Linear Sobolev Type Equations. Journal of Inverse and Ill-Posed Problems, 2004, vol. 12, no. 4, pp. 387-395. DOI: 10.1515/1569394042248210
12. Ablabekov B.S. Obratnye zadachi dlya psevdoparabolicheskikh uravneniy [Inverse Problems for Pseudoparabolic Equations]. Bishkek, Ilim, 2001. (in Russian)
13. Asanov A., Atamanov E.R. An Inverse Problem for a Pseudoparabolic Integro-Defferential Operator Equation. Siberian Mathematical Journal, 1995, vol. 38, no. 4, pp. 645-655. DOI: 10.1007/BF02107322
14. Favini A, Lorenzi A. Differential Equations: Inverse and Direct Problems. Boca Raton, London, N.Y., Chapman and Hall/CRC, 2006. 304 p. DOI: 10.1201/9781420011135
15. Zamyshlyaeva A.A., Tsyplenkova O.N. The Optimal Control over Solutions of the Initial-Finish Value Problem for the Boussinesque - Love Equation. The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 5 (264), pp. 13-24. (in Russian)
16. Zamyshlyaeva A.A., Yuzeeva A.V. The Initial-Finish Value Problem for the Boussinesq - Love Equation. The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), pp. 23-31. (in Russian)
17. Zamyshlyaeva A.A., Tsyplenkova O.N. Optimal Control of Solutions of the Showalter - Sidorov - Dirichlet Problem for the Boussinesq - Love Equation. Differential Equations, 2013, vol. 49, no. 11, pp. 1356-1365. DOI: 10.1134/S0012266113110049
18. Mehraliyev Ya.T. Inverse Problem of the Boussinesq - Love Equation with an Extra Integral Condition. Journal of Applied and Industrial Mathematics, 2013, vol. 16, no. 1 (53), pp. 75-83.
19. Mehraliyev Ya.T. On Solvability of an Inverse Boundary Value Problem for the Boussinesq - Love Equation. Journal of Siberian Federal University. Mathematics and Physics, 2013, vol. 6, no. 4, pp. 485-494.
20. Ablabekov B.S., Kasymalieva A.A. [An Inverse Problem of Recovering the Right-Hand Side for the Boussinesq - Love Equation]. Abstracts of the Second International Scientific Conference 'Theory and Numerical Methods of Solving Inverse and Ill-Posed Problems', September 21-29, 2010, Novosibirsk, Sobolev Institute of Mathematics, 2010, pp. 2-4. (in Russian)
21. Ladyzhenskaya O.A, Ural'tseva N.N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa [Linear and Quasilinear Elliptic Equations]. Moscow, Nauka, 1964. (in Russian)
22. Gilbarg D., Trudinger N. Elliptic Differential Equation with Partial Derivative of the Second Order. Berlin, Heidelberg, Springer-Verlag, 2001.
23. Pyatkov S.G., Shergin S.N. On Some Mathematical Models of Filtration Theory. The Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 105-116.
24. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik matematicki, 2000, vol. 35, no. 1, pp. 161-177.
25. Amann H. Operator-Valued Foutier Multipliers, Vector-Valued Besov Spaces and Applications. Mathematische Nachrichten, 1997, vol. 186, no. 1, pp. 5-56. DOI: 10.1002/mana.3211860102