Inverse Problem for Sobolev Type Mathematical Models

The work is devoted to the study of an inverse problem for the linear Sobolev type equation of higher order with an unknown coefficient depending on time. Since the equation might be degenerate the phase space method is used. It consists in construction of projectors splitting initial spaces into a direct sum of subspaces. Actions of operators also split. Therefore, the initial problem is reduced to two problems: regular and singular. The regular one is reduced to the first order nondegenerate problem which is solved via approximations. The needed smoothness of the solution is obtained. Then it is substituted into the singular problem which is solved using the methods of relatively polynomially bounded operator pencils theory. The main result of the work contains sufficient conditions for the existence and uniqueness of the solution to the inverse problem for a complete Sobolev type model of the second order. This technique can be used to investigate inverse problems of the considered type for Boussinesq-Love mathematical model.

Keywords

Sobolev type equation; inverse problem; mathematical models; equation of second order.

References

1. Wang S., Chen G. Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Applications, 2002, vol. 274, pp. 846-866. DOI: 10.1016/S0022-247X(02)00401-8
2. Uizem G. Linear and Nonlinear Waves. N.Y., Joyn Wiley and Sons, 1974.
3. Landau L.D., Lifshits E.M. Theoretical Phisics. Elasticity Theory. Vol. 7. London, Pergamon Press, 1970.
4. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-sectorial Operators in Space of 'Noises'. Abstract and Applied Analysis, 2015, article ID: 697410, 8 p. DOI: 10.1155/2015/697410
5. Fedorov V.E., Urazaeva A.V. Linear Inverse Evolution Problems for Sobolev Type. Non-Classical Equations of Mathematical Physics: Collection of Scientific Papers, 2010, pp. 293-310.
6. Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier-Stokes Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132-136. DOI: 10.14529/mmp150111
7. Manakova N.A., Sviridyuk G.A. An Optimal Control of the Solutions of the Initial-Final Problem for Linear Sobolev Type Equations with Strongly Relatively p-Radial Operator. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 213-224. DOI: 10.1007/978-3-319-12145-1_13
8. Zamyshlyaeva A.A., Lut A.V. Numerical Investigation of the Boussinesq-Love Mathematical Models on Geometrical Graphs. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 2, pp. 137-143. DOI: 10.14529/mmp170211
9. Keller A.V., Sagadeeva M.A. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 134-138. DOI: 10.14529/mmp140111 (in Russian)
10. Bychkov E.V. On a Semilinear Sobolev-Type Mathematical Model. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 2, pp. 111-117. DOI: 10.14529/mmp140210 (in Russian)
11. Tsyplenkova O.N. Optimal Control in Higher-Order Sobolev-Type Mathematical Models with (A,p)-bounded Operators. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 2, pp. 129-135. DOI: 10.14529/mmp140213 (in Russian)
12. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation. Mathematical Biosciences, 2007, vol. 206, no. 2, pp. 200-2015.
13. Zaynullov A.R. [An Inverse Problem for Two-Dimensional Equations of Finding the Thermal Conductivity of the Initial Distribution]. Bulletin of the Samara State Technical University. Series: Physics and Mathematics, 2015, vol. 19, no. 4, pp. 667-679. (in Russian)
14. Safiullova R.R. Inverse Problems for the Second Order Hyperbolic Equation with Unknown Time Depended Coefficient. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 4, pp. 73-86.
15. Megraliev Ya.T., Isgenderova Q.N. [Inverse Boundary Value Problem for a Second-Order Hyperbolic Equation with Integral Condition of the First Kind]. Problems of Physics, Mathematics and Technology, 2016, vol. 1, no. 26, pp. 42-47. (in Russian)
16. Pavlov S.S. Solvability of the Inverse Problem of Reconstruction of the External Action for a Multidimensional Wave Equation. Bulletin of the Chelyabinsk State University, 2011, vol. 26, pp. 27-37.
17. Zamyshlyaeva A.A, Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 5-16. DOI: 10.14529/mmph160401
18. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. N.Y., Marcel Dekker, 2000.