Multipoint Initial-Final Problem for One Class of Sobolev Type Models of Higher Order with Additive "White Noise"

Sobolev type equations theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise. A new concept of "white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equations theory and provide some practical applications. The main idea is to construct 'noise' spaces using the Nelson-Gliklikh derivative. Abstract results concerning initial-final problems for higher order Sobolev type equations are applied to the Boussinesq-Love model with additive 'white noise'. We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.

Keywords

Sobolev type equation; propagator; '"white noise".

References

1. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003.
2. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. De Gruyter, 2011.
3. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 1999.
4. Kozhanov A.I. Boundary Problems for Odd Ordered Equations of Mathematical Physics. Novosibirsk, NGU, 1990.
5. Showalter R.E. Hilbert Space Methods for Partial Differential Equations. Pitman, London, San Francisco, Melbourne, 1977.
6. Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.
7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003.
8. Zamyshlyaeva A.A., Sviridyuk G.A. The Linearized Benney-Luke Mathematical Model with Additive White Noise. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 327-337. DOI: 10.1007 / 978-3-319-12145-1_21
9. 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
10. Zamyshlyaeva A.A. The Higher Order Sobolev Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 2, pp. 5-28. DOI: 10.14529/mmp140201
11. Melnikova I.V., Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions. Journal of Mathematical Sciences, 2003, vol. 116, no. 5, pp. 3620-3656.
12. Wang S., Chen G. Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Applications, 2002, vol. 274, pp. 846-866.
13. Whitham G.B. Linear and Nonlinear Waves. N.Y., Wiley, 1974.
14. Landau L.D., Lifshits E.M. Teoreticheskaya fizika. T 7. Teoriya uprugosti [Theoretical Phisics, VII. Elasticity Theory]. Мoscow, Nauka, 1987. (in Russian)
15. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.Y., Springer, 2011.
16. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontieff Type Equations and Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 2, pp. 25-39.
17. Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 317-325. DOI: 10.1007/978-3-319-12145-1-20
18. Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions of Encyclopedia of Mathematics and Its Applications, 44. Cambridge, Cambridge University Press, 1992.
19. Arato M. Linear Stochastic Systems with Constant Coefficients. A Statistical Approach 45 of Lecture Notes in Control and Information Sciences, Berlin, Springer, 1982.
20. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of 'New Directions in the Mathematical and Computer Sciences', National Universities Commission, Abuja, Nigeria, October 8-12, 2007, Publications of the ICMCS, 2008, vol. 4, pp. 159-232.
21. Favini A., Sviridyuk G., Manakova N. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of 'Noises'. Abstract and Applied Analysis, 2015, vol. 2015, Article ID 697410, 8 p.
22. Favini A., Sviridyuk G., Sagadeeva M., Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of 'Noises'. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
23. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive 'White Noise'. Communications on Pure and Applied Analysis, 2014, vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
24. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
25. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104.
26. 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
27. Konkina A.S. Multipoint Initial-Final Value Problem for the Model of Devis with Additive White Noise. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 2, pp. 144-149. DOI: 10.14529/mmp170212
28. Manakova N.A. Mathematical Models and Optimal Control of the Filtration and Deformation Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 5-24. DOI: 10.14529/mmp150301
29. Zagrebina S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 3, pp. 5-22. DOI:10.14529/mmp140301