Volume 7, no. 1Pages 90 - 103

The Dynamical Models of Sobolev Type with Showalter - Sidorov Condition and Additive 'Noise'

G.A. Sviridyuk, N.A. Manakova
The concept of 'white noise', initially established in finite-dimensional spaces, has been transfered to infinite-dimensional spaces. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical value. The derivative of Nelson - Gliklikh is entered to reach this goal, as well as the spaces of 'noises' are developed. The equations of Sobolev type with relatively bounded operators are considered in the spaces of differentiable 'noises'. Besides, the existence and uniqueness of their classical solutions are proved. A stochastic equation of Barenblatt - Zheltov - Kochina is considered as an application in bounded domain with homogeneous boundary condition of Dirichlet and initial condition of Showalter - Sidorov.
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Keywords
the Sobolev type equations; Wiener process; Nelson - Gliklikh derivative; 'white noise'; space of 'noise'; stochastic equation of Barenblatt-Zheltov-Kochina.
References
1. Arato M. Linear Stochastic Systems with Constant Coefficients. A Statistical Approach. Berlin, Heidelberg, N.-Y., Springer, 1982. DOI: 10.1007/BFb0043631
2. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.-Y., Springer, 2011. DOI: 10.1007/978-0-85729-163-9
3. Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge, Cambridge University Press, 1992. DOI: 10.1017/CBO9780511666223
4. 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. V. 4. Lagos, Publications of the ICMCS, 2008, pp. 159-232.
5. Zamyshlyaeva A.A. Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, no. 40, issue 14, pp. 73-82. (in Russian)
6. Zagrebina S.A., Soldatova E.A. The Linear Sobolev-Type Equations With Relatively p-bounded Operators and Additive White Noise. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya 'Matematika', [The Bulletin of Irkutsk State University. Series 'Mathematics'], 2013, vol. 6, no. 1, pp. 20-34. (in Russian)
7. Melnikova I.V., Filinkov A.I., Alshansky M.A. Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions. J. of Mathematical Sciences, 2003, vol. 116, no. 5, pp. 3620-3656. DOI: 10.1023/A:1024159908410
8. Melnikova I.V., Filinkov A.I. Generalized Solutions to Abstract Stochastic Problems. J. Integ. Transf. and Special Funct., 2009, vol. 20, no. 3-4, pp. 199-206. DOI: 10.1080/10652460802567574
9. Shestakov A.L., Sviridyuk G. A. On a New Conception of White Noise. Obozrenie Prikladnoy i Promyshlennoy Matematiki, 2012, vol. 19, no. 2, pp. 287-288. (in Russian)
10. Shestakov A.L., Sviridyuk G.A. On the Measurement of the 'White Noise'. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, no. 27 (286), issue 13, pp. 99-108.
11. Gliklikh Yu.E. Investigation of Leontieff Type Equations with White Noise Protect by the Methods of Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, no. 27 (286), issue 13, pp. 24-34. (in Russian)
12. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2011, no. 17 (234), issue 8, pp. 70-75.
13. 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. DOI: 10.1134/S0005117912010079
14. Shestakov A.L., Sviridyuk G.A., Hudyakov Yu.V. Dynamic Measurement in Spaces of 'Noise'. Bulletin of the South Ural State University. Series 'Computer Technologies, Automatic Control, Radio Electronics', 2013, vol. 13, no. 2, pp. 4-11. (in Russian)
15. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as a Phenomena of the Sobolev Type Equations. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya 'Matematika' [The Bulletin of Irkutsk State University. Series 'Mathematics'], 2010, vol. 3, no. 1, pp. 104-125. (in Russian)
16. Kuropatenko V.F. Mesomechanics Single-Component and Multicomponent Materials [Mezomekhanika odnokomponentnykh i mnogokomponentnykh materialov]. Fizicheskaya mezomekhanika [Physical Mesomechanics], 2001, vol. 4, no. 3, pp. 49-55.
17. Kuropatenko V.F. Momentum and Energy Exchange in Nonequilibrium Multicomponent Media. J. of Applied Mechanics and Technical Physics, 2005, vol. 46, no. 1, pp. 1-8. DOI: 10.1007/s10808-005-0021-9
18. Kuropatenko V.F. New Models of Continuum Mechanics. J. of Engineering Physics and Thermophysics, 2011, vol. 84, no. 1, pp. 77-99. DOI: 10.1007/s10891-011-0457-0
19. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
20. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russian Mathematical Surveys, 1994, vol. 49, no. 4, pp. 45-74. DOI: 10.1070/RM1994v049n04ABEH002390
21. Triebel H. / Interpolation Theory, Function Spaces, Differential Operators. Heidelberg, Barth, 1995.
22. Sviridyuk G.A., Fedorov V.E. Lineynye uravneniya sobolevskogo tipa [Linear Sobolev Type Equations]. Chelyabinsk, Chelyabinsk State University, 2003. 179 p.