One nonclassical higher order mathematical model with additive "white noise"'

A. A. Zamyshlyaeva

Journal of computational and engineering mathematics, Tome 1 (2014) no. 1, pp. 55-68.

Résumé

Sobolev type equations theory experiences an epoch of blossoming. The majority of researches is devoted to the determined equations and systems. However in natural experiments there are the mathematical models containing accidental indignation, for example, white noise. Therefore recently even more often there arise the researches devoted to the stochastic differential equations. A new conception of "white noise", originally constructed for finite dimensional spaces, is spread here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher order Sobolev type equations theory and practical applications. The main idea is in construction of «noise» spaces using the Nelson–Gliklikh derivative. Abstract results are applied for the investigation of the Boussinesq–Lòve model with additive "white noise" within the Sobolev type equations theory. At studying the methods and results of theory of Sobolev type equations with relatively p-sectorial operators are very useful. We use already well proved at the investigation of Sobolev type equations the phase space method consisting in a reduction of singular equation to regular one, defined on some subspace of initial space. In the first part of article the spaces of noises are constructed. In the second — the Cauchy problem for the stochastic Sobolev type equation of higher order is investigated. As an example the stochastic Boussinesq–Lòve model is considered.

Détail

Keywords: Sobolev type equation, propagator, "white noise", K-Wiener process.

Bibliographie

[1] A. B. Al’shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 15, de Gruyter, Berlin, 2011, xii+648 pp. | MR | Zbl

[2] G. V. Demidenko, S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-order Derivative, Marcel Dekker Inc., New York–Basel–Hong Kong, 2003, xvi+490 pp. | MR | Zbl

[3] A. Favini, A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York–Basel–Hong Kong, 1999, 336 pp. | MR | Zbl

[4] V. E. Fedorov, “On Some Correlations in the Theory of Degenerate Semigroups of Operators”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2008, no. 15 (115), 89–99 | Zbl

[5] Yu. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics, Springer, London, 2011 | DOI | MR | Zbl

[6] Yu. E. Gliklikh, “Investigation of Leontieff Type Equations with White Noise by the Methods of Mean Derivatives of Stochastic Processes”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 24–34 | Zbl

[7] M. Kovács, S. Larsson, “Introduction to Stochastic Partial Differential Equations”, Processing of «New Directions in the Mathematical and Computer Sciences» (National Universities Commission, Abuja, Nigeria, October 8-12, 2007), Publications of the ICMCS, 4, 2008, 159–232 | Zbl

[8] A. I. Kozhanov, Boundary Problems for Odd Ordered Equations of Mathematical Physics, NGU, Novosibirsk, 1990, 132 pp. | MR

[9] L. D. Landau, E. M. Lifshits, Theoretical Phisics, v. VII, Elasticity Theory, Nauka Publ., Moscow, 1987, 248 pp. | MR

[10] I. V. Melnikova, A. I. Filinkov, M. A. Alshansky, “Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions”, Journal of Mathematical Sciences, 116:5 (2003), 3620–3656 | DOI | MR | Zbl

[11] A. L. Shestakov, A. V. Keller, E. I. Nazarova, “Numerical Solution of the Optimal Measurement Problem”, Automation and Remote Control, 73:1 (2012), 97–104 | DOI | MR | Zbl

[12] A. L. Shestakov, G. A. Sviridyuk, “On a New Conception of White Noise”, Obozrenie Prikladnoy i Promyshlennoy Matematiki, 19:2 (2012), 287–288

[13] A. L. Shestakov, G. A. Sviridyuk, “On the Measurement of the «White Noise»”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 13, 99–108 | Zbl

[14] A. L. Shestakov, G. A. Sviridyuk, “Optimal measurement of dynamically distorted signals”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2011, no. 8, 70–75 | Zbl

[15] A. L. Shestakov, G. A. Sviridyuk, Yu. V. Hudyakov, “Dynamic Measurement in Spaces of «Noise»”, Bulletin of the South Ural State University. Series «Computer Technologies, Automatic Control Radioelectronics», 13:2 (2013), 4–11

[16] R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London–San Francisco–Melbourne, 1977, 196 pp. | MR | Zbl

[17] N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev, Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002, xx+548 pp. | DOI | MR

[18] G. A. Sviridyuk, T. V. Apetova, “The Phase Spaces of Linear Dynamic Sobolev Type Equations”, Doklady Akademii Nauk, 330:6 (1993), 696–699 | MR | Zbl

[19] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston, 2003, viii+216 pp. | MR | Zbl

[20] G. A. Sviridyuk, O. V. Vakarina, “Linear Sobolev Type Equations of Higher Order”, Doklady Akademii Nauk, 393:3 (1998), 308–310 | MR

[21] G. A. Sviridyuk, A. A. Zamyshlyaeva, “The phase spaces of a class of linear higher-order Sobolev type equations”, Differ. Uravn., 42:2 (2006), 269–278 | DOI | MR | Zbl

[22] G. Uizem, Linear and Nonlinear Waves, Mir Publ., Moscow, 1977, 638 pp. | MR

[23] S. Wang, G. Chen, “Small Amplitude Solutions of the Generalized IMBq Equation”, Mathematical Analysis and Applications, 274:2 (2002), 846–866 | DOI | MR | Zbl

[24] A. A. Zamyshlyaeva, “Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 14, 73–82 | MR | Zbl

[25] S. A. Zagrebina, E. A. Soldatova, “The linear Sobolev-type Equations With Relatively $p$-bounded Operators and Additive White Noise”, IIGU Ser. Matematika, 6:1 (2013), 20–34 | Zbl