ONE NONCLASSICAL HIGHER ORDER MATHEMATICAL MODEL WITH ADDITIVE "WHITE NOISE"

Alena Aleksandrovna Zamyshlyaeva

Abstract


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 - Love 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 - Love model is considered.


Keywords


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

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References


Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Series in Nonlinear Analysis and Applications, 15, De Gruyter, 2011.

Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. New York, Basel, Hong Kong, Marcel Dekker, Inc., 2003.

Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. New York, Basel, Hong Kong, Marcel Dekker, Inc., 1999.

Fedorov V.Е. On Some Correlations in the Theory of Degenerate Semigroups of Operators. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2008, vol. 15, no. 115, pp. 89-699. (in Russian)

Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.Y., Springer, 2011.

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, vol. 27, no. 286, pp. 24-34. (in Russian)

Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Processing of "New Directions in the Mathematical and Computer Sciences", National Universities Commission. October 8-12. 2007. Abuja. Nigeria. Publications of the ICMCS, 2008, no. 4, pp. 159-232.

Kozhanov A.I. Boundary Problems for Odd Ordered Equations of Mathematical Physics. Novosibirsk, NGU, 1990. (in Russian)

Landau L.D., Lifshits E.M. Theoretical Phisics, VII. Elasticity Theory. Мoscow, Nauka Publ., 1987. (in Russian)

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.

Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2008, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079

Shestakov A.L., Sviridyuk G.A. On a New Conception of White Noise. Obozrenie Prikladnoy i Promyshlennoy Matematiki Survey of Applied and Industrial Mathematics, 2012, vol. 19, no. 2, pp. 287-288. (in Russian)

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, vol.27, no. 286, pp. 99-108. (in Russian)

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, vol. 17, no. 234, pp. 70-75. (in Russian)

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 & Radioelectronics", 2013, no. 2, pp. 4-11. (in Russian)

Showalter R.E., Hilbert Space Methods for Partial Differential Equations. Pitman London, San Francisco, Melbourne, 1977.

Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.

Sviridyuk G.A., Apetova T.V., The Phase Spaces of Linear Dynamic Sobolev Type Equations. Doklady Akademii Nauk, 1993, vol. 330, no. 6, pp. 696-699. (in Russian)

Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K"oln, Tokyo, VSP, 2003.

Sviridyuk G.A., Vakarina O.V. Linear Sobolev Type Equations of Higher Order. Doklady Akademii Nauk, 1998, vol. 393, no. 3, pp. 308-310. (in Russian)

Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Spaces of a Class of Linear Higher-Order Sobolev Type Equations. Differential Equations, 2006, vol. 42, no. 2, pp. 269-278.

Uizem G. Linear and Nonlinear Waves. Мoscow, Mir Publ., 1977. (in Russian)

Wang S., Chen G., Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Applications, 2002, no. 274, pp. 846-866.

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, vol. 40, no. 299, pp. 73-82. (in Russian)

Zagrebina S.A., Soldatova E.A. The Linear Sobolev-Type Equations with Relatively p-Bounded Operators and Additive White Noise. The Bulletin of Irkutsk State University. Series "Mathematics", 2013, vol. 6, no. 1, 20-34. (in Russian)


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