# Some Mathematical Models with a Relatively Bounded Operator and Additive 'White Noise'

K.V. Vasyuchkova, N.A. Manakova, G.A. SviridyukThe article is devoted to the research of the class of stochastic models in mathematical physics on the basis of an abstract Sobolev type equation in Banach spaces of sequences, which are the analogues of Sobolev spaces. As operators we take polynomials with real coefficients from the analogue of the Laplace operator, and carry over the theory of linear stochastic equations of Sobolev type on the Banach spaces of sequences. The spaces of sequences of differentiable 'noises' are denoted, and the existence and the uniqueness of the classical solution of Showalter - Sidorov problem for the stochastic equation of Sobolev type with a relatively bounded operator are proved. The constructed abstract scheme can be applied to the study of a wide class of stochastic models in mathematical physics, such as, for example, the Barenblatt - Zheltov - Kochina model and the Hoff model.Full text

- Keywords
- Sobolev type equations; Banach spaces of sequences; the Nelson - Gliklikh derivative; 'white noise'.
- References
- 1. Barenblatt G.I., Zheltov Iu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks [Strata]. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, issue 5, pp. 1286-1303. DOI: 10.1016/0021-8928(60)90107-6

2. Hoff N.J. Creep Buckling. The Aeronautical Quarterly, 1956, vol. 7, no. 1, pp. 1-20.

3. Keller A.V., Al-Delfi J.K. Holomorphic Degenerate Groups of Operators in Quasi-Banach Spaces. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2015, vol. 7, no. 1, pp. 20-27.

4. Zamyshlyaeva A.A., Keller A.V., Sagadeeva M.A. On Integration in Quasi-Banach Spaces of Sequences. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 1, pp. 52-56. DOI: 10.14529/jcem150106

5. Solovyova N.N., Zagrebina S.A., Sviridyuk G.A. Sobolev Type Mathematical Models with Relatively Positive Operators in the Sequence Spaces. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 4, pp. 27-35. DOI: 10.14529/mmph170404

6. Al-Isawi J.K.T., Zamyshlyaeva A.A. Computational Experiment for One Class of Evolution Mathematical Models in Quasi-Sobolev Spaces. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 4, pp. 141-147. DOI: 10.14529/mmp160413

7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K'oln, Tokyo, VSP, 2003. 216 p. DOI: 10.1515/9783110915501

8. Sviridyuk G.A., Manakova N.A. Dynamic Models of Sobolev Type with the Showalter - Sidorov Condition and Additive 'Noises'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. (in Russian) DOI: 10.14529/mmp140108

9. 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, vol. 2015, Article ID 697410, 8 p. DOI: 10.1155/2015/697410

10. 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, 2016, vol. 15, no. 1, pp. 185-196.

11. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of 'Noises'. Mediterranian Journal of Mathematics, 2016, vol. 13, no 6, pp. 4607-4621.

12. Sviridyuk G.A., Manakova N.A. The Barenblatt - Zheltov - Kochina Model with Additive White Noise in Quasi-Sobolev Spaces. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 1, pp. 61-67. DOI: 10.14529/jcem160107

13. 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 and Computer Software, 2012, no. 27 (286), issue 13, pp. 24-34.

14. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.

15. 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.

16. 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. DOI: 10.1023/A:1024159908410