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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2014, Volume 7, Issue 1, Pages 90–103 (Mi vyuru121)  

This article is cited in 13 scientific papers (total in 14 papers)

Mathematical Modelling

The Dynamical Models of Sobolev Type with Showalter–Sidorov Condition and Additive “Noise”

G. A. Sviridyuk, N. A. Manakova

South Ural State University, Chelyabinsk, Russian Federation

Abstract: 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.

Keywords: the Sobolev type equations; Wiener process; Nelson–Gliklikh derivative; “white noise”; space of “noise”; stochastic equation of Barenblatt–Zheltov–Kochina.

DOI: https://doi.org/10.14529/mmp140108

Full text: PDF file (382 kB)
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UDC: 517.9
MSC: 60H30
Received: 10.12.2013

Citation: G. A. Sviridyuk, N. A. Manakova, “The Dynamical Models of Sobolev Type with Showalter–Sidorov Condition and Additive “Noise””, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:1 (2014), 90–103

Citation in format AMSBIB
\Bibitem{SviMan14}
\by G.~A.~Sviridyuk, N.~A.~Manakova
\paper The Dynamical Models of Sobolev Type with Showalter--Sidorov Condition and Additive ``Noise''
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2014
\vol 7
\issue 1
\pages 90--103
\mathnet{http://mi.mathnet.ru/vyuru121}
\crossref{https://doi.org/10.14529/mmp140108}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. L. Shestakov, M. A. Sagadeeva, “Stochastic Leontieff-type equations with multiplicative effect in spaces of complex-valued “noises””, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:4 (2014), 132–139  mathnet  crossref
    2. A. L. Shestakov, A. V. Keller, G. A. Sviridyuk, “The theory of optimal measurements”, J. Comp. Eng. Math., 1:1 (2014), 3–16  mathnet  zmath  elib
    3. E. A. Bogatyreva, “Numerical modeling of quasi-steady process in conducting nondispersive medium with relaxation”, J. Comp. Eng. Math., 2:1 (2015), 45–51  mathnet  crossref  zmath  elib
    4. N. A. Manakova, G. A. Sviridyuk, “Neklassicheskie uravneniya matematicheskoi fiziki. Fazovye prostranstva polulineinykh uravnenii sobolevskogo tipa”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 8:3 (2016), 31–51  mathnet  crossref  elib
    5. S. I. Kadchenko, E. A. Soldatova, S. A. Zagrebina, “Numerical research of the Barenblatt–Zheltov–Kochina stochastic model”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 9:2 (2016), 117–123  mathnet  crossref  elib
    6. A. A. Zamyshlyaeva, O. N. Tsyplenkova, E. V. Bychkov, “Optimal control of solutions to the initial-final problem for the Sobolev type equation of higher order”, J. Comp. Eng. Math., 3:2 (2016), 57–67  mathnet  crossref  mathscinet  zmath  elib
    7. M. A. Sagadeeva, “Mathematical bases of optimal measurements theory in nonstationary case”, J. Comp. Eng. Math., 3:3 (2016), 19–32  mathnet  crossref  mathscinet  elib
    8. M. A. Sagadeeva, “Vyrozhdennye potoki razreshayuschikh operatorov dlya nestatsionarnykh uravnenii sobolevskogo tipa”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 9:1 (2017), 22–30  mathnet  crossref  elib
    9. E. M. Buryak, T. K. Plyshevskaya, A. B. Samarov, “Seminaru po uravneniyam sobolevskogo tipa chetvert veka”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:1 (2017), 165–169  mathnet  crossref  elib
    10. F. L. Hasan, “The bounded solutions on a semiaxis for the linearized Hoff equation in quasi-Sobolev spaces”, J. Comp. Eng. Math., 4:1 (2017), 27–37  mathnet  crossref  mathscinet  elib
    11. E. V. Kirillov, “The spectral identity for the operator with non-nuclear resolvent”, J. Comp. Eng. Math., 4:1 (2017), 69–75  mathnet  crossref  mathscinet  elib
    12. D. E. Shafranov, N. V. Adukova, “Solvability of the Showalter–Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace–Beltrami operator on differential forms”, J. Comp. Eng. Math., 4:3 (2017), 27–34  mathnet  crossref  mathscinet  elib
    13. N. N. Solovyova, S. A. Zagrebina, G. A. Sviridyuk, “Sobolev type mathematical models with relatively positive operators in the sequence spaces”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 9:4 (2017), 27–35  mathnet  crossref  elib
    14. K. V. Vasyuchkova, N. A. Manakova, G. A. Sviridyuk, “Some mathematical models with a relatively bounded operator and additive “white noise” in spaces of sequences”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:4 (2017), 5–14  mathnet  crossref  elib
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