Multipoint initial-final problem for one class of Sobolev type models of higher order with additive "white noise"

Multipoint initial-final problem for one class of Sobolev type models of higher order with additive "white noise"

Автор: Sviridyuk G.A., Zamyshlyaeva A.A., Zagrebina S.A.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp

Рубрика: Математическое моделирование

Статья в выпуске: 3 т.11, 2018 года.

Бесплатный доступ

Sobolev type equations theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise. A new concept of "white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equations theory and provide some practical applications. The main idea is to construct "noise" spaces using the Nelson-Gliklikh derivative. Abstract results concerning initial-final problems for higher order Sobolev type equations are applied to the Boussinesq-Love model with additive "white noise". We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.

Еще

Sobolev type equation, propagator, white noise, wiener k-process, multipoint initial-final problem, винеровский k-процесс

Короткий адрес: https://sciup.org/147232890

IDR: 147232890   |   DOI: 10.14529/mmp180308

Список литературы Multipoint initial-final problem for one class of Sobolev type models of higher order with additive "white noise"

  • Demidenko, G.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative/G.V. Demidenko, S.V. Uspenskii. -N.Y.; Basel; Hong Kong: Marcel Dekker, Inc., 2003.
  • Al'shin, A.B. Blow-up in Nonlinear Sobolev Type Equations/A.B. Al'shin, M.O. Korpusov, A.G. Sveshnikov. -De Gruyter, 2011.
  • Favini, A. Degenerate Differential Equations in Banach Spaces/A. Favini, A. Yagi. -N.Y.; Basel; Hong Kong: Marcel Dekker, Inc., 1999.
  • Кожанов, А.И. Краевые задачи для уравнений математической физики нечетного порядка/А.И. Кожанов. -Новосибирск: НГУ, 1990.
  • Showalter, R.E. Hilbert Space Methods for Partial Differential Equations/R.E. Showalter. -Pitman; London; San Francisco; Melbourne, 1977.
  • Lyapunov -Shmidt Methods in Nonlinear Analysis and Applications/N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev. -Dordrecht; Boston; London: Kluwer Academic Publishers, 2002.
  • Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators/G.A. Sviridyuk, V.E. Fedorov. -Utrecht; Boston; Koln; Tokyo: VSP, 2003.
  • Zamyshlyaeva, A.A. The Linearized Benney -Luke Mathematical Model with Additive White Noise/A.A. Zamyshlyaeva, G.A. Sviridyuk//Springer Proceedings in Mathematics and Statistics. -2015. -V. 113. -P. 327-337.
  • Бычков, Е.В. Об одной полулинейной математической модели соболевского типа высокого порядка/Е.В. Бычков//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2014. -Т. 7, № 2. -С. 25-39.
  • Melnikova, I.V. Abstract Stochastic Equations II. Solutions In Spaces of Abstract Stochastic Distributions/I.V. Melnikova, A.I. Filinkov, M.A. Alshansky//Journal of Mathematical Sciences. -2003. -V. 116, № 5. -P. 3620-3656.
  • Замышляева, А.А. Математические модели соболевского типа высокого порядка/А.А. Замышляева//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2014. -Т. 7, № 2. -С. 25-39.
  • Wang, S. Small Amplitude Solutions of the Generalized IMBq Equation/S. Wang, G. Chen//Mathematical Analysis and Applications. -2002. -V. 274. -P. 846-866.
  • Уизем, Дж. Линейные и нелинейные волны/Дж. Уизем. -М.: Мир, 1977.
  • Ландау, Л.Д. Теоретическая физика. Т. VII. Теория упругости/Л.Д. Ландау, Е.М. Лифшиц. -М.: Наука, 1987.
  • Gliklikh, Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics/Yu.E. Gliklikh. -London; Dordrecht; Heidelberg; N.Y.: Springer, 2011.
  • Gliklikh, Yu.E. Stochastic Leontieff Type Equations and Mean Derivatives of Stochastic Processes/Yu.E. Gliklikh, E.Yu. Mashkov//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2013. -V. 6, № 2. -P. 25-39.
  • Zagrebina, S.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline/S.A. Zagrebina, E.A. Soldatova, G.A. Sviridyuk//Springer Proceedings in Mathematics and Statistics. -2015. -V. 113. -P. 317-325.
  • Da Prato, G. Stochastic Equations in Infinite Dimensions of Encyclopedia of Mathematics and Its Applications/G. Da Prato, J. Zabczyk. -Cambridge: Cambridge University Press, 1992.
  • Arato, M. Linear Stochastic Systems with Constant Coefficients/M. Arato. -Berlin: Springer, 1982.
  • Kovacs, M. Introduction to Stochastic Partial Differential Equations/M. Kovacs, S. Larsson//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. -P. 159-232.
  • Favini, A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of Noises/A. Favini, G.A. Sviridyuk, N.A. Manakova//Abstract and Applied Analysis. -2015. -V. 2015. -Article ID 697410. -8 p.
  • Favini, A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of Noises/A. Favini, G. Sviridyuk, M. Sagadeeva//Mediterranean Journal of Mathematics. -2016. -V. 13, № 6. -P. 4607-4621.
  • Favini, A. One Class of Sobolev Type Equations of Higher Order with Additive White Noise/A. Favini, G.A. Sviridyuk, A.A. Zamyshlyaeva//Communications on Pure and Applied Analysis. -2014. -V. 15, № 1. -P. 185-196.
  • Nelson, E. Dynamical Theories of Brownian Motion/E. Nelson. -Princeton: Princeton University Press, 1967.
  • Shestakov, A.L. Numerical Solution of the Optimal Measurement Problem/A.L. Shestakov, A.V. Keller, E.I. Nazarova//Automation and Remote Control. -2012. -V. 73, 1. -P. 97-104.
  • Keller, A.V. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia/A.V. Keller, M.A. Sagadeeva//Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. -2014. -Т. 7, № 1. -P. 134-138.
  • Konkina, A.S. Multipoint Initial-Final Value Problem for the Model of Devis with Additive White Noise/A.S. Konkina//Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. -2017. -Т. 10, № 2. -P. 144-149.
  • Манакова, Н.А. Математические модели и оптимальное управление процессами фильтрации и деформации/Н.А. Манакова//Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. -2015. -Т. 8, № 3. -С. 5-24.
  • Zagrebina, S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid/S.A. Zagrebina//Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. -2014. -Т. 7, 3. -P. 5-22.
Еще
Статья научная
QR-код для установки приложения SciUp Наведите камеру и скачайте бесплатное приложение SciUp