Analysis of the Wentzell stochastic system composed of the equations of unpressurised filtration in the hemisphere and at its boundary
Автор: Goncharov N.S., Sviridyuk G.A.
Рубрика: Краткие сообщения
Статья в выпуске: 1 т.17, 2024 года.
Бесплатный доступ
The deterministic and stochastic Wentzell systems of Dzekzer equations in a hemisphere and on its boundary are studied for the first time. The deterministic case is characterised by the unambiguous solvability of the initial problem for the Wentzell system in a specific constructed Hilbert space. In the case of the stochastic hydrodynamic system ``reservoir-well-collector'', the theory of Nelson-Glicklich derivative is applied and a stochastic solution is constructed, which allows us to determine the prognoses of quantitative changes in the geochemical regime of groundwater under non-pressure filtration. It should be noted that for the filtration system under study, the non-classical Wentzell condition is considered, since it is represented by an equation with the Laplace - Beltrami operator defined on the boundary of the domain, understood as a smooth compact Riemannian manifold without an edge, and the external influence is represented by the normal derivative of the function defined in the domain.
Wentzell system, dzekzer equation, nelson-glicklich derivative
Короткий адрес: https://sciup.org/147243955
IDR: 147243955 | DOI: 10.14529/mmp240108
Список литературы Analysis of the Wentzell stochastic system composed of the equations of unpressurised filtration in the hemisphere and at its boundary
- Dzektser E.S. Generalization of the Equation of Motion of Ground Waters with free Surface. Doklady Akademii Nauk SSSR, 1972, vol. 202, no. 5, pp. 1031-1033. (in Russian)
- Goncharov N.S., Zagrebina S.A., Sviridyuk G.A. Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2021, vol. 14, no. 4, pp. 102-105. DOI: 10.14529/mmp210408
- Goncharov N.S., Sviridyuk G.A. An Analysis of the Wentzell Stochastic System of the Equations of Moisture Filtration in a Ball and on Its Boundary. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2023, vol. 16, no. 4, pp. 84-92. DOI: 10.14529/mmp230406
- Ventcel' A.D. On Boundary Conditions for Multidimensional Diffusion Processes. Theory of Probability and Its Applications, 1959, vol. 4, pp. 164-177. DOI: 10.1137/1104014
- Feller W. Diffusion Processes in One Dimension. Transactions of the American Mathematical Society, 1954, vol. 77, no. 1, pp. 1-31.
- Luo Yousong, Trudinger N.S. Linear Second Order Elliptic Equations with Venttsel Boundary Conditions. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 1991, vol. 118, no. 3-4, pp. 193-207.
- Apushkinskaya D.E., Nazarov A.I. The Initial-Boundary Value Problem for Nondivergent Parabolic Equations with Venttsel' Boundary Condition. St. Petersburg Mathematical Journal, 1995, vol. 6, no. 6, pp. 1127-1149.
- Lukyanov V.V., Nazarov A.I. Solving the Venttsel Problem for the Laplace and Helmholtz Equations with the Help of Iterated Potentials. Journal of Mathematical Sciences (New York), 2000, vol. 102, no. 4, pp. 4265-4274. DOI: 10.1007/BF02673857
- Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. Co-Semigroups Generated by Second Order Differential Operators with General Wentzell Boundary Conditions. Proceedings of the American Mathematical Society, 2000, vol. 128, no. 7, pp. 1981-1989. DOI: 10.1090/S0002-9939-00-05486-1
- Favini A., Goldstein G.R., Goldstein J.A., Romanelli S. The Heat Equation with Generalized Wentzell Boundary Condition. Journal of Evolution Equations, 2002, vol. 2, no. 1, pp. 1-19. DOI: 10.1007/s00028-002-8077-y
- Coclite G.M., Gal C.G., Goldstein G.R., Goldstein J.A., Obrecht E., Romanelli S. The Role of Wentzell Boundary Conditions in Linear and Nonlinear Analysis. Advances in Nonlinear Analysis: Theory, Methods and Applications, 2009, vol. 3, pp. 279-292.
- Engel K.J., Fragnelli G. Analyticity of Semigroups Generated by Operators with Generalized Wentzell Boundary Conditions. Advances in Differential Equations, 2005, vol. 10, no. 11, pp. 1301-1320. DOI: 10.57262/ade/1355867753
- Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problem for Dynamical Sobolev-Type Equation in the Space of Noises. Electronic Journal of Differential Equations, 2018, vol. 2018, no. 128, pp. 1-10.
- Goncharov N.S., Zagrebina S.A., Sviridyuk G.A. The Showater-Sidorov and Cauchy Problems for the Linear Dzekzer Equation with Wentzell and Robin Boundary Conditions in a Bounded Domain. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2022, vol. 14, no. 1, pp. 50-63. DOI: 10.14529/mmph220106
- Goncharov N.S. Stochastic Barenblatt-Zheltov-Kochina Model on the Interval with Wentzell Boundary Conditions. Global and Stochastic Analysis. 2020. vol. 7. no. 1. pp. 11-23._
- Denk R., Kunze M., Ploss D. The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains. Integral Equations and Operator Theory, 2021, vol. 93, no. 2, article ID: 13, 26 p. DOI: 10.1007/s00020-021-02624-w
- Lions J.L., Magenes E. Problems aux Limites non Homogenes et Applications. Paris, Dunod, 1968. (in French)
- Shestakov A.L., Sviridyuk G.A., Hudyakov Yu.V. Dinamic Measurement in Spaces of "Noises". Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control, Radio Electronics, 2013, vol. 13, no. 2, pp. 4-11.
- Sviridyuk G.A., Efremov A.A. An Optimal Control Problem for a Class of Linear Equations of Sobolev Type. Russian Mathematics, 1996, no. 12, pp. 72-80.
- Shestakov A.L., Keller A.V., Zamyshlyaeva A.A., Manakova N.A., Zagrebina S.A., Sviridyuk G.A. The Optimal Measurements Theory as a New Paradigm in the Metrology. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 3-23. DOI: 10.14529/jcem200101
