Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations
The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way u...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2021-06-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
| Subjects: | |
| Online Access: | http://mathizv.isu.ru/en/article/file?id=1375 |
| _version_ | 1819243229023830016 |
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| author | K.B. Mansimov R.O. Mastaliyev |
| author_facet | K.B. Mansimov R.O. Mastaliyev |
| author_sort | K.B. Mansimov |
| collection | DOAJ |
| description | The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied. |
| first_indexed | 2024-12-23T14:52:22Z |
| format | Article |
| id | doaj.art-93b51cfeb8c54d5bb2012fc6479fd5ea |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-12-23T14:52:22Z |
| publishDate | 2021-06-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-93b51cfeb8c54d5bb2012fc6479fd5ea2022-12-21T17:42:54ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852021-06-013612943https://doi.org/10.26516/1997-7670.2021.36.29Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential EquationsK.B. MansimovR.O. MastaliyevThe article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.http://mathizv.isu.ru/en/article/file?id=1375linear goursat-darboux stochastic systemrepresentation of solution boundary value problemriemann methodstochastic analogue of the riemann matrix |
| spellingShingle | K.B. Mansimov R.O. Mastaliyev Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations Известия Иркутского государственного университета: Серия "Математика" linear goursat-darboux stochastic system representation of solution boundary value problem riemann method stochastic analogue of the riemann matrix |
| title | Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations |
| title_full | Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations |
| title_fullStr | Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations |
| title_full_unstemmed | Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations |
| title_short | Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations |
| title_sort | representation of the solution of goursat problem for second order linear stochastic hyperbolic differential equations |
| topic | linear goursat-darboux stochastic system representation of solution boundary value problem riemann method stochastic analogue of the riemann matrix |
| url | http://mathizv.isu.ru/en/article/file?id=1375 |
| work_keys_str_mv | AT kbmansimov representationofthesolutionofgoursatproblemforsecondorderlinearstochastichyperbolicdifferentialequations AT romastaliyev representationofthesolutionofgoursatproblemforsecondorderlinearstochastichyperbolicdifferentialequations |