A solvability conditions of mixed problems for equations of parabolic type with involution
In this work the partial differential equations with involutions are considered. The mixed problems for the parabolic type equation, with constant and variable constants, corresponding to the Dirichlet type boundary conditions is investigated. The involution is contained by the second derivative wi...
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2018-12-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/247 |
| _version_ | 1797372662320726016 |
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| author | A.A. Sarsenbi |
| author_facet | A.A. Sarsenbi |
| author_sort | A.A. Sarsenbi |
| collection | DOAJ |
| description |
In this work the partial differential equations with involutions are considered. The mixed problems for the parabolic type equation, with constant and variable constants, corresponding to the Dirichlet type boundary conditions is investigated. The involution is contained by the second derivative with respect to the variable x, which is the difficult case for investigations. One-dimensional differential operators with involution have an infinite number of positive and negative eigenvalues. This means that the operator on the right-hand side of the equation under study is not semi-bounded. In the case of classical problems, ordinary differential operators usually appear on the right-hand side of the equations, which are semibounded. Therefore, the incorrectness of mixed problems for a parabolic equation with an involution is discussed in this paper. Examples are given. Sufficient conditions for the initial data are found when the problem under study has a unique solution. The representation of the solution in the form of partial sums of the Fourier series in eigenfunctions is found. The density in the space L2 (−1, 1) of the set of initial functions is proved everywhere, when the problem has a unique solution.
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| first_indexed | 2024-03-08T18:38:05Z |
| format | Article |
| id | doaj.art-8452059836694168a323e5b1930566ce |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-03-08T18:38:05Z |
| publishDate | 2018-12-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-8452059836694168a323e5b1930566ce2023-12-29T10:21:05ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112018-12-0192410.31489/2018m4/87-93A solvability conditions of mixed problems for equations of parabolic type with involutionA.A. Sarsenbi In this work the partial differential equations with involutions are considered. The mixed problems for the parabolic type equation, with constant and variable constants, corresponding to the Dirichlet type boundary conditions is investigated. The involution is contained by the second derivative with respect to the variable x, which is the difficult case for investigations. One-dimensional differential operators with involution have an infinite number of positive and negative eigenvalues. This means that the operator on the right-hand side of the equation under study is not semi-bounded. In the case of classical problems, ordinary differential operators usually appear on the right-hand side of the equations, which are semibounded. Therefore, the incorrectness of mixed problems for a parabolic equation with an involution is discussed in this paper. Examples are given. Sufficient conditions for the initial data are found when the problem under study has a unique solution. The representation of the solution in the form of partial sums of the Fourier series in eigenfunctions is found. The density in the space L2 (−1, 1) of the set of initial functions is proved everywhere, when the problem has a unique solution. http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/247Fourier methodmixed probleminvolutioneigenfunctionsbasis |
| spellingShingle | A.A. Sarsenbi A solvability conditions of mixed problems for equations of parabolic type with involution Қарағанды университетінің хабаршысы. Математика сериясы Fourier method mixed problem involution eigenfunctions basis |
| title | A solvability conditions of mixed problems for equations of parabolic type with involution |
| title_full | A solvability conditions of mixed problems for equations of parabolic type with involution |
| title_fullStr | A solvability conditions of mixed problems for equations of parabolic type with involution |
| title_full_unstemmed | A solvability conditions of mixed problems for equations of parabolic type with involution |
| title_short | A solvability conditions of mixed problems for equations of parabolic type with involution |
| title_sort | solvability conditions of mixed problems for equations of parabolic type with involution |
| topic | Fourier method mixed problem involution eigenfunctions basis |
| url | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/247 |
| work_keys_str_mv | AT aasarsenbi asolvabilityconditionsofmixedproblemsforequationsofparabolictypewithinvolution AT aasarsenbi solvabilityconditionsofmixedproblemsforequationsofparabolictypewithinvolution |