The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients
We study the Dirichlet problem in a parallelepiped for a three-dimensional equation of mixed type with three singular coefficients. Separation of variables with Fourier series and spectral analysis are used to investigate this problem. Two one-dimensional spectral problems are obtained for the posse...
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| Format: | Article |
| Language: | English |
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Samara State Technical University
2017-12-01
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| Series: | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
| Subjects: | |
| Online Access: | https://journals.eco-vector.com/1991-8615/article/viewFile/20567/16814 |
| _version_ | 1818024945505009664 |
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| author | Akhmadjon K Urinov Kamoliddin T Karimov |
| author_facet | Akhmadjon K Urinov Kamoliddin T Karimov |
| author_sort | Akhmadjon K Urinov |
| collection | DOAJ |
| description | We study the Dirichlet problem in a parallelepiped for a three-dimensional equation of mixed type with three singular coefficients. Separation of variables with Fourier series and spectral analysis are used to investigate this problem. Two one-dimensional spectral problems are obtained for the possed problem using the Fourier method. On the basis of the completeness property of the eigenfunction systems of these problems, the uniqueness theorem is proved. The solution of the problem is constructed as the sum of a double Fourier-Bessel series. In justification of the uniform convergence of the series constructed, asymptotic estimates of the Bessel functions of the real and imaginary argument are used. On their basis, estimates are obtained for each member of the series. The estimates obtained made it possible to prove the convergence of the series and its derivatives up to the second order inclusive, and also the existence theorem in the class of regular solutions. |
| first_indexed | 2024-12-10T04:08:17Z |
| format | Article |
| id | doaj.art-b805cb96d6a8410dbc83d1a41d6d3d6d |
| institution | Directory Open Access Journal |
| issn | 1991-8615 2310-7081 |
| language | English |
| last_indexed | 2024-12-10T04:08:17Z |
| publishDate | 2017-12-01 |
| publisher | Samara State Technical University |
| record_format | Article |
| series | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-b805cb96d6a8410dbc83d1a41d6d3d6d2022-12-22T02:02:47ZengSamara State Technical UniversityVestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki1991-86152310-70812017-12-0121466568310.14498/vsgtu155917987The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficientsAkhmadjon K Urinov0Kamoliddin T Karimov1Fergana State UniversityFergana State UniversityWe study the Dirichlet problem in a parallelepiped for a three-dimensional equation of mixed type with three singular coefficients. Separation of variables with Fourier series and spectral analysis are used to investigate this problem. Two one-dimensional spectral problems are obtained for the possed problem using the Fourier method. On the basis of the completeness property of the eigenfunction systems of these problems, the uniqueness theorem is proved. The solution of the problem is constructed as the sum of a double Fourier-Bessel series. In justification of the uniform convergence of the series constructed, asymptotic estimates of the Bessel functions of the real and imaginary argument are used. On their basis, estimates are obtained for each member of the series. The estimates obtained made it possible to prove the convergence of the series and its derivatives up to the second order inclusive, and also the existence theorem in the class of regular solutions.https://journals.eco-vector.com/1991-8615/article/viewFile/20567/16814dirichlet problemmixed-type equationsspectral methoduniqueness of solutionexistence of solution |
| spellingShingle | Akhmadjon K Urinov Kamoliddin T Karimov The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki dirichlet problem mixed-type equations spectral method uniqueness of solution existence of solution |
| title | The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients |
| title_full | The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients |
| title_fullStr | The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients |
| title_full_unstemmed | The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients |
| title_short | The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients |
| title_sort | dirichlet problem for a three dimensional equation of mixed type with three singular coefficients |
| topic | dirichlet problem mixed-type equations spectral method uniqueness of solution existence of solution |
| url | https://journals.eco-vector.com/1991-8615/article/viewFile/20567/16814 |
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