Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic
We consider the system of $n$ partial differential equations in matrix notation (the system of Euler-Poisson-Darboux equations). For the system we formulate the Cauchy-Goursat and Darboux problems for the case when the eigenvalues of the coefficient matrix lie in $(0; 1/2)$. The coefficient matrix i...
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| Format: | Article |
| Language: | English |
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Samara State Technical University
2015-12-01
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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/20440/16687 |
| _version_ | 1811330277232869376 |
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| author | Aleksander A Andreev Ekaterina A Maksimova |
| author_facet | Aleksander A Andreev Ekaterina A Maksimova |
| author_sort | Aleksander A Andreev |
| collection | DOAJ |
| description | We consider the system of $n$ partial differential equations in matrix notation (the system of Euler-Poisson-Darboux equations). For the system we formulate the Cauchy-Goursat and Darboux problems for the case when the eigenvalues of the coefficient matrix lie in $(0; 1/2)$. The coefficient matrix is reduced to the Jordan form, which allows to separate the system to the $r$ independent systems, one for each Jordan cell. The coefficient matrix in the obtained systems has the only one eigenvalue in the considered interval. For a system of equations having the only coefficient matrix in form of Jordan cell, which is the diagonal or triangular matrix, we can construct the solution using the properties of matrix functions. We form the Riemann-Hadamard matrices for each of $r$ systems using the Riemann matrix for the considered system, constructed before. That allow to find out the solutions of the Cauchy-Goursat and Darboux problems for each system of matrix partial differential equations. The solutions of the original problems are represented in form of the direct sum of the solutions of systems for Jordan cells. The correctness theorem for the obtained solutions is formulated. |
| first_indexed | 2024-04-13T15:58:49Z |
| format | Article |
| id | doaj.art-8accf492ed20430bbcbc7fdfb9a7de5f |
| institution | Directory Open Access Journal |
| issn | 1991-8615 2310-7081 |
| language | English |
| last_indexed | 2024-04-13T15:58:49Z |
| publishDate | 2015-12-01 |
| publisher | Samara State Technical University |
| record_format | Article |
| series | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-8accf492ed20430bbcbc7fdfb9a7de5f2022-12-22T02:40:36ZengSamara State Technical UniversityVestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki1991-86152310-70812015-12-0119460361210.14498/vsgtu142417860Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristicAleksander A Andreev0Ekaterina A Maksimova1Samara State Technical UniversitySamara State Technical UniversityWe consider the system of $n$ partial differential equations in matrix notation (the system of Euler-Poisson-Darboux equations). For the system we formulate the Cauchy-Goursat and Darboux problems for the case when the eigenvalues of the coefficient matrix lie in $(0; 1/2)$. The coefficient matrix is reduced to the Jordan form, which allows to separate the system to the $r$ independent systems, one for each Jordan cell. The coefficient matrix in the obtained systems has the only one eigenvalue in the considered interval. For a system of equations having the only coefficient matrix in form of Jordan cell, which is the diagonal or triangular matrix, we can construct the solution using the properties of matrix functions. We form the Riemann-Hadamard matrices for each of $r$ systems using the Riemann matrix for the considered system, constructed before. That allow to find out the solutions of the Cauchy-Goursat and Darboux problems for each system of matrix partial differential equations. The solutions of the original problems are represented in form of the direct sum of the solutions of systems for Jordan cells. The correctness theorem for the obtained solutions is formulated.https://journals.eco-vector.com/1991-8615/article/viewFile/20440/16687riemann methodcauchy-goursat problemdarboux problempartial differential equationssystem of euler-poisson-darboux equations |
| spellingShingle | Aleksander A Andreev Ekaterina A Maksimova Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki riemann method cauchy-goursat problem darboux problem partial differential equations system of euler-poisson-darboux equations |
| title | Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic |
| title_full | Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic |
| title_fullStr | Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic |
| title_full_unstemmed | Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic |
| title_short | Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic |
| title_sort | boundary value problems for matrix euler poisson darboux equation with data on a characteristic |
| topic | riemann method cauchy-goursat problem darboux problem partial differential equations system of euler-poisson-darboux equations |
| url | https://journals.eco-vector.com/1991-8615/article/viewFile/20440/16687 |
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