Solving eigenvalues problems for Helmholtz equation by point-source method
A method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analys...
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| Format: | Article |
| Language: | Russian |
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Don State Technical University
2016-09-01
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| Series: | Advanced Engineering Research |
| Subjects: | |
| Online Access: | https://www.vestnik-donstu.ru/jour/article/view/102 |
| _version_ | 1797881397455618048 |
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| author | Elena E. Shcherbakova |
| author_facet | Elena E. Shcherbakova |
| author_sort | Elena E. Shcherbakova |
| collection | DOAJ |
| description | A method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analysis of the condition number of the PSM system or error of the problem numerical solution. The concept of “eigenvalues criterion” is introduced. The research result is a developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, the dependence of the condition number of the PSM system or the error of the problem numerical solution can be calculated from the Helmholtz parameter. Then, according to the position of the maximum of the obtained dependences, the eigenvalues of the Helmholtz equation in a given domain are found. It allows searching the eigenvalues. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made. |
| first_indexed | 2024-04-10T03:18:23Z |
| format | Article |
| id | doaj.art-e00ca7235cdc4e53925b601a370cb430 |
| institution | Directory Open Access Journal |
| issn | 2687-1653 |
| language | Russian |
| last_indexed | 2024-04-10T03:18:23Z |
| publishDate | 2016-09-01 |
| publisher | Don State Technical University |
| record_format | Article |
| series | Advanced Engineering Research |
| spelling | doaj.art-e00ca7235cdc4e53925b601a370cb4302023-03-13T07:31:26ZrusDon State Technical UniversityAdvanced Engineering Research2687-16532016-09-01163879510.12737/20227102Solving eigenvalues problems for Helmholtz equation by point-source methodElena E. Shcherbakova0Донской государственный технический университетA method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analysis of the condition number of the PSM system or error of the problem numerical solution. The concept of “eigenvalues criterion” is introduced. The research result is a developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, the dependence of the condition number of the PSM system or the error of the problem numerical solution can be calculated from the Helmholtz parameter. Then, according to the position of the maximum of the obtained dependences, the eigenvalues of the Helmholtz equation in a given domain are found. It allows searching the eigenvalues. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made.https://www.vestnik-donstu.ru/jour/article/view/102point source methodeigenvalueseigenfunctionshelmholtz equationfundamental solutionmethod of fundamental solutionsметод точечных источниковсобственные значениясобственные функцииуравнение гельмгольцафундаментальное решениеметод фундаментальных решений |
| spellingShingle | Elena E. Shcherbakova Solving eigenvalues problems for Helmholtz equation by point-source method Advanced Engineering Research point source method eigenvalues eigenfunctions helmholtz equation fundamental solution method of fundamental solutions метод точечных источников собственные значения собственные функции уравнение гельмгольца фундаментальное решение метод фундаментальных решений |
| title | Solving eigenvalues problems for Helmholtz equation by point-source method |
| title_full | Solving eigenvalues problems for Helmholtz equation by point-source method |
| title_fullStr | Solving eigenvalues problems for Helmholtz equation by point-source method |
| title_full_unstemmed | Solving eigenvalues problems for Helmholtz equation by point-source method |
| title_short | Solving eigenvalues problems for Helmholtz equation by point-source method |
| title_sort | solving eigenvalues problems for helmholtz equation by point source method |
| topic | point source method eigenvalues eigenfunctions helmholtz equation fundamental solution method of fundamental solutions метод точечных источников собственные значения собственные функции уравнение гельмгольца фундаментальное решение метод фундаментальных решений |
| url | https://www.vestnik-donstu.ru/jour/article/view/102 |
| work_keys_str_mv | AT elenaeshcherbakova solvingeigenvaluesproblemsforhelmholtzequationbypointsourcemethod |