vmaitComputational Mathematics and Information TechnologiesComputational Mathematics and Information Technologies2587-8999Донской государственный технический университетvmait-81Research ArticleСтатьиРешение задач на собственные значения и собственные функции для уравнения Гельмгольца методом точечных источников поляSolving the eigenvalues and eigenfunctions problems for the Helmholtz equation by the point-sources methodhttps://orcid.org/0000-0002-9239-1955ЩербаковаЕлена ЕвгеньевнаShcherbakovaElena E.

Щербакова Елена Евгеньевна, доцент каф. «Физическое и прикладное материаловедение» Донского государственного технического университета (РФ, 344000, Ростов-на-Дону, пл. Гагарина, 1), кандидат технических наук, доцент

Shcherbakova, Elena E., associate professor of the Material Physics and Applied Hylology De-partment, Don State Technical University (RF,344000, Rostov-on-Don, Gagarin sq., 1), Cand. Sci. (Eng.), associate professor

Sherbakovaee@mail.ruДонской государственный технический университет (РФ, 344000, Ростов-на-Дону, пл. Гагарина, 1)РоссияDon State Technical University (RF,344000, Rostov-on-Don, Gagarin sq., 1)Russian Federation2017290320231181Copyright © Щербакова Е.Е., 20232023Щербакова Е.Е.Shcherbakova E.E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.cmit-journal.ru/jour/article/view/81

Разработан способ решения задач вычисления собственных значений и собственных функций для уравнения Гельмгольца в областях с произвольной конфигурацией. При разработке способа численного решения задач используется метод точечных источников поля (МТИ). Предлагаемый способ основан на анализе числа обусловленности системы МТИ или по-грешности численного решения задачи. Вводится понятие «критерий собственных значений». Результатом работы является разработанный эффективный способ — алгоритм реше-ния задач на нахождение собственных значений и собственных функций для уравнения Гельмгольца. Показано, что при приближении параметра Гельмгольца к собственному значению задачи число обусловленности системы МТИ и погрешность численного решения резко возрастают. Определив зависимость погрешности численного решения задачи или числа обусловленности системы МТИ от параметра Гельмгольца, можно по расположению максимума для полученных зависимостей найти собственные значения уравнения Гельмгольца в заданной области. После нахождения собственного значения можно приступить к нахождению собственных функций. При этом, если собственное значение оказывается вырожденным, то есть ему соответствует несколько собственных функций, то, с учетом симметрии области решения, возможно нахождение всех собственных функций. Приведены результаты решения тестовых двумерных и трехмерных задач, на основании которых делается вывод об эффективности предложенного метода.

The paper provides the developed approach to solve the eigenvalues and eigenfunctions problemsfor the Helmholtz equation in domains with an arbitrary configuration. In developing the approachfor numerical solution of problems, the point-sources method (PSM) was used. The proposedmethod is based on the analysis of the condition number of the PSM system or the error of the numericalsolution of problems. The concept of "eigenvalues criteria" is introduced. The research result is adeveloped effective method - an algorithm for solving problems of eigenvalues and eigenfunctionsfor the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problemeigenvalue, the condition number of the PSM system and the error of the numerical solution risesharply. Therefore, we calculate the dependence of the condition number of the PSM system or errorof the problem numerical solution on the Helmholtz parameter. Then, according to position of themaximum of the received dependences we find the eigenvalues of the Helmholtz equation in a givendomain.. 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, thenit 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, theconclusion about the efficiency of the proposed method is made.

метод точечных источниковсобственные значениясобственные функ-цииуравнение Гельмгольцафундаментальное решениеметод фундаментальных решенийpoint-sources methodeigenvalueseigenfunctionsHelmholtz equationfundamental solutionthe method of fundamental solutionsРабота выполнена в рамках инициативной НИРThe research is done within the frame independent R&D.ReferencesFairweather G. The method of fundamental solutions for elliptic boundary value problems / G. Fairweather, A. Karageorghis // Ad. Vol. Comput. Math. – 1998. – Vol. 9. – P. 69-95.Fairweather G. The method of fundamental solutions for elliptic boundary value problems / G. Fairweather, A. Karageorghis // Ad. Vol. Comput. Math. – 1998. – Vol. 9. – P. 69-95.Alves C.J.S. A new method of fundamental solutions applied to nonhomogeneous elliptic problems / C.J.S. Alves, C.S. Chen // Advances in Computational Mathematics. – 2005. – Vol. 23 – P. 125-142.Alves C.J.S. 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The authors declare that there are no conflicts of interest present.