Solving the eigenvalues and eigenfunctions problems for the Helmholtz equation by the point-sources method
Abstract
The paper provides the developed approach to solve the eigenvalues and eigenfunctions problems
for the Helmholtz equation in domains with an arbitrary configuration. In developing the approach
for numerical solution of problems, the point-sources method (PSM) was used. The proposed
method is based on the analysis of the condition number of the PSM system or the error of the numerical
solution of problems. The concept of "eigenvalues criteria" 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, we calculate the dependence of the condition number of the PSM system or error
of the problem numerical solution on the Helmholtz parameter. Then, according to position of the
maximum of the received dependences we find the eigenvalues of the Helmholtz equation in a given
domain.. 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.
Keywords
point-sources method,
eigenvalues,
eigenfunctions,
Helmholtz equation,
fundamental solution,
the method of fundamental solutions
Supporting agencies: The research is done within the frame independent R&D.
About the Author
Elena E. Shcherbakova
Don State Technical University
(RF,344000, Rostov-on-Don, Gagarin sq., 1)
Russian Federation
Russian Federation
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
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Review
For citations:
Shcherbakova E.E. Solving the eigenvalues and eigenfunctions problems for the Helmholtz equation by the point-sources method. Computational Mathematics and Information Technologies. 2017;1(1).
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