Open AccessFluctuations of the membrane with piecewise smooth contour and mixed boundary conditions
Author(s) -
S. D. Algazin
Publication year - 2015
Publication title -
vestnik mgsu
Language(s) - English
Resource type - Journals
eISSN - 2304-6600
pISSN - 1997-0935
DOI - 10.22227/1997-0935.2015.11.29-37
Subject(s) - eigenvalues and eigenvectors , piecewise , conformal map , mathematics , methods of contour integration , operator (biology) , smoothness , mathematical analysis , laplace transform , a priori and a posteriori , boundary value problem , algorithm , physics , biochemistry , chemistry , philosophy , epistemology , repressor , quantum mechanics , transcription factor , gene
The eigenvalue problem for the two-dimensional operator Laplace is classical in mathematics and physics. However, computing methods for calculation of eigenvalues have still many problems, especially in applications to acoustic and electromagnetic wave guides. The investigated below two-dimensional spectral for the Laplace operator have been previously considered by the author only in smooth areas. The solutions of these tasks (eigen functions) are infinitely differentiated or. even analytical and therefore in order to create effective algorithms it is necessary to consider this enormous a priori information. Traditional methods of finite differences and finite elements almost do not practically use the information on smoothness of the decision, i.e. these are methods with saturation. The term “saturation” was entered by K.I. Babenko. Using the method of computing experiment the author investigates the task about fluctuations of the membrane with the piecewise smooth contour for two-dimensional area, obtained by conformal representation of the square. It is shown that eigen functions are infinitely differentiated. Therefore, numerical algorithms without saturation are applicable. In article the calculation algorithm of eigenvalues in this two-dimensional area is developed, which allows determining up to 10 natural frequencies with the accuracy acceptable for practice on the grid 10×10.