### Fluctuations of the membrane with piecewise smooth contour and mixed boundary conditions

Pages 29-37

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.

DOI: 10.22227/1997-0935.2015.11.29-37

- Algazin S.D. Chislennye algoritmy klassicheskoy matematicheskoy fiziki [Numerical Algorithms of Classical Mathematical Physics]. Moscow, Dialog-MIFI Publ., 2010, 240 p. (In Russian)
- Babenko K.I. Osnovy chislennogo analiza [Fundamentals of Numerical Analysis]. 2nd edition, revised and enlarged. Moscow; Izhevsk, RKhD Publ., 2002, 847 p. (In Russian)
- Algazin S.D., Babenko K.I., Kosorukov A.L. O chislennom reshenii zadachi na sobstvennye znacheniya [On the Numerical Solution of the Task on Eigenvalues]. Moscow, 1975, 57 p. (Preprint. IPM; no. 108, 1975). (In Russian)
- Algazin S.D. Vychislenie sobstvennykh chisel i sobstvennykh funktsiy operatora Laplasa (Lap123) [Calculation of Eigenvalues and Eigenfunctions of Laplace Operator]. SVIDETEL’’STVO o gosudarstvennoy registratsii programmy dlya EVM № 2012617739. Zaregistrirovana v Reestre programm dlya EVM [Certificate on State Registration of the Computer Program № 2012617739. Registered in Software Registration Book]. August 27, 2012, 18 p. (In Russian)
- Kuttler J.R., Sigillito V.G. Eigenvalues of the Laplacian in Two Dimensions. SIAM Review. Apr. 1984, vol. 26, no. 2, pp. 163—193. DOI: http://dx.doi.org/10.1137/1026033