zbMATH — the first resource for mathematics

Spectral collocation solutions to multiparameter Mathieu’s system. (English) Zbl 1280.65078
Summary: Our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of Mathieu’s system as a multiparameter eigenvalue problem (MEP). The reduced wave equation, for small deflections, is solved directly without approximations introduced by the classical Mathieu functions. We show how for moderate values of the cut-off collocation parameter the QR algorithm and the Arnoldi method may be applied successfully, while for larger values the Jacobi-Davidson method is the method of choice with respect to convergence, accuracy and memory usage.

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Full Text: DOI
[1] Volkmer, H., Multiparameter problems and expansion theorems, Lecture notes in mathematics, vol. 1356, (1988), Springer-Verlag New York
[2] Meixner, J.; Schäfke, F.W., Mathieusche funktionen und sph äroidfunktionen, (1954), Springer-Verlag
[3] Ruby, L., Applications of the Mathieu equation, Am. J. phys., 64, 39-44, (1996)
[4] Igbokoyi, A.O.; Tiab, D., New method of well test analysis in naturally fractured reservoirs based on elliptical flow, J. can. pet. technol., 49, 1-15, (2010)
[5] Neves, A.G.M., Eigenmodes and eigenfrequencies of vibrating elliptic membranes: A Klein oscillation theorem and numerical calculations, Commun. pure appl. anal., 9, 611-624, (2004) · Zbl 1190.33020
[6] Troesch, B.A.; Troesch, H.R., Eigenfrequencies of an elliptic membrane, Math. comput., 27, 755-765, (1973) · Zbl 0271.35017
[7] Gutiérrez-Vega, J.; Chávez-Cerda, S.; Rodríguez-Dagnino, R., Free oscillations in an elliptic mambrane, Rev. mex. fiz., 45, 613-622, (1999) · Zbl 1291.74091
[8] Wilson, H.B.; Scharstein, R.W., Computing elliptic membrane high frequencies by Mathieu and Galerkin methods, J. eng. math., 57, 41-55, (2007) · Zbl 1107.74021
[9] Trefethen, L.N., Spectral methods in MATLAB, (2000), SIAM Philadelphia · Zbl 0953.68643
[10] Weideman, J.A.C.; Reddy, S.C., A MATLAB differentiation matrix suite, ACM trans. math. softw., 26, 465-519, (2000)
[11] Boyd, J.P., ()
[12] Shen, J.; Wang, L.-L., On spectral approximation in elliptical geometries using Mathieu functions, Math. comput., 78, 815-884, (2009) · Zbl 1198.65236
[13] S.R. Finch, Mathieu eigenvalues, algo.inria.fr/csolve/mthu.pdf (2008).
[14] Sleeman, B.D., Multiparameter spectral theory and separation of variables, J. phys. A math. theor., 41, 1-20, (2008) · Zbl 1141.34008
[15] Atkinson, F.V., Multiparameter eigenvalue problems, (1972), Academic Press New York · Zbl 0555.47001
[16] Hochstenbach, M.E.; Košir, T.; Plestenjak, B., A jacobi – davidson type method for the nonsingular two-parameter eigenvalue problem, SIAM J. matrix anal. appl., 26, 477-497, (2005) · Zbl 1077.65036
[17] Plestenjak, B., A continuation method for a right definite two-parameter eigenvalue problem, SIAM J. matrix anal. appl., 21, 1163-1184, (2000) · Zbl 0995.65036
[18] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), The Johns Hopkins University Press Baltimore · Zbl 0865.65009
[19] Hochstenbach, M.E.; Plestenjak, B., Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem, Electron. trans. numer. anal., 29, 81-96, (2008) · Zbl 1171.65378
[20] Rommes, J., Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems \(\mathit{Ax} = \lambda \mathit{Bx}\) with B singular, Math. comput., 77, 995-1015, (2008) · Zbl 1133.65020
[21] Dragomirescu, F.I.; Gheorghiu, C.I., Analytical and numerical solutions to an electrohydrodynamic stability problem, Appl. math. comput., 216, 3718-3727, (2010) · Zbl 1197.78006
[22] Gheorghiu, C.I., Spectral methods for differential problems, (2007), Casa Cartii de Stiinta Cluj-Napoca · Zbl 1122.65118
[23] Gheorghiu, C.I.; Dragomirescu, F.I., Spectral methods in linear stability. application to thermal convection with variable gravity field, Appl. numer. math., 59, 1290-1302, (2009) · Zbl 1160.76036
[24] Fornberg, B., A practical guide to pseudospectral methods, (1998), Cambridge University Press Cambridge · Zbl 0912.65091
[25] J. Hoepffner, Implementation of boundary conditions, <www.fukagata.mech.keio.ac.jp/jerome/> (2007).
[26] Hochstenbach, M.E.; Plestenjak, B., Backward error, condition numbers, and pseudospectra for the multiparameter eigenvalue problem, Linear algebra appl., 375, 63-81, (2003) · Zbl 1048.65034
[27] Quarteroni, A.; Sacco, R.; Saleri, F., Numerical mathematics, Texts in applied mathematics, 47, (2007), Springer Berlin Heidelberg · Zbl 0913.65002
[28] Wilson, H.B.; Turcotte, L.S.; Halpern, D., Advanced mathematics and mechanics applications using MATLAB, (2003), Chapman and Hall/CRC Boca Raton · Zbl 1020.65005
[29] H. B. Wilson, Vibration modes of an elliptic membrane, MATLAB File Exchange, The MathWorks, Natick, 2004. Available from: <http://www.mathworks.com/matlabcentral/fileexchange>.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.