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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.

MSC:
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
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