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On well-conditioned spectral collocation and spectral methods by the integral reformulation. (English) Zbl 1351.65053

Summary: Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T. A. Driscoll, J. Comput. Phys. 229, No. 17, 5980–5998 (2010; Zbl 1195.65225)] and [L.-L. Wang et al., SIAM J. Sci. Comput. 36, No. 3, A907-A929 (2014; Zbl 1297.65086)], and the ultraspherical spectral method proposed in [S. Olver and A. Townsend, SIAM Rev. 55, No. 3, 462–489 (2013; Zbl 1273.65182)] for an \(m\)th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a compact perturbation of the identity. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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