Abstract
Spectral methods using generalized Laguerre functions are proposed for second-order equations under polar (resp. spherical) coordinates in ℝ2 (resp. ℝ3) and fourth-order equations on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for our Laguerre spectral methods for elliptic problems. Optimal error estimates of the Laguerre spectral methods are obtained for both second-order and fourth-order elliptic equations. Numerical experiments demonstrate the effectiveness and the spectral accuracy.
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