Spectral methods using generalized Laguerre functions for second and fourth order problems

Spectral methods using generalized Laguerre functions are proposed for second-order equations under polar (resp. spherical) coordinates in ℝ 2 (resp. ℝ 3 ) 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.