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A fully diagonalized spectral method using generalized Laguerre functions isproposed and analyzed for solving elliptic equations on the half line. We firstdefine the generalized Laguerre functions which are complete and mutuallyorthogonal with respect to an equivalent Sobolev inner product. Then theFourier-like Sobolev orthogonal basis functions are constructed for thediagonalized Laguerre spectral method of elliptic equations. Besides, a unifiedorthogonal Laguerre projection is established for various elliptic equations.On the basis of this orthogonal Laguerre projection, we obtain optimal errorestimates of the fully diagonalized Laguerre spectral method for both Dirichletand Robin boundary value problems. Finally, numerical experiments, which are inagreement with the theoretical analysis, demonstrate the effectiveness and thespectral accuracy of our diagonalized method.

A fully diagonalized spectral method using generalized Laguerre functions on the half line