Hermite–Sobolev orthogonal functions and spectral methods for second- and fourth-order problems on unbounded domains

ABSTRACT Hermite spectral methods using Sobolev orthogonal/biorthogonal basis functions for solving second and fourth-order differential equations on unbounded domains are proposed. Some Hermite–Sobolev orthogonal/biorthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. The convergence is analyzed and some numerical results are presented to illustrate the effectiveness and the spectral accuracy of this approach.

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