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.

Keywords:

• Spectral method
• Herimite–Sobolev orthogonal functions
• elliptic boundary value problems
• unbounded domains
• numerical results

2010 Mathematics subject classifications:

• Primary: 65N35
• 33C45
• Secondary: 41A30
• 35J25
• 35J40
• 35J25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by the National Natural Science Foundation of China [Nos. 11571238, 11601332, 91130014, 11471312 and 91430216].