A New Collocation Scheme Using Non-polynomial Basis Functions

• Main Authors: Zhang, Chao, Liu, Wenjie, Wang, Li-Lian
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2017
• Subjects: Generalised Birkhoff Interpolation Problem; Non-polynomial Basis
• Online Access: https://hdl.handle.net/10356/85589; http://hdl.handle.net/10220/43728
• Institution: Nanyang Technological University

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: Lλ=d2/dx2−λ2Lλ=d2/dx2−λ2 with constant λ.λ. With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator LλLλ being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.