Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm–Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm–Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm–Liouville operator are approximated by the eigenvalues of the corr...

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Bibliographic Details
Published in:Applied mathematics and computation 2010-11, Vol.217 (5), p.2266-2276
Main Author: Zhang, Xuecang
Format: Article
Language:English
Subjects:
Accuracy
Barycentric rational interpolation
Chebyshev approximation
Computation
Conformal map
Differentiation
Eigenvalue
Eigenvalues
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical models
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Operators
Ordinary differential equations
Sciences and techniques of general use
Sequences, series, summability
Spectral differentiation matrix
Sturm–Liouville problems
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm–Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm–Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm–Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2010.07.027