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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

An efficient spectral method for ordinary differential equations with rational function coefficients


Authors: Evangelos A. Coutsias, Thomas Hagstrom and David Torres
Journal: Math. Comp. 65 (1996), 611-635
MSC (1991): Primary 65Q05, 65L60, 65P05, 76M25, 33A45, 33C55, 33C45
MathSciNet review: 1333309
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Abstract: We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation $N$, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.


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Additional Information

Evangelos A. Coutsias
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
Email: vageli@math.unm.edu

Thomas Hagstrom
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
Email: hagstrom@math.unm.edu

David Torres
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
Email: dtorres@math.unm.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00704-1
PII: S 0025-5718(96)00704-1
Keywords: Spectral methods, orthogonal polynomials, boundary value problems
Received by editor(s): August 9, 1994
Received by editor(s) in revised form: February 12, 1995
Additional Notes: Part of the work of the first author was performed at Risø\ National Laboratory, DK–4000 Roskilde, Denmark. All authors supported in part by DOE Grant DE-FG03-92ER25128.
The work of the second author was partially supported by NSF Grants DMS-9108072, DMS-9304406 and by ICOMP, NASA Lewis Res. Ctr., Cleveland, OH, USA
Article copyright: © Copyright 1996 American Mathematical Society