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AN EFFICIENT SPECTRAL METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS WITH RATIONAL FUNCTION COEFFICIENTS

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Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/20187

AN EFFICIENT SPECTRAL METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS WITH RATIONAL FUNCTION COEFFICIENTS

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Title: AN EFFICIENT SPECTRAL METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS WITH RATIONAL FUNCTION COEFFICIENTS
Author: Coutsias, Evangelos A.; HAGSTROM, THOMAS; TORRES, DAVID
Subject: Spectral methods
orthogonal polynomials
boundary value problems
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.
Date: 1996-04
Publisher: American Mathematical Society
Citation: Mathematics of Computation, 65(214): 611-635
URI: http://hdl.handle.net/1928/20187
ISSN: 0025-5718


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