Faculty and Staff Publications

Document Type

Article

Publication Date

4-1-1996

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.

Publisher

American Mathematical Society

Publication Title

Mathematics of Computation

ISSN

0025-5718

Volume

65

Issue

214

First Page

611

Last Page

635

Language (ISO)

English

Sponsorship

American Mathematical Society

Keywords

Spectral methods, orthogonal polynomials, boundary value problems

Included in

Mathematics Commons

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