On a high order numerical method for

functions with singularities

Author:
Knut S. Eckhoff

Journal:
Math. Comp. **67** (1998), 1063-1087

MSC (1991):
Primary 65M70, 65N35

DOI:
https://doi.org/10.1090/S0025-5718-98-00949-1

MathSciNet review:
1459387

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

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Additional Information

**Knut S. Eckhoff**

Affiliation:
Department of Mathematics, University of Bergen, Johannes Bruns gate 12, N-5008 Bergen Norway

Email:
reske@mi.uib.no

DOI:
https://doi.org/10.1090/S0025-5718-98-00949-1

Keywords:
Spectral methods,
Fourier series,
discontinuous functions,
Bernoulli polynomials,
singular functions,
quadrature,
partial differential equations

Received by editor(s):
December 11, 1996

Received by editor(s) in revised form:
March 26, 1997

Additional Notes:
This paper is partly based on work done while the author was engaged at the SINTEF Multiphase Flow Laboratory, Trondheim, Norway. The paper is also partly based on work done while the author was in residence at the Division of Applied Mathematics, Brown University, Providence, R.I., U.S.A. supported by AFOSR grant 95-1-0074 and NSF grant DMS-9500814.

Article copyright:
© Copyright 1998
American Mathematical Society