University of Oulu

J. Nissilä, "Error analysis of numerical Weyl fractional derivatives in the case of certain Hölder continuous functions," 2018 14th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA), Oulu, Finland, 2018, pp. 1-7. doi: 10.1109/MESA.2018.8449193

Error analysis of numerical Weyl fractional derivatives in the case of certain Hölder continuous functions

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Author: Nissilä, Juhani1
Organizations: 1Mechatronics and Machine Diagnostics, Faculty of Technology, University of Oulu
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Persistent link: http://urn.fi/urn:nbn:fi-fe2018091035359
Language: English
Published: Institute of Electrical and Electronics Engineers, 2018
Publish Date: 2018-09-10
Description:

Abstract

The calculation of fractional or integer order derivatives and integrals has been demonstrated to be simple and fast in the frequency domain. It is also the most sensible method if one wishes to calculate derivatives or integrals of periodic signals. In this paper, error analysis is carried out for the numerical algorithm for Weyl fractional derivatives. To derive an upper bound for the numerical error, some knowledge of the smoothness of the signal must be known in advance or it must be estimated. The derived error analysis is tested with sampled functions with known regularity and with real vibration measurements from rotating machines. Compared to previous publications which deal with error analysis of integer order numerical derivatives in the frequency domain using L 2 errors, the result of this paper is in terms of maximum absolute error and it is based on a novel result on the signal’s regularity. The general conclusion using either error estimates is the same: the error of numerical Weyl derivatives is bounded by some constant times the sequence length raised to a negative power. The exponent depends on the smoothness of the signal. This contrasts with using difference quotients in numerical differentiation, in which case the error is bounded by a constant times the sequence length raised to a some fixed negative power and the order of the method defines that exponent.

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ISBN Print: 978-1-5386-4643-4
DOI: 10.1109/MESA.2018.8449193
OADOI: https://oadoi.org/10.1109/MESA.2018.8449193
Conference: 2018 14th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA)
Type of Publication: A4 Article in conference proceedings
Field of Science: 111 Mathematics
214 Mechanical engineering
Subjects:
Funding: The author is currently doing research under a grant from the Finnish Cultural Foundation, North Ostrobothnia Regional Fund. This work has also been supported by the Riitta and Jorma J. Takanen foundation, Tauno Tönning foundation and Otto A. Malm foundation.
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