Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coeff...

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Main Authors: Liu, Wenjie, Wang, Li-Lian, Wu, Boying
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2022
Subjects:
Science::Mathematics
Optimal Estimates
Fractional Taylor Formula
Online Access:https://hdl.handle.net/10356/160560
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1605602022-07-26T08:31:33Z Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation Liu, Wenjie Wang, Li-Lian Wu, Boying School of Physical and Mathematical Sciences Science::Mathematics Optimal Estimates Fractional Taylor Formula We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L∞-estimates and L2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature. Ministry of Education (MOE) The research of the first author was supported by the National Natural Science Foundation of China (Nos. 11801120 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2020081), the Natural Science Foundation of Heilongjiang Province (Nos. LH2020A004 and LH2021A011), and the Guangdong Basic and Applied Basic Research Foundation (No.2020B1515310006). The research of the second author is partially supported by Singapore MOE AcRF Tier 2 Grant: MOE2018-T2-1-059 and Tier 1 Grant: RG15/21. The research of the third author was supported by the National Natural Science Foundation of China (Nos. 11971131, U1637208, 61873071, 51476047). 2022-07-26T08:31:32Z 2022-07-26T08:31:32Z 2021 Journal Article Liu, W., Wang, L. & Wu, B. (2021). Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation. Advances in Computational Mathematics, 47(6), 79-. https://dx.doi.org/10.1007/s10444-021-09905-3 1019-7168 https://hdl.handle.net/10356/160560 10.1007/s10444-021-09905-3 2-s2.0-85117569287 6 47 79 en MOE2018-T2-1-059 RG15/21 Advances in Computational Mathematics © 2021 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature. All rights reserved.
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Optimal Estimates
Fractional Taylor Formula
spellingShingle Science::Mathematics
Optimal Estimates
Fractional Taylor Formula
Liu, Wenjie
Wang, Li-Lian
Wu, Boying
Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
description We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L∞-estimates and L2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Liu, Wenjie
Wang, Li-Lian
Wu, Boying
format Article
author Liu, Wenjie
Wang, Li-Lian
Wu, Boying
author_sort Liu, Wenjie
title Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
title_short Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
title_full Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
title_fullStr Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
title_full_unstemmed Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
title_sort optimal error estimates for legendre expansions of singular functions with fractional derivatives of bounded variation
publishDate 2022
url https://hdl.handle.net/10356/160560
_version_ 1739837449784262656

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