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...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2022
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| Online Access: | https://hdl.handle.net/10356/160560 |
| _version_ | 1811692636614950912 |
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| author | Liu, Wenjie Wang, Li-Lian Wu, Boying |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Liu, Wenjie Wang, Li-Lian Wu, Boying |
| author_sort | Liu, Wenjie |
| collection | NTU |
| 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. |
| first_indexed | 2024-10-01T06:38:56Z |
| format | Journal Article |
| id | ntu-10356/160560 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:38:56Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | ntu-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. |
| 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 |
| title | 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_short | 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 |
| topic | Science::Mathematics Optimal Estimates Fractional Taylor Formula |
| url | https://hdl.handle.net/10356/160560 |
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