Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions
In this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–L...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2020-11-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://www.journals.vu.lt/nonlinear-analysis/article/view/20566 |
| _version_ | 1818343577265111040 |
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| author | Vitalii Makogin Yuliya Mishura |
| author_facet | Vitalii Makogin Yuliya Mishura |
| author_sort | Vitalii Makogin |
| collection | DOAJ |
| description | In this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–Landsberg functions of order H > 0 on [0; 1] coincides with the class of H-Hölder continuous functions on [0; 1]. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows us to get a new formula of a Riemann–Stieltjes integral. The application of such series representation is a new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function. |
| first_indexed | 2024-12-13T16:32:48Z |
| format | Article |
| id | doaj.art-a0c81a91f705465988de3d4447d212c1 |
| institution | Directory Open Access Journal |
| issn | 1392-5113 2335-8963 |
| language | English |
| last_indexed | 2024-12-13T16:32:48Z |
| publishDate | 2020-11-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj.art-a0c81a91f705465988de3d4447d212c12022-12-21T23:38:29ZengVilnius University PressNonlinear Analysis1392-51132335-89632020-11-0125610.15388/namc.2020.25.20566Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functionsVitalii Makogin0Yuliya Mishura1Ulm UniversityTaras Shevchenko National University of KyivIn this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–Landsberg functions of order H > 0 on [0; 1] coincides with the class of H-Hölder continuous functions on [0; 1]. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows us to get a new formula of a Riemann–Stieltjes integral. The application of such series representation is a new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function.https://www.journals.vu.lt/nonlinear-analysis/article/view/20566Takagi–Landsberg functionsfractional derivativesSchauder basisVolterra equation |
| spellingShingle | Vitalii Makogin Yuliya Mishura Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions Nonlinear Analysis Takagi–Landsberg functions fractional derivatives Schauder basis Volterra equation |
| title | Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions |
| title_full | Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions |
| title_fullStr | Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions |
| title_full_unstemmed | Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions |
| title_short | Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions |
| title_sort | fractional integrals derivatives and integral equations with weighted takagi landsberg functions |
| topic | Takagi–Landsberg functions fractional derivatives Schauder basis Volterra equation |
| url | https://www.journals.vu.lt/nonlinear-analysis/article/view/20566 |
| work_keys_str_mv | AT vitaliimakogin fractionalintegralsderivativesandintegralequationswithweightedtakagilandsbergfunctions AT yuliyamishura fractionalintegralsderivativesandintegralequationswithweightedtakagilandsbergfunctions |