Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions
Articles
Vitalii Makogin
Ulm University
https://orcid.org/0000-0002-5849-0132
Yuliya Mishura
Taras Shevchenko National University of Kyiv
https://orcid.org/0000-0002-6877-1800
Published 2020-11-01
https://doi.org/10.15388/namc.2020.25.20566
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Keywords

Takagi–Landsberg functions
fractional derivatives
Schauder basis
Volterra equation

How to Cite

Makogin V. and Mishura Y. (2020) “Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions”, Nonlinear Analysis: Modelling and Control, 25(6), pp. 1079-1106. doi: 10.15388/namc.2020.25.20566.

Abstract

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.

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