Riemann-Liouville derivative over the space of integrable distributions

María Guadalupe Morales; Zuzana Došlá; Francisco J. Mendoza; María Guadalupe Morales; Zuzana Došlá; Francisco J. Mendoza

doi:10.3934/era.2020030

Electronic Research Archive

2020, Volume 28, Issue 2: 567-587. doi: 10.3934/era.2020030

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Riemann-Liouville derivative over the space of integrable distributions

1.
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2,611 37 Brno, Czech Republic
2.
Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur S/N, Puebla, Puebla, 72570, México

Received: 01 December 2019 Revised: 01 February 2020
Primary: 26A33, 26A39; Secondary: 46F12, 47G20

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, $ D_{HK} $. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.
Citation: María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions[J]. Electronic Research Archive, 2020, 28(2): 567-587. doi: 10.3934/era.2020030

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Abstract

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, $ D_{HK} $. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

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