Generalized fractional derivatives and fourier transforms in tempered distributions with applications

Amin Benaissa Cherif; Fatima Zohra Ladrani; Dalal Alhwikem; Ahmed Hammoudi; Khaled Zennir; Keltoum Bouhali; Amin Benaissa Cherif; Fatima Zohra Ladrani; Dalal Alhwikem; Ahmed Hammoudi; Khaled Zennir; Keltoum Bouhali

doi:10.3934/nhm.2025037

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 868-884. doi: 10.3934/nhm.2025037

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Research article

Generalized fractional derivatives and fourier transforms in tempered distributions with applications

1.
Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Bir El Djir, Oran 31000, Algeria
2.
Department of Exact Sciences, Oran Higher Training Teacher's School (ENSO), Oran, Algeria
3.
Department of Mathematics, College of science, Qassim University, Saudi Arabia
4.
Department of Mathematics and Informatics, Faculty of Science and Technology, University Ain Temouchent Belhadj Bouchaib, BP 284, Ain Temouchent 46000, Algeria

Received: 07 April 2025 Revised: 29 May 2025 Accepted: 05 June 2025 Published: 18 July 2025

The purpose of this paper is to define and prove that the Riemann–Liouville and Caputo fractional derivatives can be computed for tempered distributions, such that the fractional derivative of a tempered distribution remains a tempered distribution. The fact that the Fourier transform operator is an isomorphism in the dual of the Schwartz space is used, and we found that the fractional Riemann–Liouville and Caputo derivatives can be written as a Fourier transform composition and inverse. In this way, we are able to generalize both fractional Riemann–Liouville and Caputo derivatives for the tempered distributions. Moreover, certain examples of fractional derivatives for some tempered distributions are provided, such as the distribution of Dirac, the distribution of Heaviside, and the distribution of principal value.
- fractional derivatives,
- fourier transform,
- Schwartz space,
- tempered distributions
Citation: Amin Benaissa Cherif, Fatima Zohra Ladrani, Dalal Alhwikem, Ahmed Hammoudi, Khaled Zennir, Keltoum Bouhali. Generalized fractional derivatives and fourier transforms in tempered distributions with applications[J]. Networks and Heterogeneous Media, 2025, 20(3): 868-884. doi: 10.3934/nhm.2025037

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Abstract

The purpose of this paper is to define and prove that the Riemann–Liouville and Caputo fractional derivatives can be computed for tempered distributions, such that the fractional derivative of a tempered distribution remains a tempered distribution. The fact that the Fourier transform operator is an isomorphism in the dual of the Schwartz space is used, and we found that the fractional Riemann–Liouville and Caputo derivatives can be written as a Fourier transform composition and inverse. In this way, we are able to generalize both fractional Riemann–Liouville and Caputo derivatives for the tempered distributions. Moreover, certain examples of fractional derivatives for some tempered distributions are provided, such as the distribution of Dirac, the distribution of Heaviside, and the distribution of principal value.

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