THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS

Sabri T. M. Thabet, Imed Kedim (), Bahaaeldin Abdalla () and Thabet Abdeljawad
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Sabri T. M. Thabet: Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602 105, Tamil Nadu, India†Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen‡Department of Mathematics, College of Science, Korea University, Seoul 02814, South Korea
Imed Kedim: �Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
Bahaaeldin Abdalla: �Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
Thabet Abdeljawad: �Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia∥Department of Medical Research, China Medical University, Taichung 40402, Taiwan**Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea††Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa

FRACTALS (fractals), 2024, vol. 32, issue 07n08, 1-14

Abstract: This paper is devoted to introducing a new q-fractional calculus in the framework of Atangana–Baleanu (𠒜ℬ) and Caputo–Fabrizio (𠒞ℱ) operators. First, an appropriate q-Mittag-Leffler function is defined, and then q-analogues of fractional derivatives of Atangana–Baleanu–Riemann (𠒜ℬℛ) and Atangana–Baleanu–Caputo (𠒜ℬ𠒞) are derived. Next, the q-analogues of proper fractional integrals in the 𠒜ℬ sense are proved. Several important properties of these definitions are investigated by using the q-Laplace transform. Additionally, a suitable q-exponential function is defined, and the q-analogues of 𠒞ℱ fractional derivatives with their inverse operators are introduced. The higher-order extension of the q-analogues of 𠒜ℬ and 𠒞ℱ fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of q-𠒜ℬ𠒞 calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.

Keywords: q-Atangana–Baleanu Fractional Operator; q-Caputo–Fabrizio Fractional Operator; q-Mittag-Leffler Function; q-Laplace Transformation (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X24400449

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