TY - JOUR
T1 - The Q-Analogues Of Nonsingular Fractional Operators With Mittag-Leffler And Exponential Kernels
AU - Thabet, Sabri T.M.
AU - Kedim, Imed
AU - Abdalla, Bahaaeldin
AU - Abdeljawad, Thabet
N1 - Publisher Copyright:
© 2024 The Author(s).
PY - 2024
Y1 - 2024
N2 - This paper is devoted to introducing a new q-fractional calculus in the framework of Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) operators. First, an appropriate q-Mittag-Leffler function is defined, and then q-analogues of fractional derivatives of Atangana-Baleanu-Riemann (ABR) and Atangana-Baleanu-Caputo (ABC) are derived. Next, the q-analogues of proper fractional integrals in the AB 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 CF fractional derivatives with their inverse operators are introduced. The higher-order extension of the q-analogues of AB and CF fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of q-ABC calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.
AB - This paper is devoted to introducing a new q-fractional calculus in the framework of Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) operators. First, an appropriate q-Mittag-Leffler function is defined, and then q-analogues of fractional derivatives of Atangana-Baleanu-Riemann (ABR) and Atangana-Baleanu-Caputo (ABC) are derived. Next, the q-analogues of proper fractional integrals in the AB 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 CF fractional derivatives with their inverse operators are introduced. The higher-order extension of the q-analogues of AB and CF fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of q-ABC calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.
KW - q-Atangana-Baleanu Fractional Operator
KW - q-Caputo-Fabrizio Fractional Operator
KW - q-Laplace Transformation
KW - q-Mittag-Leffler Function
UR - http://www.scopus.com/inward/record.url?scp=85197615524&partnerID=8YFLogxK
U2 - 10.1142/S0218348X24400449
DO - 10.1142/S0218348X24400449
M3 - Article
AN - SCOPUS:85197615524
SN - 0218-348X
VL - 32
JO - Fractals
JF - Fractals
IS - 7-8
M1 - 2440044
ER -